Загрузил feyke02

# Математические рекомендации к практическим занятиям курса "Методы математической физики"

```&Ecirc;&Egrave;&Acirc;&Ntilde;&Uuml;&Ecirc;&Egrave;&Eacute; &Iacute;&Agrave;&Ouml;I&Icirc;&Iacute;&Agrave;&Euml;&Uuml;&Iacute;&Egrave;&Eacute; &Oacute;&Iacute;I&Acirc;&Aring;&ETH;&Ntilde;&Egrave;&Ograve;&Aring;&Ograve;
i&igrave;&aring;&iacute;i &Ograve;&Agrave;&ETH;&Agrave;&Ntilde;&Agrave; &Oslash;&Aring;&Acirc;&times;&Aring;&Iacute;&Ecirc;&Agrave;
I.&Ntilde;. &Auml;&icirc;&ouml;&aring;&iacute;&ecirc;&icirc;, &Icirc;.I. &szlig;&ecirc;&egrave;&igrave;&aring;&iacute;&ecirc;&icirc;
&Igrave;&aring;&ograve;&icirc;&auml;&egrave;&divide;&iacute;i &eth;&aring;&ecirc;&icirc;&igrave;&aring;&iacute;&auml;&agrave;&ouml;i&uml;
&auml;&icirc; &iuml;&eth;&agrave;&ecirc;&ograve;&egrave;&divide;&iacute;&egrave;&otilde; &ccedil;&agrave;&iacute;&yuml;&ograve;&uuml; &ccedil; &ecirc;&oacute;&eth;&ntilde;&oacute;
&quot;&Igrave;&aring;&ograve;&icirc;&auml;&egrave; &igrave;&agrave;&ograve;&aring;&igrave;&agrave;&ograve;&egrave;&divide;&iacute;&icirc;&uml; &ocirc;i&ccedil;&egrave;&ecirc;&egrave;&quot;
&auml;&euml;&yuml; &ntilde;&ograve;&oacute;&auml;&aring;&iacute;&ograve;i&acirc; &ocirc;i&ccedil;&egrave;&divide;&iacute;&icirc;&atilde;&icirc; &ocirc;&agrave;&ecirc;&oacute;&euml;&uuml;&ograve;&aring;&ograve;&oacute;
&Ecirc;&egrave;&uml;&acirc; 2006
&Igrave;&aring;&ograve;&icirc;&auml;&egrave;&divide;&iacute;i &eth;&aring;&ecirc;&icirc;&igrave;&aring;&iacute;&auml;&agrave;&ouml;i&uml; &auml;&icirc; &iuml;&eth;&agrave;&ecirc;&ograve;&egrave;&divide;&iacute;&egrave;&otilde; &ccedil;&agrave;&iacute;&yuml;&ograve;&uuml; &ccedil; &ecirc;&oacute;&eth;&ntilde;&oacute; &quot;&Igrave;&aring;&ograve;&icirc;&auml;&egrave; &igrave;&agrave;&ograve;&aring;&igrave;&agrave;&ograve;&egrave;&divide;&iacute;&icirc;&uml; &ocirc;i&ccedil;&egrave;&ecirc;&egrave;&quot;&auml;&euml;&yuml; &ntilde;&ograve;&oacute;&auml;&aring;&iacute;&ograve;i&acirc; &ocirc;i&ccedil;&egrave;&divide;&iacute;&icirc;&atilde;&icirc; &ocirc;&agrave;&ecirc;&oacute;&euml;&uuml;&ograve;&aring;&ograve;&oacute;/ I.&Ntilde;.&Auml;&icirc;&ouml;&aring;&iacute;&ecirc;&icirc;,
&Icirc;.I.&szlig;&ecirc;&egrave;&igrave;&aring;&iacute;&ecirc;&icirc;, - &Ecirc;.: &ETH;&Acirc;&Ouml; &quot;&Ecirc;&egrave;&uml;&acirc;&ntilde;&uuml;&ecirc;&egrave;&eacute; &oacute;&iacute;i&acirc;&aring;&eth;&ntilde;&egrave;&ograve;&aring;&ograve;&quot;, 2006. - 50 &ntilde;.
&ETH;&aring;&ouml;&aring;&iacute;&ccedil;&aring;&iacute;&ograve;&egrave;:
&Ecirc;&egrave;&uml;&acirc;&ntilde;&uuml;&ecirc;&egrave;&eacute; &oacute;&iacute;i&acirc;&aring;&eth;&ntilde;&egrave;&ograve;&aring;&ograve; i&igrave;&aring;&iacute;i &Ograve;&agrave;&eth;&agrave;&ntilde;&agrave; &Oslash;&aring;&acirc;&divide;&aring;&iacute;&ecirc;&agrave;, 2006.
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&Ccedil;&igrave;i&ntilde;&ograve;
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&Acirc;&ntilde;&ograve;&oacute;&iuml;. . . . . . . . . . . . . . . . . . . . . .
&Igrave;&aring;&ograve;&icirc;&auml; &otilde;&agrave;&eth;&agrave;&ecirc;&ograve;&aring;&eth;&egrave;&ntilde;&ograve;&egrave;&ecirc;. . . . . . . . . . . . .
&Igrave;&aring;&ograve;&icirc;&auml; &eth;&icirc;&ccedil;&auml;i&euml;&aring;&iacute;&iacute;&yuml; &ccedil;&igrave;i&iacute;&iacute;&egrave;&otilde; . . . . . . . . . .
&Ccedil;&agrave;&auml;&agrave;&divide;i &ccedil; &acirc;&egrave;&ecirc;&icirc;&eth;&egrave;&ntilde;&ograve;&agrave;&iacute;&iacute;&yuml;&igrave; δ -&ocirc;&oacute;&iacute;&ecirc;&ouml;i&eacute;. . . . . .
&Igrave;&aring;&ograve;&icirc;&auml; &eth;&icirc;&ccedil;&auml;i&euml;&aring;&iacute;&iacute;&yuml; &ccedil;&igrave;i&iacute;&iacute;&egrave;&otilde; &ccedil; &acirc;&egrave;&ecirc;&icirc;&eth;&egrave;&ntilde;&ograve;&agrave;&iacute;&iacute;&yuml;&igrave;
&ntilde;&iuml;&aring;&ouml;i&agrave;&euml;&uuml;&iacute;&egrave;&otilde; &ocirc;&oacute;&iacute;&ecirc;&ouml;i&eacute;. . . . . . . . . . . . .
I&iacute;&ograve;&aring;&atilde;&eth;&agrave;&euml;&uuml;&iacute;i &eth;i&acirc;&iacute;&iacute;&yuml;&iacute;&iacute;&yuml;. . . . . . . . . . . . .
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1. &Acirc;&ntilde;&ograve;&oacute;&iuml;.
&Ograve;&aring;&icirc;&eth;&aring;&ograve;&egrave;&divide;&iacute;i &auml;&icirc;&ntilde;&euml;i&auml;&aelig;&aring;&iacute;&iacute;&yuml; &eth;i&ccedil;&iacute;&icirc;&igrave;&agrave;&iacute;i&ograve;&iacute;&egrave;&otilde; &ocirc;i&ccedil;&egrave;&divide;&iacute;&egrave;&otilde; &iuml;&eth;&icirc;&ouml;&aring;&ntilde;i&acirc; i &yuml;&acirc;&egrave;&ugrave; &acirc; &aacute;&agrave;&atilde;&agrave;&ograve;&uuml;&icirc;&otilde; &acirc;&egrave;&iuml;&agrave;&auml;&ecirc;&agrave;&otilde; &ccedil;&acirc;&icirc;&auml;&yuml;&ograve;&uuml;&ntilde;&yuml; &auml;&icirc; &eth;&icirc;&ccedil;&acirc;'&yuml;&ccedil;&oacute;&acirc;&agrave;&iacute;&iacute;&yuml; &ccedil;&agrave;&auml;&agrave;&divide;, &icirc;&ntilde;&iacute;&icirc;&acirc;&iacute;&egrave;&igrave;&egrave; &ntilde;&ecirc;&euml;&agrave;&auml;&icirc;&acirc;&egrave;&igrave;&egrave;
&yuml;&ecirc;&egrave;&otilde; &sup1; &auml;&egrave;&ocirc;&aring;&eth;&aring;&iacute;&ouml;i&agrave;&euml;&uuml;&iacute;i &eth;i&acirc;&iacute;&yuml;&iacute;&iacute;&yuml; &ccedil; &divide;&agrave;&ntilde;&ograve;&egrave;&iacute;&iacute;&egrave;&igrave;&egrave; &iuml;&icirc;&otilde;i&auml;&iacute;&egrave;&igrave;&egrave; &auml;&eth;&oacute;&atilde;&icirc;&atilde;&icirc; &iuml;&icirc;&eth;&yuml;&auml;&ecirc;&oacute;. &Iuml;&eth;&egrave;&ecirc;&euml;&agrave;&auml;&agrave;&igrave;&egrave; &ograve;&agrave;&ecirc;&egrave;&otilde; &eth;i&acirc;&iacute;&yuml;&iacute;&uuml; &sup1;: &otilde;&acirc;&egrave;&euml;&uuml;&icirc;&acirc;&aring; &eth;i&acirc;&iacute;&yuml;&iacute;&iacute;&yuml;, &ugrave;&icirc; &icirc;&iuml;&egrave;&ntilde;&oacute;&sup1; &auml;&egrave;&iacute;&agrave;&igrave;i&ecirc;&oacute;
&iuml;&icirc;&oslash;&egrave;&eth;&aring;&iacute;&iacute;&yuml; &otilde;&acirc;&egrave;&euml;&uuml; &eth;i&ccedil;&iacute;&icirc;&uml; &iuml;&eth;&egrave;&eth;&icirc;&auml;&egrave;, &eth;i&acirc;&iacute;&yuml;&iacute;&iacute;&yuml; &ograve;&aring;&iuml;&euml;&icirc;&iuml;&eth;&icirc;&acirc;i&auml;&iacute;&icirc;&ntilde;&ograve;i i &auml;&egrave;&ocirc;&oacute;&ccedil;i&uml;,
&ugrave;&icirc; &icirc;&iuml;&egrave;&ntilde;&oacute;&thorn;&ograve;&uuml; &iuml;&eth;&icirc;&ouml;&aring;&ntilde;&egrave; &iuml;&aring;&eth;&aring;&iacute;&aring;&ntilde;&aring;&iacute;&iacute;&yuml; &acirc; &ntilde;&aring;&eth;&aring;&eth;&aring;&auml;&icirc;&acirc;&egrave;&ugrave;i &ograve;&aring;&iuml;&euml;&icirc;&ograve;&egrave; &agrave;&aacute;&icirc; &eth;&aring;&divide;&icirc;&acirc;&egrave;&iacute;&egrave;,
&eth;i&acirc;&iacute;&yuml;&iacute;&iacute;&yuml; &Euml;&agrave;&iuml;&euml;&agrave;&ntilde;&agrave; i &Iuml;&oacute;&agrave;&ntilde;&icirc;&iacute;&agrave;, &auml;&icirc;&aacute;&eth;&aring; &acirc;i&auml;&icirc;&igrave;i &ccedil; &eth;&icirc;&ccedil;&auml;i&euml;&oacute; &quot;&Aring;&euml;&aring;&ecirc;&ograve;&eth;&icirc;&ntilde;&ograve;&agrave;&ograve;&egrave;&ecirc;&agrave;&quot;,
&ograve;&icirc;&ugrave;&icirc;. &Ecirc;&euml;&thorn;&divide;&icirc;&acirc;&aring; &eth;i&acirc;&iacute;&yuml;&iacute;&iacute;&yuml; &ecirc;&acirc;&agrave;&iacute;&ograve;&icirc;&acirc;&icirc;&uml; &igrave;&aring;&otilde;&agrave;&iacute;i&ecirc;&egrave; &eth;i&acirc;&iacute;&yuml;&iacute;&iacute;&yuml; &Oslash;&eth;&uuml;&icirc;&auml;i&iacute;&atilde;&aring;&eth;&agrave;
&ograve;&agrave;&ecirc;&icirc;&aelig; &yuml;&acirc;&euml;&yuml;&sup1; &ntilde;&icirc;&aacute;&icirc;&thorn; &auml;&egrave;&ocirc;&aring;&eth;&aring;&iacute;&ouml;i&agrave;&euml;&uuml;&iacute;&aring; &eth;i&acirc;&iacute;&yuml;&iacute;&iacute;&yuml; &auml;&eth;&oacute;&atilde;&icirc;&atilde;&icirc; &iuml;&icirc;&eth;&yuml;&auml;&ecirc;&oacute; &ccedil; &divide;&agrave;&ntilde;&ograve;&egrave;&iacute;&iacute;&egrave;&igrave;&egrave; &iuml;&icirc;&otilde;i&auml;&iacute;&egrave;&igrave;&egrave;.
&Icirc;&ntilde;&iacute;&icirc;&acirc;&iacute;i &acirc;&euml;&agrave;&ntilde;&ograve;&egrave;&acirc;&icirc;&ntilde;&ograve;i &auml;&egrave;&ocirc;&aring;&eth;&aring;&iacute;&ouml;i&agrave;&euml;&uuml;&iacute;&egrave;&otilde; &eth;i&acirc;&iacute;&yuml;&iacute;&uuml; &ccedil; &divide;&agrave;&ntilde;&ograve;&egrave;&iacute;&iacute;&egrave;&igrave;&egrave; &iuml;&icirc;&otilde;i&auml;&iacute;&egrave;&igrave;&egrave; &eth;&icirc;&ccedil;&atilde;&euml;&yuml;&iacute;&aring;&igrave;&icirc; &ntilde;&iuml;&icirc;&divide;&agrave;&ograve;&ecirc;&oacute; &iacute;&agrave; &iuml;&eth;&egrave;&ecirc;&euml;&agrave;&auml;i &eth;i&acirc;&iacute;&yuml;&iacute;&iacute;&yuml; &auml;&eth;&oacute;&atilde;&icirc;&atilde;&icirc; &iuml;&icirc;&eth;&yuml;&auml;&ecirc;&oacute; &ccedil; &auml;&acirc;&icirc;&igrave;&agrave;
&iacute;&aring;&ccedil;&agrave;&euml;&aring;&aelig;&iacute;&egrave;&igrave;&egrave; &ccedil;&igrave;i&iacute;&iacute;&egrave;&igrave;&egrave;, &ccedil;&agrave;&atilde;&agrave;&euml;&uuml;&iacute;&egrave;&eacute; &acirc;&egrave;&eth;&agrave;&ccedil; &yuml;&ecirc;&icirc;&atilde;&icirc; &igrave;&icirc;&aelig;&iacute;&agrave; &iuml;&eth;&aring;&auml;&ntilde;&ograve;&agrave;&acirc;&egrave;&ograve;&egrave; &oacute; &acirc;&egrave;&atilde;&euml;&yuml;&auml;i:
F (x, y, u, ux , uy , uxx , uxy , uyy ) = 0,
(1.1)
&ograve;&icirc;&aacute;&ograve;&icirc; &ouml;&aring; &auml;&aring;&yuml;&ecirc;&aring; &ntilde;&iuml;i&acirc;&acirc;i&auml;&iacute;&icirc;&oslash;&aring;&iacute;&iacute;&yuml; &igrave;i&aelig; &iacute;&aring;&ccedil;&agrave;&euml;&aring;&aelig;&iacute;&egrave;&igrave;&egrave; &ccedil;&igrave;i&iacute;&iacute;&egrave;&igrave;&egrave; x, y , &oslash;&oacute;&ecirc;&agrave;&iacute;&icirc;&thorn; &ocirc;&oacute;&iacute;&ecirc;&ouml;i&sup1;&thorn; u(x, y), &divide;&agrave;&ntilde;&ograve;&egrave;&iacute;&iacute;&egrave;&igrave;&egrave; &iuml;&icirc;&otilde;i&auml;&iacute;&egrave;&igrave;&egrave; &iuml;&aring;&eth;&oslash;&icirc;&atilde;&icirc; ux , uy &ograve;&agrave; &auml;&eth;&oacute;&atilde;&icirc;&atilde;&icirc;
uxx , uxy , uyy &iuml;&icirc;&eth;&yuml;&auml;&ecirc;i&acirc; &acirc;i&auml; &oslash;&oacute;&ecirc;&agrave;&iacute;&icirc;&uml; &ocirc;&oacute;&iacute;&ecirc;&ouml;i&uml;. &Ograve;&oacute;&ograve; i &iacute;&agrave;&auml;&agrave;&euml;i &auml;&euml;&yuml; &ntilde;&ecirc;&icirc;&eth;&icirc;&divide;&aring;&iacute;∂u
∂2u
&iacute;&yuml; &acirc;&egrave;&ecirc;&icirc;&eth;&egrave;&ntilde;&ograve;&icirc;&acirc;&oacute;&thorn;&ograve;&uuml;&ntilde;&yuml; &ograve;&agrave;&ecirc;i &iuml;&icirc;&ccedil;&iacute;&agrave;&divide;&aring;&iacute;&iacute;&yuml;: ux ≡ ∂u
∂x , uy ≡ ∂y , uxx ≡ ∂x2 ,
2
2
∂ u
, uyy ≡ ∂∂yu2 . &ETH;&icirc;&ccedil;&acirc;'&yuml;&ccedil;&ecirc;&icirc;&igrave; &eth;i&acirc;&iacute;&yuml;&iacute;&iacute;&yuml; (1.1) &iacute;&agrave;&ccedil;&egrave;&acirc;&agrave;&sup1;&ograve;&uuml;&ntilde;&yuml; &aacute;&oacute;&auml;&uuml;-&yuml;&ecirc;&agrave;
uxy ≡ ∂x∂y
&ocirc;&oacute;&iacute;&ecirc;&ouml;i&yuml;, &ugrave;&icirc; &iuml;&aring;&eth;&aring;&ograve;&acirc;&icirc;&eth;&thorn;&sup1; &auml;&agrave;&iacute;&aring; &eth;i&acirc;&iacute;&yuml;&iacute;&iacute;&yuml; &acirc; &ograve;&icirc;&ograve;&icirc;&aelig;&iacute;i&ntilde;&ograve;&uuml;.
&Acirc; &oacute;&iacute;i&acirc;&aring;&eth;&ntilde;&egrave;&ograve;&aring;&ograve;&ntilde;&uuml;&ecirc;&icirc;&igrave;&oacute; &ecirc;&oacute;&eth;&ntilde;i &ocirc;i&ccedil;&egrave;&ecirc;&egrave; &ccedil;&agrave;&ntilde;&ograve;&icirc;&ntilde;&icirc;&acirc;&oacute;&thorn;&ograve;&uuml;&ntilde;&yuml; &iuml;&aring;&eth;&aring;&acirc;&agrave;&aelig;&iacute;&icirc; &euml;i&iacute;i&eacute;&iacute;i
&auml;&egrave;&ocirc;&aring;&eth;&aring;&iacute;&ouml;i&agrave;&euml;&uuml;&iacute;i &eth;i&acirc;&iacute;&yuml;&iacute;&iacute;&yuml;. &Euml;i&iacute;i&eacute;&iacute;i &auml;&egrave;&ocirc;&aring;&eth;&aring;&iacute;&ouml;i&agrave;&euml;&uuml;&iacute;i &eth;i&acirc;&iacute;&yuml;&iacute;&iacute;&yuml; &auml;&eth;&oacute;&atilde;&icirc;&atilde;&icirc; &iuml;&icirc;&eth;&yuml;&auml;&ecirc;&oacute; &ccedil; &divide;&agrave;&ntilde;&ograve;&egrave;&iacute;&iacute;&egrave;&igrave;&egrave; &iuml;&icirc;&otilde;i&auml;&iacute;&egrave;&igrave;&egrave; &ccedil; &auml;&acirc;&icirc;&igrave;&agrave; &iacute;&aring;&ccedil;&agrave;&euml;&aring;&aelig;&iacute;&egrave;&igrave;&egrave; &ccedil;&igrave;i&iacute;&iacute;&egrave;&igrave;&egrave; &ntilde;&agrave;&igrave;&icirc;&atilde;&icirc;
&ccedil;&agrave;&atilde;&agrave;&euml;&uuml;&iacute;&icirc;&atilde;&icirc; &acirc;&egrave;&atilde;&euml;&yuml;&auml;&oacute; &iuml;&eth;&aring;&auml;&ntilde;&ograve;&agrave;&acirc;&euml;&yuml;&thorn;&ograve;&uuml;&ntilde;&yuml; &acirc;&egrave;&eth;&agrave;&ccedil;&icirc;&igrave;:
a11 uxx + 2a12 uxy + a22 uyy + b1 ux + b2 uy + cu = f,
(1.2)
&auml;&aring; &ecirc;&icirc;&aring;&ocirc;i&ouml;i&sup1;&iacute;&ograve;&egrave; aij , bi , c &ograve;&agrave; &ocirc;&oacute;&iacute;&ecirc;&ouml;i&yuml; f &ccedil;&agrave;&euml;&aring;&aelig;&agrave;&ograve;&uuml; &ograve;i&euml;&uuml;&ecirc;&egrave; &acirc;i&auml; &ccedil;&igrave;i&iacute;&iacute;&egrave;&otilde; x, y i
&iacute;&aring; &ccedil;&agrave;&euml;&aring;&aelig;&agrave;&ograve;&uuml; &acirc;i&auml; &oslash;&oacute;&ecirc;&agrave;&iacute;&icirc;&uml; &ocirc;&oacute;&iacute;&ecirc;&ouml;i&uml; u &ograve;&agrave; &uml;&uml; &iuml;&icirc;&otilde;i&auml;&iacute;&egrave;&otilde;. &Ocirc;&oacute;&iacute;&ecirc;&ouml;i&yuml; f (x, y) &acirc;&acirc;&agrave;&aelig;&agrave;&sup1;&ograve;&uuml;&ntilde;&yuml; &ccedil;&agrave;&auml;&agrave;&iacute;&icirc;&thorn;. &szlig;&ecirc;&ugrave;&icirc; f (x, y) ≡ 0, &ograve;&icirc; &eth;i&acirc;&iacute;&yuml;&iacute;&iacute;&yuml; (1.2) &iacute;&agrave;&ccedil;&egrave;&acirc;&agrave;&sup1;&ograve;&uuml;&ntilde;&yuml; &icirc;&auml;&iacute;&icirc;&eth;i&auml;&iacute;&egrave;&igrave;. &Ecirc;&icirc;&aring;&ocirc;i&ouml;i&sup1;&iacute;&ograve;&egrave; a11 , a12 i a22 &iacute;&agrave;&ccedil;&egrave;&acirc;&agrave;&thorn;&ograve;&uuml; &ecirc;&icirc;&aring;&ocirc;i&ouml;i&sup1;&iacute;&ograve;&agrave;&igrave;&egrave; &iuml;&eth;&egrave; &ntilde;&ograve;&agrave;&eth;&oslash;&egrave;&otilde;
&iuml;&icirc;&otilde;i&auml;&iacute;&egrave;&otilde; (&ograve;&icirc;&aacute;&ograve;&icirc; &iuml;&eth;&egrave; &iuml;&icirc;&otilde;i&auml;&iacute;&egrave;&otilde; &iacute;&agrave;&eacute;&acirc;&egrave;&ugrave;&icirc;&atilde;&icirc;, &acirc; &auml;&agrave;&iacute;&icirc;&igrave;&oacute; &eth;i&acirc;&iacute;&yuml;&iacute;&iacute;i, &iuml;&icirc;&eth;&yuml;&auml;&ecirc;&oacute;).
&times;&agrave;&ntilde;&ograve;&icirc; &ccedil;&agrave;&igrave;i&ntilde;&ograve;&uuml; &ccedil;&igrave;i&iacute;&iacute;&egrave;&otilde; x, y &acirc; &eth;i&acirc;&iacute;&yuml;&iacute;&iacute;i (1.2) &auml;&icirc;&ouml;i&euml;&uuml;&iacute;&icirc; &icirc;&aacute;&eth;&agrave;&ograve;&egrave; i&iacute;&oslash;i, &iacute;&icirc;&acirc;i
&iacute;&aring;&ccedil;&agrave;&euml;&aring;&aelig;&iacute;i &ccedil;&igrave;i&iacute;&iacute;i ξ, η , &ugrave;&icirc; &iuml;&icirc;&acirc;'&yuml;&ccedil;&agrave;&iacute;i &ccedil;i &quot;&ntilde;&ograve;&agrave;&eth;&egrave;&igrave;&egrave;&quot; &ccedil;&igrave;i&iacute;&iacute;&egrave;&igrave;&egrave; &iuml;&aring;&acirc;&iacute;&egrave;&igrave; &divide;&egrave;&iacute;&icirc;&igrave;:
ξ = ξ(x, y), η = η(x, y). &Iacute;&aring;&ccedil;&agrave;&euml;&aring;&aelig;&iacute;i&ntilde;&ograve;&uuml; &iacute;&icirc;&acirc;&egrave;&otilde; &ccedil;&igrave;i&iacute;&iacute;&egrave;&otilde; ξ i η &igrave;i&aelig; &ntilde;&icirc;&aacute;&icirc;&thorn;
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&acirc;&egrave;&ccedil;&iacute;&agrave;&divide;&agrave;&sup1;&ograve;&uuml;&ntilde;&yuml; &oacute;&igrave;&icirc;&acirc;&icirc;&thorn; &acirc;i&auml;&igrave;i&iacute;&iacute;&icirc;&ntilde;&ograve;i &acirc;i&auml; &iacute;&oacute;&euml;&yuml; &szlig;&ecirc;&icirc;&aacute;i&agrave;&iacute;&oacute; &iuml;&aring;&eth;&aring;&otilde;&icirc;&auml;&oacute; &auml;&icirc; &iacute;&icirc;&acirc;&egrave;&otilde;
&ccedil;&igrave;i&iacute;&iacute;&egrave;&otilde;:
&macr;
&macr;
&macr; ξx ξy &macr;
&macr;
&macr;
&macr; ηx ηy &macr; = ξx ηy − ξy ηx /≡ 0.
(1.3)
&Iuml;&eth;&egrave; &iuml;&aring;&eth;&aring;&otilde;&icirc;&auml;i &acirc;i&auml; &ntilde;&ograve;&agrave;&eth;&egrave;&otilde; &ccedil;&igrave;i&iacute;&iacute;&egrave;&otilde; &auml;&icirc; &iacute;&icirc;&acirc;&egrave;&otilde; &eth;i&acirc;&iacute;&yuml;&iacute;&iacute;&yuml; (1.1) &agrave;&aacute;&icirc; (1.2) &iuml;&aring;&acirc;&iacute;&egrave;&igrave;
&divide;&egrave;&iacute;&icirc;&igrave; &iuml;&aring;&eth;&aring;&ograve;&acirc;&icirc;&eth;&thorn;&sup1;&ograve;&uuml;&ntilde;&yuml;. &Iuml;&eth;&egrave; &ouml;&uuml;&icirc;&igrave;&oacute; &euml;i&iacute;i&eacute;&iacute;&aring; &eth;i&acirc;&iacute;&yuml;&iacute;&iacute;&yuml; &ccedil;&agrave;&acirc;&aelig;&auml;&egrave; &iuml;&aring;&eth;&aring;&ograve;&acirc;&icirc;&eth;&thorn;&sup1;&ograve;&uuml;&ntilde;&yuml; &acirc; &euml;i&iacute;i&eacute;&iacute;&aring;. &Acirc; &iuml;&aring;&eth;&aring;&ograve;&acirc;&icirc;&eth;&aring;&iacute;&icirc;&igrave;&oacute; &eth;i&acirc;&iacute;&yuml;&iacute;&iacute;i &oslash;&oacute;&ecirc;&agrave;&iacute;&agrave; &ocirc;&oacute;&iacute;&ecirc;&ouml;i&yuml;, &uml;&uml; &iuml;&icirc;&otilde;i&auml;&iacute;i
i &ecirc;&icirc;&aring;&ocirc;i&ouml;i&sup1;&iacute;&ograve;&egrave; &eth;i&acirc;&iacute;&yuml;&iacute;&iacute;&yuml; &ccedil;&agrave;&euml;&aring;&aelig;&agrave;&ograve;&uuml; &acirc;i&auml; &iacute;&icirc;&acirc;&egrave;&otilde; &ccedil;&igrave;i&iacute;&iacute;&egrave;&otilde;. &Oslash;&euml;&yuml;&otilde;&icirc;&igrave; &iuml;&aring;&acirc;&iacute;&icirc;&atilde;&icirc;
&ouml;i&euml;&aring;&ntilde;&iuml;&eth;&yuml;&igrave;&icirc;&acirc;&agrave;&iacute;&icirc;&atilde;&icirc; &acirc;&egrave;&aacute;&icirc;&eth;&oacute; &iacute;&icirc;&acirc;&egrave;&otilde; &ccedil;&igrave;i&iacute;&iacute;&egrave;&otilde; &eth;i&acirc;&iacute;&yuml;&iacute;&iacute;&yuml; (1.2) &igrave;&icirc;&aelig;&iacute;&agrave; &ntilde;&oacute;&ograve;&ograve;&sup1;&acirc;&icirc;
&ntilde;&iuml;&eth;&icirc;&ntilde;&ograve;&egrave;&ograve;&egrave;, &ccedil;&acirc;i&acirc;&oslash;&egrave; &eacute;&icirc;&atilde;&icirc; &auml;&icirc; &ograve;&egrave;&iuml;&icirc;&acirc;&icirc;&atilde;&icirc;, &ograve;&agrave;&ecirc; &ccedil;&acirc;&agrave;&iacute;&iacute;&icirc;&atilde;&icirc; &ecirc;&agrave;&iacute;&icirc;&iacute;i&divide;&iacute;&icirc;&atilde;&icirc; &acirc;&egrave;&auml;&oacute;. &Acirc;
&ccedil;&agrave;&euml;&aring;&aelig;&iacute;&icirc;&ntilde;&ograve;i &acirc;i&auml; &ccedil;&iacute;&agrave;&divide;&aring;&iacute;&iacute;&yuml; &acirc;&egrave;&eth;&agrave;&ccedil;&oacute; (&auml;&egrave;&ntilde;&ecirc;&eth;&egrave;&igrave;i&iacute;&agrave;&iacute;&ograve;&oacute; ) D = a212 − a11 a22 , &ntilde;&ecirc;&euml;&agrave;&auml;&aring;&iacute;&icirc;&atilde;&icirc; &ccedil; &ecirc;&icirc;&aring;&ocirc;i&ouml;i&sup1;&iacute;&ograve;i&acirc; &eth;i&acirc;&iacute;&yuml;&iacute;&iacute;&yuml; (1.2), &icirc;&ntilde;&ograve;&agrave;&iacute;&iacute;&sup1; &acirc;i&auml;&iacute;&icirc;&ntilde;&egrave;&ograve;&uuml;&ntilde;&yuml; &auml;&icirc; &icirc;&auml;&iacute;&icirc;&atilde;&icirc; &ccedil;
&ograve;&eth;&uuml;&icirc;&otilde; &ograve;&egrave;&iuml;i&acirc;.
&Iuml;&eth;&egrave; D &gt; 0 &eth;i&acirc;&iacute;&yuml;&iacute;&iacute;&yuml; (1.2) &igrave;&agrave;&sup1; &iacute;&agrave;&ccedil;&acirc;&oacute; &eth;i&acirc;&iacute;&yuml;&iacute;&iacute;&yuml; &atilde;i&iuml;&aring;&eth;&aacute;&icirc;&euml;i&divide;&iacute;&icirc;&atilde;&icirc; &ograve;&egrave;&iuml;&oacute;.
&Aacute;&oacute;&auml;&uuml;-&yuml;&ecirc;&aring; &eth;i&acirc;&iacute;&yuml;&iacute;&iacute;&yuml; &atilde;i&iuml;&aring;&eth;&aacute;&icirc;&euml;i&divide;&iacute;&icirc;&atilde;&icirc; &ograve;&egrave;&iuml;&oacute; &oslash;&euml;&yuml;&otilde;&icirc;&igrave; &iacute;&agrave;&euml;&aring;&aelig;&iacute;&icirc;&atilde;&icirc; &acirc;&egrave;&aacute;&icirc;&eth;&oacute; &iacute;&icirc;&acirc;&egrave;&otilde;
&ccedil;&igrave;i&iacute;&iacute;&egrave;&otilde; ξ = ξ(x, y) i η = η(x, y) &igrave;&icirc;&aelig;&iacute;&agrave; &ccedil;&acirc;&aring;&ntilde;&ograve;&egrave; &auml;&icirc; &acirc;i&auml;&iuml;&icirc;&acirc;i&auml;&iacute;&icirc;&atilde;&icirc; &ecirc;&agrave;&iacute;&icirc;&iacute;i&divide;&iacute;&icirc;&atilde;&icirc; &acirc;&egrave;&auml;&oacute; (&eth;&icirc;&ccedil;&atilde;&euml;&yuml;&auml;&agrave;&sup1;&ograve;&uuml;&ntilde;&yuml; &acirc;&egrave;&iuml;&agrave;&auml;&icirc;&ecirc; &icirc;&auml;&iacute;&icirc;&eth;i&auml;&iacute;&icirc;&atilde;&icirc; &eth;i&acirc;&iacute;&yuml;&iacute;&iacute;&yuml;, &ecirc;&icirc;&euml;&egrave; f = 0):
uξη + b̄1 uξ + b̄2 uη + c̄u = 0.
(1.4)
&Iacute;&agrave;&ntilde;&ograve;&oacute;&iuml;&iacute;&egrave;&igrave; &iuml;&aring;&eth;&aring;&ograve;&acirc;&icirc;&eth;&aring;&iacute;&iacute;&yuml;&igrave;, &oslash;&euml;&yuml;&otilde;&icirc;&igrave; &acirc;&egrave;&aacute;&icirc;&eth;&oacute; i&iacute;&oslash;&egrave;&otilde; &ccedil;&igrave;i&iacute;&iacute;&egrave;&otilde; α = 12 (ξ + η),
β = 12 (ξ − η) &eth;i&acirc;&iacute;&yuml;&iacute;&iacute;&yuml; (1.4) &igrave;&icirc;&aelig;&iacute;&agrave; &iuml;&eth;&aring;&auml;&ntilde;&ograve;&agrave;&acirc;&egrave;&ograve;&egrave; &acirc; i&iacute;&oslash;i&eacute; &ocirc;&icirc;&eth;&igrave;i
uαα − uββ + b̃1 uα + b̃2 uβ + c̃u = 0,
(1.5)
&ugrave;&icirc; &ograve;&agrave;&ecirc;&icirc;&aelig; &acirc;&acirc;&agrave;&aelig;&agrave;&sup1;&ograve;&uuml;&ntilde;&yuml; &ecirc;&agrave;&iacute;&icirc;&iacute;i&divide;&iacute;&icirc;&thorn;.
&szlig;&ecirc;&ugrave;&icirc; D &lt; 0, &ograve;&icirc; &eth;i&acirc;&iacute;&yuml;&iacute;&iacute;&yuml; (1.2) &iacute;&agrave;&ccedil;&egrave;&acirc;&agrave;&sup1;&ograve;&uuml;&ntilde;&yuml; &eth;i&acirc;&iacute;&yuml;&iacute;&iacute;&yuml;&igrave; &aring;&euml;i&iuml;&ograve;&egrave;&divide;&iacute;&icirc;&atilde;&icirc;
&ograve;&egrave;&iuml;&oacute; i &ecirc;&agrave;&iacute;&icirc;&iacute;i&divide;&iacute;&egrave;&eacute; &acirc;&egrave;&auml; &ouml;&uuml;&icirc;&atilde;&icirc; &eth;i&acirc;&iacute;&yuml;&iacute;&iacute;&yuml; &iuml;&eth;&aring;&auml;&ntilde;&ograve;&agrave;&acirc;&euml;&yuml;&sup1;&ograve;&uuml;&ntilde;&yuml; &acirc;&egrave;&eth;&agrave;&ccedil;&icirc;&igrave;:
uξξ + uηη + b̄1 uα + b̄2 uβ + c̄u = 0.
(1.6)
&szlig;&ecirc;&ugrave;&icirc; D = 0, &ograve;&icirc; &eth;i&acirc;&iacute;&yuml;&iacute;&iacute;&yuml; (1.2) &igrave;&agrave;&sup1; &iacute;&agrave;&ccedil;&acirc;&oacute; &eth;i&acirc;&iacute;&yuml;&iacute;&iacute;&yuml; &iuml;&agrave;&eth;&agrave;&aacute;&icirc;&euml;i&divide;&iacute;&icirc;&atilde;&icirc; &ograve;&egrave;&iuml;&oacute;,
i &ecirc;&agrave;&iacute;&icirc;&iacute;i&divide;&iacute;&agrave; &ocirc;&icirc;&eth;&igrave;&agrave; &ograve;&agrave;&ecirc;&icirc;&atilde;&icirc; &eth;i&acirc;&iacute;&yuml;&iacute;&iacute;&yuml; &ccedil;&agrave;&iuml;&egrave;&ntilde;&oacute;&sup1;&ograve;&uuml;&ntilde;&yuml; &oacute; &acirc;&egrave;&atilde;&euml;&yuml;&auml;i
uηη + b̄1 uα + b̄2 uβ + c̄u = 0.
(1.7)
&Iacute;&agrave;&ccedil;&acirc;&egrave; &quot;&atilde;i&iuml;&aring;&aacute;&icirc;&euml;i&divide;&iacute;&egrave;&eacute;&quot;, &quot;&aring;&euml;i&iuml;&ograve;&egrave;&divide;&iacute;&egrave;&eacute;&quot; &ograve;&agrave; &quot;&iuml;&agrave;&eth;&agrave;&aacute;&icirc;&euml;i&divide;&iacute;&egrave;&eacute;&quot; &ograve;&egrave;&iuml; &iacute;&agrave;&auml;&agrave;&iacute;&icirc;
&eth;i&acirc;&iacute;&yuml;&iacute;&iacute;&yuml;&igrave;, &acirc;&egrave;&otilde;&icirc;&auml;&yuml;&divide;&egrave; &ccedil; &agrave;&iacute;&agrave;&euml;&icirc;&atilde;i&uml;, &iuml;&eth;&egrave; &iuml;&icirc;&eth;i&acirc;&iacute;&yuml;&iacute;&iacute;i &ccedil; &iacute;&agrave;&ccedil;&acirc;&agrave;&igrave;&egrave; &ecirc;&eth;&egrave;&acirc;&egrave;&otilde; &auml;&eth;&oacute;&atilde;&icirc;&atilde;&icirc;
&iuml;&icirc;&eth;&yuml;&auml;&ecirc;&oacute;, &ugrave;&icirc; &icirc;&iuml;&egrave;&ntilde;&oacute;&thorn;&ograve;&uuml;&ntilde;&yuml; &ecirc;&acirc;&agrave;&auml;&eth;&agrave;&ograve;&egrave;&divide;&iacute;&icirc;&thorn; &ocirc;&icirc;&eth;&igrave;&icirc;&thorn; &ccedil;&agrave;&atilde;&agrave;&euml;&uuml;&iacute;&icirc;&atilde;&icirc; &acirc;&egrave;&auml;&oacute;
a11 x2 + 2a12 xy + a22 y 2 + b1 x + b2 y + c = 0.
7
(1.8)
&szlig;&ecirc;&ugrave;&icirc; D = a212 − a11 a22 &gt; 0, &ograve;&icirc; &eth;i&acirc;&iacute;&yuml;&iacute;&iacute;&yuml; (1.8) &sup1; &eth;i&acirc;&iacute;&yuml;&iacute;&iacute;&yuml;&igrave; &atilde;i&iuml;&aring;&eth;&aacute;&icirc;&euml;&egrave;,
&yuml;&ecirc;&ugrave;&icirc; D &lt; 0, &ograve;&icirc; &acirc;i&auml;&iuml;&icirc;&acirc;i&auml;&iacute;&agrave; &ecirc;&eth;&egrave;&acirc;&agrave; &sup1; &aring;&euml;i&iuml;&ntilde;&icirc;&igrave;, &yuml;&ecirc;&ugrave;&icirc; D = 0 &iuml;&agrave;&eth;&agrave;&aacute;&icirc;&euml;&icirc;&thorn;.
&Ntilde;&euml;i&auml; &ccedil;&agrave;&oacute;&acirc;&agrave;&aelig;&egrave;&ograve;&egrave;, &ugrave;&icirc; &eth;i&acirc;&iacute;&yuml;&iacute;&iacute;&yuml; &eth;i&ccedil;&iacute;&icirc;&atilde;&icirc; &ograve;&egrave;&iuml;&oacute; &acirc;i&auml;&eth;i&ccedil;&iacute;&yuml;&thorn;&ograve;&uuml;&ntilde;&yuml; &icirc;&auml;&iacute;&aring; &acirc;i&auml;
&icirc;&auml;&iacute;&icirc;&atilde;&icirc; &iacute;&aring; &ograve;i&euml;&uuml;&ecirc;&egrave; &ccedil;&icirc;&acirc;&iacute;i&oslash;&iacute;i&igrave; &acirc;&egrave;&atilde;&euml;&yuml;&auml;&icirc;&igrave;, &agrave; &ugrave;&icirc; &aacute;i&euml;&uuml;&oslash; &ntilde;&oacute;&ograve;&ograve;&sup1;&acirc;&icirc;, &acirc;&icirc;&iacute;&egrave; &yuml;&ecirc;i&ntilde;&iacute;&icirc; &acirc;i&auml;&eth;i&ccedil;&iacute;&yuml;&thorn;&ograve;&uuml;&ntilde;&yuml; &otilde;&agrave;&eth;&agrave;&ecirc;&ograve;&aring;&eth;&icirc;&igrave; &eth;&icirc;&ccedil;&acirc;'&yuml;&ccedil;&ecirc;i&acirc; i &icirc;&iuml;&egrave;&ntilde;&oacute;&thorn;&ograve;&uuml; &iuml;&eth;&egrave;&iacute;&ouml;&egrave;&iuml;&icirc;&acirc;&icirc; &eth;i&ccedil;&iacute;i
&ocirc;i&ccedil;&egrave;&divide;&iacute;i &iuml;&eth;&icirc;&ouml;&aring;&ntilde;&egrave; i &yuml;&acirc;&egrave;&ugrave;&agrave;. &ETH;i&acirc;&iacute;&yuml;&iacute;&iacute;&yuml; &atilde;i&iuml;&aring;&eth;&aacute;&icirc;&euml;i&divide;&iacute;&icirc;&atilde;&icirc; &ograve;&egrave;&iuml;&oacute; &acirc;&egrave;&eth;&agrave;&aelig;&agrave;&thorn;&ograve;&uuml; &auml;&egrave;&iacute;&agrave;&igrave;i&ecirc;&oacute; &otilde;&acirc;&egrave;&euml;&uuml;&icirc;&acirc;&egrave;&otilde; &iuml;&eth;&icirc;&ouml;&aring;&ntilde;i&acirc;, &eth;&icirc;&ccedil;&acirc;'&yuml;&ccedil;&ecirc;&egrave; &ograve;&agrave;&ecirc;&egrave;&otilde; &eth;i&acirc;&iacute;&yuml;&iacute;&uuml;, &ccedil;&icirc;&ecirc;&eth;&aring;&igrave;&agrave;, &icirc;&iuml;&egrave;&ntilde;&oacute;&thorn;&ograve;&uuml;
&iuml;&icirc;&oslash;&egrave;&eth;&aring;&iacute;&iacute;&yuml; &ccedil;&acirc;&oacute;&ecirc;&icirc;&acirc;&egrave;&otilde; &agrave;&aacute;&icirc; &aring;&euml;&aring;&ecirc;&ograve;&eth;&icirc;&igrave;&agrave;&atilde;&iacute;i&ograve;&iacute;&egrave;&otilde; &otilde;&acirc;&egrave;&euml;&uuml; &agrave; &ograve;&agrave;&ecirc;&icirc;&aelig; &ntilde;&ograve;&icirc;&yuml;&divide;i &otilde;&acirc;&egrave;&euml;i &acirc; &eth;i&ccedil;&iacute;&egrave;&otilde; &ntilde;&aring;&eth;&aring;&auml;&icirc;&acirc;&egrave;&ugrave;&agrave;&otilde;. &ETH;i&acirc;&iacute;&yuml;&iacute;&iacute;&yuml; &iuml;&agrave;&eth;&agrave;&aacute;&icirc;&euml;i&divide;&iacute;&icirc;&atilde;&icirc; &ograve;&egrave;&iuml;&oacute; &icirc;&iuml;&egrave;&ntilde;&oacute;&thorn;&ograve;&uuml; &iuml;&eth;&icirc;&ouml;&aring;&ntilde;&egrave; &auml;&egrave;&ocirc;&oacute;&ccedil;i&uml; &agrave;&aacute;&icirc; &iuml;&icirc;&oslash;&egrave;&eth;&aring;&iacute;&iacute;&yuml; &ograve;&aring;&iuml;&euml;&icirc;&ograve;&egrave;. &ETH;&icirc;&ccedil;&acirc;'&yuml;&ccedil;&ecirc;&egrave; &eth;i&acirc;&iacute;&yuml;&iacute;&uuml; &aring;&euml;i&iuml;&ograve;&egrave;&divide;&iacute;&icirc;&atilde;&icirc; &ograve;&egrave;&iuml;&oacute;,
&yuml;&ecirc; &iuml;&eth;&agrave;&acirc;&egrave;&euml;&icirc;, &icirc;&iuml;&egrave;&ntilde;&oacute;&thorn;&ograve;&uuml; &ntilde;&ograve;&agrave;&ouml;i&icirc;&iacute;&agrave;&eth;&iacute;&egrave;&eacute; &eth;&icirc;&ccedil;&iuml;&icirc;&auml;i&euml; &ograve;&aring;&igrave;&iuml;&aring;&eth;&agrave;&ograve;&oacute;&eth;&egrave; &acirc; &ntilde;&aring;&eth;&aring;&auml;&icirc;&acirc;&egrave;&ugrave;i,
&ntilde;&ograve;&agrave;&ouml;i&icirc;&iacute;&agrave;&eth;&iacute;&egrave;&eacute; &eth;&icirc;&ccedil;&iuml;&icirc;&auml;i&euml; &ecirc;&icirc;&iacute;&ouml;&aring;&iacute;&ograve;&eth;&agrave;&ouml;i&uml; &eth;&aring;&divide;&icirc;&acirc;&egrave;&iacute;&egrave;, &agrave;&aacute;&icirc; &eth;&icirc;&ccedil;&iuml;&icirc;&auml;i&euml; &acirc; &iuml;&eth;&icirc;&ntilde;&ograve;&icirc;&eth;i
&aring;&euml;&aring;&ecirc;&ograve;&eth;&icirc;&ntilde;&ograve;&agrave;&ograve;&egrave;&divide;&iacute;&icirc;&atilde;&icirc; &iuml;&icirc;&ograve;&aring;&iacute;&ouml;i&agrave;&euml;&oacute; &iuml;&eth;&egrave; &ccedil;&agrave;&auml;&agrave;&iacute;&icirc;&igrave;&oacute; &eth;&icirc;&ccedil;&iuml;&icirc;&auml;i&euml;i &ccedil;&agrave;&eth;&yuml;&auml;i&acirc;.
&szlig;&ecirc;&ugrave;&icirc; &ecirc;&icirc;&aring;&ocirc;i&ouml;i&sup1;&iacute;&ograve;&egrave; aij , bi &ograve;&agrave; c &acirc; &eth;i&acirc;&iacute;&yuml;&iacute;&iacute;i (1.2) &sup1; &ntilde;&ograve;&agrave;&euml;&egrave;&igrave;&egrave; &acirc;&aring;&euml;&egrave;&divide;&egrave;&iacute;&agrave;&igrave;&egrave;, &ograve;&icirc;&aacute;&ograve;&icirc; &iacute;&aring; &ccedil;&agrave;&euml;&aring;&aelig;&agrave;&ograve;&uuml; &acirc;i&auml; x i y , &ograve;&icirc; &iuml;i&ntilde;&euml;&yuml; &ccedil;&acirc;&aring;&auml;&aring;&iacute;&iacute;&yuml; &auml;&icirc; &ecirc;&agrave;&iacute;&icirc;&iacute;i&divide;&iacute;&icirc;&atilde;&icirc; &acirc;&egrave;&auml;&oacute;
&eth;i&acirc;&iacute;&yuml;&iacute;&iacute;&yuml; &ccedil;&agrave;&euml;&egrave;&oslash;&agrave;&thorn;&ograve;&uuml;&ntilde;&yuml; &eth;i&acirc;&iacute;&yuml;&iacute;&iacute;&yuml;&igrave;&egrave; &ccedil;i &ntilde;&ograve;&agrave;&euml;&egrave;&igrave;&egrave; &ecirc;&icirc;&aring;&ocirc;i&ouml;i&sup1;&iacute;&ograve;&agrave;&igrave;&egrave;.
&Ecirc;&agrave;&iacute;&icirc;&iacute;i&divide;&iacute;&oacute; &ocirc;&icirc;&eth;&igrave;&oacute; &eth;i&acirc;&iacute;&yuml;&iacute;&iacute;&yuml; i&ccedil; &ntilde;&ograve;&agrave;&euml;&egrave;&igrave;&egrave; &ecirc;&icirc;&aring;&ocirc;i&ouml;i&sup1;&iacute;&ograve;&agrave;&igrave;&egrave; &igrave;&icirc;&aelig;&iacute;&agrave; &ugrave;&aring; &auml;&icirc;&auml;&agrave;&ograve;&ecirc;&icirc;&acirc;&icirc; &ntilde;&iuml;&eth;&icirc;&ntilde;&ograve;&egrave;&ograve;&egrave; &oslash;&euml;&yuml;&otilde;&icirc;&igrave; &ccedil;&agrave;&igrave;i&iacute;&egrave; &oslash;&oacute;&ecirc;&agrave;&iacute;&icirc;&uml; &ocirc;&oacute;&iacute;&ecirc;&ouml;i&uml;: u(ξ, η) = eλξ+&micro;η v(ξ, η).
&Iacute;&agrave;&euml;&aring;&aelig;&iacute;&egrave;&igrave; &iuml;i&auml;&aacute;&icirc;&eth;&icirc;&igrave; &ccedil;&iacute;&agrave;&divide;&aring;&iacute;&uuml; &iuml;&agrave;&eth;&agrave;&igrave;&aring;&ograve;&eth;i&acirc; λ i &micro; &eth;i&acirc;&iacute;&yuml;&iacute;&iacute;&yuml; &ccedil;&acirc;&icirc;&auml;&egrave;&ograve;&uuml;&ntilde;&yuml; &auml;&icirc; &iacute;&agrave;&ntilde;&ograve;&oacute;&iuml;&iacute;&icirc;&atilde;&icirc; &icirc;&ntilde;&ograve;&agrave;&ograve;&icirc;&divide;&iacute;&icirc;&atilde;&icirc; &acirc;&egrave;&atilde;&euml;&yuml;&auml;&oacute;.
vξη + γv = 0, &agrave;&aacute;&icirc; vξξ − vηη + γv = 0 &eth;i&acirc;&iacute;&yuml;&iacute;&iacute;&yuml; &atilde;i&iuml;&aring;&eth;&aacute;&icirc;&euml;i&divide;&iacute;&icirc;&atilde;&icirc; &ograve;&egrave;&iuml;&oacute;.
vξξ + vηη + γv = 0 &eth;i&acirc;&iacute;&yuml;&iacute;&iacute;&yuml; &aring;&euml;i&iuml;&ograve;&egrave;&divide;&iacute;&icirc;&atilde;&icirc; &ograve;&egrave;&iuml;&oacute;.
vηη + b̄2 vη + γv = 0 &eth;i&acirc;&iacute;&yuml;&iacute;&iacute;&yuml; &iuml;&agrave;&eth;&agrave;&aacute;&icirc;&euml;i&divide;&iacute;&icirc;&atilde;&icirc; &ograve;&egrave;&iuml;&oacute;.
&Ograve;&aring;&icirc;&eth;&aring;&ograve;&egrave;&divide;&iacute;&aring; &icirc;&aacute;&atilde;&eth;&oacute;&iacute;&ograve;&oacute;&acirc;&agrave;&iacute;&iacute;&yuml; &ccedil;&acirc;&aring;&auml;&aring;&iacute;&iacute;&yuml; &eth;i&acirc;&iacute;&yuml;&iacute;&iacute;&yuml; &auml;&icirc; &ecirc;&agrave;&iacute;&icirc;&iacute;i&divide;&iacute;&icirc;&atilde;&icirc; &acirc;&egrave;&auml;&oacute; &iacute;&agrave;&acirc;&aring;&auml;&aring;&iacute;&aring; &acirc; [2].
&Iacute;&agrave; &auml;&icirc;&iuml;&icirc;&igrave;&icirc;&atilde;&oacute; &iuml;&eth;&agrave;&ecirc;&ograve;&egrave;&divide;&iacute;i&eacute; &iuml;&eth;&icirc;&ouml;&aring;&auml;&oacute;&eth;i &ccedil;&acirc;&aring;&auml;&aring;&iacute;&iacute;&yuml; &eth;i&acirc;&iacute;&yuml;&iacute;&uuml; &auml;&icirc; &ecirc;&agrave;&iacute;&icirc;&iacute;i&divide;&iacute;&icirc;&atilde;&icirc;
&acirc;&egrave;&auml;&oacute; &iacute;&agrave;&acirc;&icirc;&auml;&egrave;&igrave;&icirc; &ntilde;&otilde;&aring;&igrave;&oacute;, &acirc; &yuml;&ecirc;i&eacute; &acirc;i&auml;&icirc;&aacute;&eth;&agrave;&aelig;&aring;&iacute;&agrave; &iuml;&icirc;&ntilde;&euml;i&auml;&icirc;&acirc;&iacute;i&ntilde;&ograve;&uuml; &auml;i&eacute; &auml;&euml;&yuml; &icirc;&ograve;&eth;&egrave;&igrave;&agrave;&iacute;&iacute;&yuml; &ecirc;&agrave;&iacute;&icirc;&iacute;i&divide;&iacute;&icirc;&atilde;&icirc; &acirc;&egrave;&auml;&oacute; &eth;i&acirc;&iacute;&yuml;&iacute;&iacute;&yuml;.
8
&Ntilde;&otilde;&aring;&igrave;&agrave; &ccedil;&acirc;&aring;&auml;&aring;&iacute;&iacute;&yuml; &auml;&egrave;&ocirc;&aring;&eth;&aring;&iacute;&ouml;i&agrave;&euml;&uuml;&iacute;&icirc;&atilde;&icirc; &eth;i&acirc;&iacute;&yuml;&iacute;&iacute;&yuml; &auml;&eth;&oacute;&atilde;&icirc;&atilde;&icirc; &iuml;&icirc;&eth;&yuml;&auml;&ecirc;&oacute;
&ccedil; &auml;&acirc;&icirc;&igrave;&agrave; &iacute;&aring;&ccedil;&agrave;&euml;&aring;&aelig;&iacute;&egrave;&igrave;&egrave; &ccedil;&igrave;i&iacute;&iacute;&egrave;&igrave;&egrave; &auml;&icirc; &ecirc;&agrave;&iacute;&icirc;&iacute;i&divide;&iacute;&icirc;&atilde;&icirc; &acirc;&egrave;&atilde;&euml;&yuml;&auml;&oacute;.
9
&Ccedil;&agrave;&ntilde;&ograve;&icirc;&ntilde;&oacute;&acirc;&agrave;&iacute;&iacute;&yuml; &ograve;&agrave;&aacute;&euml;&egrave;&ouml;i &auml;&aring;&igrave;&icirc;&iacute;&ntilde;&ograve;&eth;&oacute;&sup1;&ograve;&uuml;&ntilde;&yuml; &acirc; &iacute;&agrave;&ntilde;&ograve;&oacute;&iuml;&iacute;&icirc;&igrave;&oacute; &iuml;&eth;&egrave;&ecirc;&euml;&agrave;&auml;i.
&Iuml;&eth;&egrave;&ecirc;&euml;&agrave;&auml; 1
&Ccedil;&acirc;&aring;&ntilde;&ograve;&egrave; &auml;&icirc; &ecirc;&agrave;&iacute;&icirc;&iacute;i&divide;&iacute;&icirc;&atilde;&icirc; &acirc;&egrave;&auml;&oacute; &eth;i&acirc;&iacute;&yuml;&iacute;&iacute;&yuml;:
1
1
xuxx − yuyy + ux − uy = 0
2
2
• &Iuml;&icirc;&eth;i&acirc;&iacute;&thorn;&sup1;&igrave;&icirc; &auml;&agrave;&iacute;&aring; &eth;i&acirc;&iacute;&yuml;&iacute;&iacute;&yuml; i&ccedil; &ccedil;&agrave;&atilde;&agrave;&euml;&uuml;&iacute;&egrave;&igrave; &acirc;&egrave;&eth;&agrave;&ccedil;&icirc;&igrave; (1.2) &auml;&euml;&yuml; &euml;i&iacute;i&eacute;&iacute;&icirc;&atilde;&icirc;
&auml;&egrave;&ocirc;&aring;&eth;&aring;&iacute;&ouml;i&agrave;&euml;&uuml;&iacute;&icirc;&atilde;&icirc; &eth;i&acirc;&iacute;&yuml;&iacute;&iacute;&yuml; &auml;&eth;&oacute;&atilde;&icirc;&atilde;&icirc; &iuml;&icirc;&eth;&yuml;&auml;&ecirc;&oacute; &ccedil; &auml;&acirc;&icirc;&igrave;&agrave; &iacute;&aring;&ccedil;&agrave;&euml;&aring;&aelig;&iacute;&egrave;&igrave;&egrave;
&ccedil;&igrave;i&iacute;&iacute;&egrave;&igrave;&egrave; i &acirc;&egrave;&ccedil;&iacute;&agrave;&divide;&agrave;&sup1;&igrave;&icirc; &ecirc;&icirc;&aring;&ocirc;i&ouml;i&sup1;&iacute;&ograve;&egrave; &eth;i&acirc;&iacute;&yuml;&iacute;&iacute;&yuml;: a11 = x, a12 = 0 ,
a22 = −y , b1 = 21 , b2 = − 12 , c = 0.
• &Acirc;&egrave;&ccedil;&iacute;&agrave;&divide;&agrave;&sup1;&igrave;&icirc; &auml;&egrave;&ntilde;&ecirc;&eth;&egrave;&igrave;i&iacute;&agrave;&iacute;&ograve; D i &acirc;&ntilde;&ograve;&agrave;&iacute;&icirc;&acirc;&euml;&thorn;&sup1;&igrave;&icirc; &ograve;&egrave;&iuml; &eth;i&acirc;&iacute;&yuml;&iacute;&iacute;&yuml;. D =
a212 − a11 a22 ⇒ D = xy. &Ccedil;&acirc;i&auml;&ntilde;&egrave; &acirc;&egrave;&auml;&iacute;&icirc;, &ugrave;&icirc; &acirc; &ccedil;&agrave;&euml;&aring;&aelig;&iacute;&icirc;&ntilde;&ograve;i &acirc;i&auml; &ccedil;&iacute;&agrave;&ecirc;i&acirc;
x &ograve;&agrave; y &auml;&egrave;&ntilde;&ecirc;&eth;&egrave;&igrave;i&iacute;&agrave;&iacute;&ograve; &igrave;&icirc;&aelig;&aring; &aacute;&oacute;&ograve;&egrave; &auml;&icirc;&auml;&agrave;&ograve;&iacute;i&igrave; &agrave;&aacute;&icirc; &acirc;i&auml;'&sup1;&igrave;&iacute;&egrave;&igrave;.
y
Еліптичний
тип
Гіперболічний
тип
x
0
Гіперболічний
тип
Еліптичний
тип
&Iacute;&agrave; &iuml;&euml;&icirc;&ugrave;&egrave;&iacute;i (x, y) &acirc;&ntilde;&ograve;&agrave;&iacute;&icirc;&acirc;&euml;&thorn;&sup1;&igrave;&icirc; &icirc;&aacute;&euml;&agrave;&ntilde;&ograve;i &acirc;, &yuml;&ecirc;&egrave;&otilde; &auml;&egrave;&ntilde;&ecirc;&eth;&egrave;&igrave;i&iacute;&agrave;&iacute;&ograve; &igrave;&agrave;&sup1;
&iuml;&icirc;&ntilde;&ograve;i&eacute;&iacute;&egrave;&eacute; &ccedil;&iacute;&agrave;&ecirc;. &Acirc; &iuml;&aring;&eth;&oslash;&icirc;&igrave;&oacute; &ecirc;&acirc;&agrave;&auml;&eth;&agrave;&iacute;&ograve;i (x &gt; 0, y &gt; 0) &ograve;&agrave; &acirc; &ograve;&eth;&aring;&ograve;&uuml;&icirc;&igrave;&oacute;
&ecirc;&acirc;&agrave;&auml;&eth;&agrave;&iacute;&ograve;i (x &lt; 0, y &lt; 0) &auml;&egrave;&ntilde;&ecirc;&eth;&egrave;&igrave;i&iacute;&agrave;&iacute;&ograve; D &gt; 0 i, &icirc;&ograve;&aelig;&aring;, &auml;&agrave;&iacute;&aring; &eth;i&acirc;&iacute;&yuml;&iacute;&iacute;&yuml; &sup1; &eth;i&acirc;&iacute;&yuml;&iacute;&iacute;&yuml;&igrave; &atilde;i&iuml;&aring;&eth;&aacute;&icirc;&euml;i&divide;&iacute;&icirc;&atilde;&icirc; &ograve;&egrave;&iuml;&oacute;. &Acirc; &auml;&eth;&oacute;&atilde;&icirc;&igrave;&oacute; (x &lt; 0, y &gt; 0) i &acirc;
&divide;&aring;&ograve;&acirc;&aring;&eth;&ograve;&icirc;&igrave;&oacute; (x &gt; 0, y &lt; 0) &ecirc;&acirc;&agrave;&auml;&eth;&agrave;&iacute;&ograve;&agrave;&otilde; &auml;&egrave;&ntilde;&ecirc;&eth;&egrave;&igrave;i&iacute;&agrave;&iacute;&ograve; D &lt; 0. &Icirc;&ograve;&aelig;&aring;,
&eth;i&acirc;&iacute;&yuml;&iacute;&iacute;&yuml; &acirc;i&auml;&iacute;&icirc;&ntilde;&egrave;&ograve;&uuml;&ntilde;&yuml; &auml;&icirc; &eth;i&acirc;&iacute;&yuml;&iacute;&uuml; &aring;&euml;i&iuml;&ograve;&egrave;&divide;&iacute;&icirc;&atilde;&icirc; &ograve;&egrave;&iuml;&oacute;. &Acirc;&ccedil;&auml;&icirc;&acirc;&aelig; &ecirc;&icirc;&icirc;&eth;&auml;&egrave;&iacute;&agrave;&ograve;&iacute;&egrave;&otilde; &icirc;&ntilde;&aring;&eacute; &eth;i&acirc;&iacute;&yuml;&iacute;&iacute;&yuml; &igrave;&agrave;&sup1; &iuml;&agrave;&eth;&agrave;&aacute;&icirc;&euml;i&divide;&iacute;&egrave;&eacute; &ograve;&egrave;&iuml; (D = 0).
&Acirc;&egrave;&ccedil;&iacute;&agrave;&divide;&agrave;&sup1;&igrave;&icirc;&ntilde;&uuml;, &ccedil; &icirc;&aacute;&euml;&agrave;&ntilde;&ograve;&thorn; &acirc; &yuml;&ecirc;i&eacute; &ograve;&eth;&aring;&aacute;&agrave; &ccedil;&acirc;&aring;&ntilde;&ograve;&egrave; &eth;i&acirc;&iacute;&yuml;&iacute;&iacute;&yuml; &auml;&icirc; &ecirc;&agrave;&iacute;&icirc;&iacute;i&divide;&iacute;&icirc;&atilde;&icirc; &acirc;&egrave;&auml;&oacute;. &Icirc;&aacute;&aring;&eth;&aring;&igrave;&icirc;, &iacute;&agrave;&iuml;&eth;&egrave;&ecirc;&euml;&agrave;&auml;, &iuml;&aring;&eth;&oslash;&egrave;&eacute; &ecirc;&acirc;&agrave;&auml;&eth;&agrave;&iacute;&ograve;, &ograve;&icirc;&aacute;&ograve;&icirc; &icirc;&aacute;&euml;&agrave;&ntilde;&ograve;&uuml;
&auml;&aring; x &gt; 0 i y &gt; 0.
• &Ccedil;&agrave;&iuml;&egrave;&ntilde;&oacute;&sup1;&igrave;&icirc; &otilde;&agrave;&eth;&agrave;&ecirc;&ograve;&aring;&eth;&egrave;&ntilde;&ograve;&egrave;&divide;&iacute;&aring; &eth;i&acirc;&iacute;&yuml;&iacute;&iacute;&yuml;:
p
√
√
xy
y
a12 &plusmn; a212 − a11 a22
dy
dy
dy
=
⇒
=&plusmn;
, &agrave;&aacute;&icirc;
= &plusmn;√
dx
a11
dx
x
dx
x
&Auml;&acirc;&icirc;&igrave; &ccedil;&iacute;&agrave;&ecirc;&agrave;&igrave; &quot;+&quot; &ograve;&agrave; &quot;−&quot; &acirc;i&auml;&iuml;&icirc;&acirc;i&auml;&agrave;&thorn;&ograve;&uuml; &auml;&acirc;&agrave; &eth;i&ccedil;&iacute;i &otilde;&agrave;&eth;&agrave;&ecirc;&ograve;&aring;&eth;&egrave;&ntilde;&ograve;&egrave;&divide;&iacute;i
&eth;i&acirc;&iacute;&yuml;&iacute;&iacute;&yuml;:
1) y 0 =
√
√
√ √
y/ x, 2) y 0 = − y/ x.
10
• I&iacute;&ograve;&aring;&atilde;&eth;&oacute;&sup1;&igrave;&icirc; &auml;&egrave;&ocirc;&aring;&eth;&aring;&iacute;&ouml;i&agrave;&euml;&uuml;&iacute;i &eth;i&acirc;&iacute;&yuml;&iacute;&iacute;&yuml; &iuml;&aring;&eth;&oslash;&icirc;&atilde;&icirc; &iuml;&icirc;&eth;&yuml;&auml;&ecirc;&oacute; i &ccedil;&iacute;&agrave;&otilde;&icirc;&auml;&egrave;&igrave;&icirc;
&acirc;i&auml;&iuml;&icirc;&acirc;i&auml;&iacute;i &ccedil;&agrave;&atilde;&agrave;&euml;&uuml;&iacute;i i&iacute;&ograve;&aring;&atilde;&eth;&agrave;&euml;&egrave;:
&frac12; √
&frac12;√
√
√
2 x − 2 y = C̃1
x − y = C1
√
√
&agrave;&aacute;&icirc;
√
√
x + y = C2 .
2 x + 2 y = C̃2 ,
&Ccedil;&atilde;i&auml;&iacute;&icirc; &ograve;&aring;&icirc;&eth;i&uml;, &iacute;&icirc;&acirc;i &ccedil;&igrave;i&iacute;&iacute;i ξ &ograve;&agrave; η &icirc;&aacute;&egrave;&eth;&agrave;&thorn;&ograve;&uuml;&ntilde;&yuml; &ocirc;&icirc;&eth;&igrave;&agrave;&euml;&uuml;&iacute;&icirc;&thorn; &ccedil;&agrave;&igrave;i&iacute;&icirc;&thorn;
√
√
√
√
C1 → ξ , C2 → η : ξ = x − y , η = x + y , &agrave;&aacute;&icirc;
&frac12;
1
1
ξ = x2 − y 2
1
1
η = x2 + y 2 .
• &Auml;&euml;&yuml; &ccedil;&iacute;&agrave;&otilde;&icirc;&auml;&aelig;&aring;&iacute;&iacute;&yuml; &ecirc;&icirc;&aring;&ocirc;i&ouml;i&sup1;&iacute;&ograve;i&acirc; &iuml;&aring;&eth;&aring;&ograve;&acirc;&icirc;&eth;&aring;&iacute;&icirc;&atilde;&icirc; &eth;i&acirc;&iacute;&yuml;&iacute;&iacute;&yuml; &iuml;&icirc;&iuml;&aring;&eth;&aring;&auml;&iacute;&uuml;&icirc;
&ccedil;&iacute;&agrave;&otilde;&icirc;&auml;&egrave;&igrave;&icirc; &divide;&agrave;&ntilde;&ograve;&egrave;&iacute;&iacute;i &iuml;&icirc;&otilde;i&auml;&iacute;i:
ξx = 12 x−1/2
ξxx = − 14 x−3/2
ξy = − 21 y −1/2
ξyy = 41 y −3/2
ξxy = 0
ηx = 12 x−1/2
ηxx = − 14 x−3/2
ηy = 12 y −1/2
ηyy = − 41 y −3/2
ηxy = 0.
• &Icirc;&aacute;&divide;&egrave;&ntilde;&euml;&thorn;&sup1;&igrave;&icirc; &ecirc;&icirc;&aring;&ocirc;i&ouml;i&sup1;&iacute;&ograve;&egrave; &iuml;&aring;&eth;&aring;&ograve;&acirc;&icirc;&eth;&aring;&iacute;&icirc;&atilde;&icirc; &eth;i&acirc;&iacute;&yuml;&iacute;&iacute;&yuml;:
&micro;
&para;2
&micro;
&para;2
1
1 1
1
ā11 = a11 ξx2 +2a12 ξx ηy +a22 ξy2 = x
x−1/2 +0−y − y −1/2 = − = 0,
2
2
4 4
&micro;
&para;2
&micro;
&para;2
1 −1/2
1 −1/2
1 1
2
2
+0−y
= − = 0,
ā22 = a11 ηx +2a12 ηx ηy +a22 ηy = x
x
y
2
2
4 4
ā12 = a11 ξx ηx + a12 (ξx ηy + ξy ηx ) + a22 ξy ηy =
&micro;
&para;&micro;
&para;
&para;&micro;
&para;
1 −1/2
1 1 1
1 −1/2
1 −1/2
1 −1/2
x
x
+0−y − y
y
= + = ,
=x
2
2
2
2
4 4 2
1
1
b̄1 = L̂ξ = xξxx − yξyy + ξx − ξy =
2
2
&micro;
&para;
&micro;
&para;
&micro;
&para;
&micro;
&para;
1 −3/2
1 −3/2
1 1 −1/2
1
1 −1/2
=x − x
y
x
−y
+
−
− y
= 0,
4
4
2 2
2
2
1
1
b̄2 = L̂η = xηxx − yηyy + ηx − ηy =
2
2
&micro;
&para;
&micro;
&para;
&micro;
&para;
&micro;
&para;
1 −3/2
1 −3/2
1 1 −1/2
1 1 −1/2
=x − x
x
y
−y − y
+
−
= 0,
4
4
2 2
2 2
&micro;
11
c̄ = c = 0.
&Ccedil;&agrave;&oacute;&acirc;&agrave;&aelig;&egrave;&igrave;&icirc;, &ugrave;&icirc; &acirc; &auml;&agrave;&iacute;&iacute;&icirc;&igrave;&oacute; &acirc;&egrave;&iuml;&agrave;&auml;&ecirc;&oacute; &icirc;&aacute;&divide;&egrave;&ntilde;&euml;&aring;&iacute;&iacute;&yuml; &ecirc;&icirc;&aring;&ocirc;i&ouml;i&sup1;&iacute;&ograve;i&acirc; ā11 &ograve;&agrave;
ā22 &iacute;&aring; &sup1; &icirc;&aacute;&icirc;&acirc;'&yuml;&ccedil;&ecirc;&icirc;&acirc;&egrave;&igrave;, &icirc;&ntilde;&ecirc;i&euml;&uuml;&ecirc;&egrave;, &ccedil;&atilde;i&auml;&iacute;&icirc; &ograve;&aring;&icirc;&eth;i&uml; &oacute; &acirc;&egrave;&iuml;&agrave;&auml;&ecirc;&oacute; &eth;i&acirc;&iacute;&yuml;&iacute;&uuml; &atilde;i&iuml;&aring;&eth;&aacute;&icirc;&euml;i&divide;&iacute;&icirc;&atilde;&icirc; &ograve;&egrave;&iuml;&oacute; &ouml;i &ecirc;&icirc;&aring;&ocirc;i&ouml;i&sup1;&iacute;&ograve;&egrave; &ccedil;&agrave;&acirc;&aelig;&auml;&egrave; &auml;&icirc;&eth;i&acirc;&iacute;&thorn;&thorn;&ograve;&uuml; &iacute;&oacute;&euml;&thorn;. &Agrave;&euml;&aring; &aacute;&aring;&ccedil;&iuml;&icirc;&ntilde;&aring;&eth;&aring;&auml;&iacute;&sup1; &icirc;&aacute;&divide;&egrave;&ntilde;&euml;&aring;&iacute;&iacute;&yuml; &ecirc;&icirc;&aring;&ocirc;i&ouml;i&sup1;&iacute;&ograve;i&acirc; ā11 &ograve;&agrave; ā22 &auml;&icirc;&ccedil;&acirc;&icirc;&euml;&yuml;&sup1; &iuml;&aring;&eth;&aring;&acirc;i&eth;&egrave;&ograve;&egrave;
&iuml;&eth;&agrave;&acirc;&egrave;&euml;&uuml;&iacute;i&ntilde;&ograve;&uuml; &ccedil;&iacute;&agrave;&otilde;&icirc;&auml;&aelig;&aring;&iacute;&iacute;&yuml; &ccedil;&agrave;&atilde;&agrave;&euml;&uuml;&iacute;&egrave;&otilde; i&iacute;&ograve;&aring;&atilde;&eth;&agrave;&euml;i&acirc; &otilde;&agrave;&eth;&agrave;&ecirc;&ograve;&aring;&eth;&egrave;&ntilde;&ograve;&egrave;&divide;&iacute;&egrave;&otilde;
&eth;i&acirc;&iacute;&yuml;&iacute;&uuml;.
• &Acirc;&eth;&agrave;&otilde;&icirc;&acirc;&oacute;&thorn;&divide;&egrave; &icirc;&ograve;&eth;&egrave;&igrave;&agrave;&iacute;i &ccedil;&iacute;&agrave;&divide;&aring;&iacute;&iacute;&yuml; &iacute;&icirc;&acirc;&egrave;&otilde; &ecirc;&icirc;&aring;&ocirc;i&ouml;i&sup1;&iacute;&ograve;i&acirc;, &ccedil;&agrave;&iuml;&egrave;&oslash;&aring;&igrave;&icirc; &iuml;&aring;&eth;&aring;&ograve;&acirc;&icirc;&eth;&aring;&iacute;&aring; &eth;i&acirc;&iacute;&yuml;&iacute;&iacute;&yuml;:
2ā12 uξη = 0,
&agrave;&aacute;&icirc; &icirc;&ntilde;&ograve;&agrave;&ograve;&icirc;&divide;&iacute;&icirc;:
uξη = 0.
&Acirc; &auml;&agrave;&iacute;&icirc;&igrave;&oacute; &acirc;&egrave;&iuml;&agrave;&auml;&ecirc;&oacute;, &ccedil;&agrave;&acirc;&auml;&yuml;&ecirc;&egrave; &ntilde;&iuml;&aring;&ouml;i&agrave;&euml;&uuml;&iacute;&icirc;&igrave;&oacute; &acirc;&egrave;&aacute;&icirc;&eth;&oacute; &ecirc;&icirc;&aring;&ocirc;i&ouml;i&sup1;&iacute;&ograve;i&acirc; aij i bi ,
&ecirc;i&iacute;&ouml;&aring;&acirc;&egrave;&eacute; &acirc;&egrave;&eth;&agrave;&ccedil; &auml;&euml;&yuml; &eth;i&acirc;&iacute;&yuml;&iacute;&iacute;&yuml; &acirc;&egrave;&yuml;&acirc;&egrave;&acirc;&ntilde;&yuml; &iacute;&agrave;&eacute;&iuml;&eth;&icirc;&ntilde;&ograve;i&oslash;&egrave;&igrave; &ccedil; &oacute;&ntilde;i&otilde; &igrave;&icirc;&aelig;&euml;&egrave;&acirc;&egrave;&otilde;
&auml;&euml;&yuml; &eth;i&acirc;&iacute;&yuml;&iacute;&uuml; &atilde;i&iuml;&aring;&eth;&aacute;&icirc;&euml;i&divide;&iacute;&icirc;&atilde;&icirc; &ograve;&egrave;&iuml;&oacute;. &Acirc; &aacute;i&euml;&uuml;&oslash; &ccedil;&agrave;&atilde;&agrave;&euml;&uuml;&iacute;&icirc;&igrave;&oacute; &acirc;&egrave;&iuml;&agrave;&auml;&ecirc;&oacute;, &iuml;i&ntilde;&euml;&yuml; &acirc;&ntilde;i&otilde;
&iacute;&agrave;&acirc;&aring;&auml;&aring;&iacute;&egrave;&otilde; &acirc;&egrave;&ugrave;&aring; &icirc;&iuml;&aring;&eth;&agrave;&ouml;i&eacute;, &eth;i&acirc;&iacute;&yuml;&iacute;&iacute;&yuml; &atilde;i&iuml;&aring;&eth;&aacute;&icirc;&euml;i&divide;&iacute;&icirc;&atilde;&icirc; &ograve;&egrave;&iuml;&oacute; &iacute;&agrave;&aacute;&oacute;&acirc;&agrave;&sup1; &acirc;&egrave;&atilde;&euml;&yuml;&auml;&oacute;:
2ā12 uξη + b̄1 uξ + b̄2 uη + c̄u = 0,
&agrave;&aacute;&icirc; &iuml;i&ntilde;&euml;&yuml; &auml;i&euml;&aring;&iacute;&iacute;&yuml; &iacute;&agrave; 2a12 :
uξη + b̃1 uξ + b̃2 uη + c̃u = 0.
&Ograve;&oacute;&ograve; &ecirc;&icirc;&aring;&ocirc;i&ouml;i&sup1;&iacute;&ograve;&egrave; b̃1 , b̃2 &ograve;&agrave; c̃ &igrave;&icirc;&aelig;&oacute;&ograve;&uuml; &ccedil;&agrave;&euml;&aring;&aelig;&agrave;&ograve;&egrave; &acirc;i&auml; &ntilde;&ograve;&agrave;&eth;&egrave;&otilde; &ccedil;&igrave;i&iacute;&iacute;&egrave;&otilde; x &ograve;&agrave; y . &Ccedil;&agrave;
&auml;&icirc;&iuml;&icirc;&igrave;&icirc;&atilde;&icirc;&thorn; &icirc;&aacute;&aring;&eth;&iacute;&aring;&iacute;&icirc;&atilde;&icirc; &iuml;&aring;&eth;&aring;&ograve;&acirc;&icirc;&eth;&aring;&iacute;&iacute;&yuml;, &ccedil; &eth;i&acirc;&iacute;&icirc;&ntilde;&ograve;&aring;&eacute; ξ = ξ(x, y), η = η(x, y)
&ntilde;&ograve;&agrave;&eth;i &ccedil;&igrave;i&iacute;&iacute;i &acirc;&egrave;&eth;&agrave;&aelig;&agrave;&sup1;&igrave;&icirc; &divide;&aring;&eth;&aring;&ccedil; &iacute;&icirc;&acirc;i: x = x(ξ, η), y = y(ξ, η) i &iuml;i&auml;&ntilde;&ograve;&agrave;&acirc;&euml;&yuml;&sup1;&igrave;&icirc;
&acirc; &ecirc;&icirc;&aring;&ocirc;i&ouml;i&sup1;&iacute;&ograve;&egrave; b̃1 , b̃2 &ograve;&agrave; c̃. &Ograve;&icirc;&auml;i &eth;i&acirc;&iacute;&yuml;&iacute;&iacute;&yuml; &ccedil;&agrave;&iuml;&egrave;&oslash;&aring;&ograve;&uuml;&ntilde;&yuml; &acirc;&egrave;&ecirc;&euml;&thorn;&divide;&iacute;&icirc; &divide;&aring;&eth;&aring;&ccedil; &iacute;&icirc;&acirc;i
&ccedil;&igrave;i&iacute;&iacute;i ξ &ograve;&agrave; η .
&Agrave;&iacute;&agrave;&euml;&icirc;&atilde;i&divide;&iacute;&icirc; &igrave;&icirc;&aelig;&iacute;&agrave; &ccedil;&acirc;&aring;&ntilde;&ograve;&egrave; &eth;i&acirc;&iacute;&yuml;&iacute;&iacute;&yuml; &auml;&icirc; &ecirc;&agrave;&iacute;&icirc;&iacute;i&divide;&iacute;&icirc;&atilde;&icirc; &acirc;&egrave;&auml;&oacute; i &acirc; i&iacute;&oslash;&egrave;&otilde; &ecirc;&acirc;&agrave;&auml;&eth;&agrave;&iacute;&ograve;&agrave;&otilde;.
&Auml;&euml;&yuml; &acirc;&iuml;&aring;&acirc;&iacute;&aring;&iacute;&icirc;&atilde;&icirc; &ccedil;&agrave;&ntilde;&acirc;&icirc;&sup1;&iacute;&iacute;&yuml; &igrave;&aring;&ograve;&icirc;&auml;&oacute; &eth;&aring;&ecirc;&icirc;&igrave;&aring;&iacute;&auml;&oacute;&sup1;&ograve;&uuml;&ntilde;&yuml; &ntilde;&agrave;&igrave;&icirc;&ntilde;&ograve;i&eacute;&iacute;&icirc; &icirc;&iuml;&eth;&agrave;&ouml;&thorn;&acirc;&agrave;&ograve;&egrave; &iacute;&agrave;&acirc;&aring;&auml;&aring;&iacute;i &iacute;&egrave;&aelig;&divide;&aring; &ccedil;&agrave;&acirc;&auml;&agrave;&iacute;&iacute;&yuml;.
12
&Ccedil;&acirc;&aring;&ntilde;&ograve;&egrave; &auml;&icirc; &ecirc;&agrave;&iacute;&icirc;&iacute;i&divide;&iacute;&icirc;&atilde;&icirc; &acirc;&egrave;&auml;&oacute; &ograve;&agrave; &ntilde;&iuml;&eth;&icirc;&ntilde;&ograve;&egrave;&ograve;&egrave; &eth;i&acirc;&iacute;&yuml;&iacute;&iacute;&yuml;:
1.1. uxx − 2 sin xuxy + (2 − cos2 x)uyy = 0
1.2. x2 uxx − y 2 uyy − 2uy = 0
1.3. xuxx + yuyy + 2ux + 2uy = 0
1.4.
1 ∂
x ∂x
&iexcl; ∂u &cent;
x ∂x +
1 ∂2u
x2 ∂y 2
=0
1.5. (1 + x2 )2 uxx + uyy + 2x(1 + x2 )ux = 0
1.6. x2 uxx − 2xyuxy − y 2 uyy + xux + uy = 0
1.7. uxx + yuyy = 0
1.8. xuxx + 2xuxy + (x − 1)uyy = 0
1.9. yuxx + xuyy = 0
1.10. signx uxx + 2uxy + signy uyy = 0
1.11. x2 uxx + 2xyuxy + y 2 uyy = 0
1.12. e2x uxx + 2ex+y uxy + e2y uyy − u = 0
1.13. xuxx + 2xuxy + (x − 1)uyy = 0
1.14. xuxx + yuyy + 2ux + 2uy = 0
1.15. uxx + xyuyy = 0
1.16. uxx + uxy − 2uyy − 3ux − 15uy + 27x = 0
1.17. uxx + 2uxy + 5uyy − 32u = 0
1.18. uxx − 2uxy + uyy + ux + uy − u = 0
1.19. uxy + 2uyy − ux + 4uy + u = 0
1.20. uxx − 4uxy + 5uyy − 3ux + uy + u = 0
13
2. &Igrave;&aring;&ograve;&icirc;&auml; &otilde;&agrave;&eth;&agrave;&ecirc;&ograve;&aring;&eth;&egrave;&ntilde;&ograve;&egrave;&ecirc;.
2.1. &Acirc; &igrave;&icirc;&igrave;&aring;&iacute;&ograve; &divide;&agrave;&ntilde;&oacute; t = 0 &iacute;&aring;&icirc;&aacute;&igrave;&aring;&aelig;&aring;&iacute;&agrave; &ntilde;&ograve;&eth;&oacute;&iacute;&agrave; &aacute;&oacute;&euml;&agrave; &ccedil;&aacute;&oacute;&auml;&aelig;&aring;&iacute;&agrave; &acirc;i&auml;&otilde;&egrave;&euml;&aring;&iacute;&iacute;&yuml;&igrave;,
&ccedil;&icirc;&aacute;&eth;&agrave;&aelig;&aring;&iacute;&egrave;&igrave; &iacute;&agrave; &eth;&egrave;&ntilde;&oacute;&iacute;&ecirc;&oacute;. &Iacute;&agrave;&igrave;&agrave;&euml;&thorn;&acirc;&agrave;&ograve;&egrave; &iuml;&eth;&icirc;&ocirc;i&euml;i &ntilde;&ograve;&eth;&oacute;&iacute;&egrave; &auml;&euml;&yuml; &igrave;&icirc;&igrave;&aring;&iacute;&ograve;i&acirc;
kc
&divide;&agrave;&ntilde;&oacute; tk = 4a
, k = 0, 1, 2, 3, 5.
2.2. &Iuml;&icirc; &iacute;&aring;&icirc;&aacute;&igrave;&aring;&aelig;&aring;&iacute;i&eacute; &ntilde;&ograve;&eth;&oacute;&iacute;i &aacute;i&aelig;&egrave;&ograve;&uuml; &otilde;&acirc;&egrave;&euml;&yuml; ϕ(x − at), &auml;&aring; ϕ &iacute;&aring;&ccedil;&agrave;&euml;&aring;&aelig;&iacute;&agrave;
&ocirc;&oacute;&iacute;&ecirc;&ouml;i&yuml;. &Iuml;&eth;&egrave;&eacute;&iacute;&yuml;&acirc;&oslash;&egrave; &ouml;&thorn; &otilde;&acirc;&egrave;&euml;&thorn; &ccedil;&agrave; &iuml;&icirc;&divide;&agrave;&ograve;&ecirc;&icirc;&acirc;&aring; &ccedil;&aacute;&oacute;&eth;&aring;&iacute;&iacute;&yuml; &ntilde;&ograve;&eth;&oacute;&iacute;&egrave; &iuml;&eth;&egrave;
t = 0, &ccedil;&iacute;&agrave;&eacute;&ograve;&egrave; &ntilde;&ograve;&agrave;&iacute; &ntilde;&ograve;&eth;&oacute;&iacute;&egrave; &iuml;&eth;&egrave; t &gt; 0.
2.3. &Aacute;&aring;&ccedil;&igrave;&aring;&aelig;&iacute;&agrave; &ntilde;&ograve;&eth;&oacute;&iacute;&agrave; &acirc; &iuml;&icirc;&divide;&agrave;&ograve;&ecirc;&icirc;&acirc;&egrave;&eacute; &igrave;&icirc;&igrave;&aring;&iacute;&ograve; &iacute;&agrave; &auml;i&euml;&yuml;&iacute;&ouml;i x ∈ [−c, c] &igrave;&agrave;&sup1;
&ocirc;&icirc;&eth;&igrave;&oacute; &ntilde;&egrave;&igrave;&aring;&ograve;&eth;&egrave;&divide;&iacute;&icirc;&uml; &iuml;&agrave;&eth;&agrave;&aacute;&icirc;&euml;&egrave;. (a) &Ccedil;&iacute;&agrave;&eacute;&ograve;&egrave; &iuml;&eth;&icirc;&ocirc;i&euml;&uuml; &ntilde;&ograve;&eth;&oacute;&iacute;&egrave; (&agrave;&iacute;&agrave;&euml;i&ograve;&egrave;&divide;&iacute;&egrave;&eacute; &acirc;&egrave;&eth;&agrave;&ccedil; &auml;&euml;&yuml; &ccedil;&igrave;i&ugrave;&aring;&iacute;&iacute;&yuml; u(x, t)) &oacute; &aacute;&oacute;&auml;&uuml;-&yuml;&ecirc;&egrave;&eacute; &igrave;&icirc;&igrave;&aring;&iacute;&ograve; t &gt; 0. (&aacute;)
&Ccedil;&iacute;&agrave;&eacute;&ograve;&egrave; &ccedil;&igrave;i&ugrave;&aring;&iacute;&iacute;&yuml; &ograve;&icirc;&divide;&icirc;&ecirc; &ntilde;&ograve;&eth;&oacute;&iacute;&egrave; &ccedil; &eth;i&ccedil;&iacute;&egrave;&igrave;&egrave; &ccedil;&iacute;&agrave;&divide;&aring;&iacute;&iacute;&yuml;&igrave;&egrave; &ecirc;&icirc;&icirc;&eth;&auml;&egrave;&iacute;&agrave;&ograve;&egrave; x.
2.4. &Acirc; &iuml;&icirc;&divide;&agrave;&ograve;&ecirc;&icirc;&acirc;&egrave;&eacute; &igrave;&icirc;&igrave;&aring;&iacute;&ograve; &divide;&agrave;&ntilde;&oacute; &divide;&agrave;&ntilde;&ograve;&egrave;&iacute;&agrave; x ∈ [−c, c] &iacute;&aring;&icirc;&aacute;&igrave;&aring;&aelig;&aring;&iacute;&icirc;&uml; &ntilde;&ograve;&eth;&oacute;&iacute;&egrave;
&igrave;&agrave;&euml;&agrave; &iuml;&icirc;&iuml;&aring;&eth;&aring;&divide;&iacute;&oacute; &oslash;&acirc;&egrave;&auml;&ecirc;i&ntilde;&ograve;&uuml; v0 , &acirc;&ntilde;i i&iacute;&oslash;i &divide;&agrave;&ntilde;&ograve;&egrave;&iacute;&egrave; &ntilde;&ograve;&eth;&oacute;&iacute;&egrave; &igrave;&agrave;&euml;&egrave; &iacute;&oacute;&euml;&uuml;&icirc;&acirc;&oacute;
&oslash;&acirc;&egrave;&auml;&ecirc;i&ntilde;&ograve;&uuml;. (&agrave;) &Ccedil;&iacute;&agrave;&eacute;&ograve;&egrave; &agrave;&iacute;&agrave;&euml;i&ograve;&egrave;&divide;&iacute;i &ocirc;&icirc;&eth;&igrave;&oacute;&euml;&egrave; &auml;&euml;&yuml; &ccedil;&igrave;i&ugrave;&aring;&iacute;&iacute;&yuml; &ntilde;&ograve;&eth;&oacute;&iacute;&egrave; &ccedil;
&eth;i&ccedil;&iacute;&egrave;&igrave;&egrave; &ccedil;&iacute;&agrave;&divide;&aring;&iacute;&iacute;&yuml;&igrave;&egrave; &ecirc;&icirc;&icirc;&eth;&auml;&egrave;&iacute;&agrave;&ograve;&egrave; &iuml;&eth;&egrave; t &gt; 0. (&aacute;) &Iacute;&agrave;&igrave;&agrave;&euml;&thorn;&acirc;&agrave;&ograve;&egrave; &iuml;&eth;&icirc;&ocirc;i&euml;i
kc
&ntilde;&ograve;&eth;&oacute;&iacute;&egrave; &auml;&euml;&yuml; &igrave;&icirc;&igrave;&aring;&iacute;&ograve;i&acirc; &divide;&agrave;&ntilde;&oacute; tk = 4a
, k = 0, 2, 4, 6.
2.5. &Acirc; &iuml;&icirc;&divide;&agrave;&ograve;&ecirc;&icirc;&acirc;&egrave;&eacute; &igrave;&icirc;&igrave;&aring;&iacute;&ograve; &divide;&agrave;&ntilde;&oacute; &acirc; &ograve;&icirc;&divide;&ouml;i x = x0 &iuml;&icirc; &iacute;&aring;&icirc;&aacute;&igrave;&aring;&aelig;&aring;&iacute;i&eacute; &ntilde;&ograve;&eth;&oacute;&iacute;i
&acirc;&auml;&agrave;&eth;&egrave;&euml;&egrave; &acirc;&oacute;&ccedil;&uuml;&ecirc;&egrave;&igrave; &igrave;&icirc;&euml;&icirc;&ograve;&icirc;&divide;&ecirc;&icirc;&igrave;, &iacute;&agrave;&auml;&agrave;&acirc;&oslash;&egrave; &ntilde;&ograve;&eth;&oacute;&iacute;i i&igrave;&iuml;&oacute;&euml;&uuml;&ntilde; I . &Ccedil;&iacute;&agrave;&eacute;&ograve;&egrave;
&acirc;i&auml;&otilde;&egrave;&euml;&aring;&iacute;&iacute;&yuml; u(x, t) &ograve;&icirc;&divide;&icirc;&ecirc; &ntilde;&ograve;&eth;&oacute;&iacute;&egrave; &acirc;i&auml; &iuml;&icirc;&euml;&icirc;&aelig;&aring;&iacute;&iacute;&yuml; &eth;i&acirc;&iacute;&icirc;&acirc;&agrave;&atilde;&egrave; &iuml;&eth;&egrave; t &gt; 0,
&yuml;&ecirc;&ugrave;&icirc; &iuml;&icirc;&divide;&agrave;&ograve;&ecirc;&icirc;&acirc;i &ccedil;&igrave;i&ugrave;&aring;&iacute;&iacute;&yuml; &ograve;&icirc;&divide;&icirc;&ecirc; &ntilde;&ograve;&eth;&oacute;&iacute;&egrave; &auml;&icirc;&eth;i&acirc;&iacute;&thorn;&thorn;&ograve;&uuml; &iacute;&oacute;&euml;&thorn;.
14
2.6. &Iacute;&agrave;&iuml;i&acirc;&icirc;&aacute;&igrave;&aring;&aelig;&aring;&iacute;&agrave; &ntilde;&ograve;&eth;&oacute;&iacute;&agrave; &acirc; &iuml;&icirc;&divide;&agrave;&ograve;&ecirc;&icirc;&acirc;&egrave;&eacute; &igrave;&icirc;&igrave;&aring;&iacute;&ograve; &divide;&agrave;&ntilde;&oacute; &igrave;&agrave;&sup1; &ocirc;&icirc;&eth;&igrave;&oacute;, &ugrave;&icirc;
kc
&ccedil;&icirc;&aacute;&eth;&agrave;&aelig;&aring;&iacute;&agrave; &iacute;&agrave; &eth;&egrave;&ntilde;&oacute;&iacute;&ecirc;&oacute;. &Iacute;&agrave;&igrave;&agrave;&euml;&thorn;&acirc;&agrave;&ograve;&egrave; &iuml;&eth;&icirc;&ocirc;i&euml;&uuml; &ntilde;&ograve;&eth;&oacute;&iacute;&egrave; &iuml;&eth;&egrave; tk = 2a
,
k = 0, 2, 3, 4, 7, &yuml;&ecirc;&ugrave;&icirc; (&agrave;) &ntilde;&ograve;&eth;&oacute;&iacute;&agrave; &ccedil;&agrave;&ecirc;&eth;i&iuml;&euml;&aring;&iacute;&agrave; &iacute;&agrave; &ecirc;i&iacute;&ouml;i; (&aacute;) &ntilde;&ograve;&eth;&oacute;&iacute;&agrave; &igrave;&agrave;&sup1;
&acirc;i&euml;&uuml;&iacute;&egrave;&eacute; &ecirc;i&iacute;&aring;&ouml;&uuml;.
2.7. &Ccedil;&iacute;&agrave;&eacute;&ograve;&egrave; &acirc;i&auml;&otilde;&egrave;&euml;&aring;&iacute;&iacute;&yuml; &acirc;i&auml; &iuml;&icirc;&euml;&icirc;&aelig;&aring;&iacute;&iacute;&yuml; &eth;i&acirc;&iacute;&icirc;&acirc;&agrave;&atilde;&egrave; u(x, t) &ntilde;&ograve;&eth;&oacute;&iacute;&egrave; &auml;&icirc;&acirc;&aelig;&egrave;-
&iacute;&icirc;&thorn; l, &yuml;&ecirc;&ugrave;&icirc; &ecirc;i&iacute;&ouml;i x = 0, x = l &ccedil;&agrave;&ecirc;&eth;i&iuml;&euml;&aring;&iacute;i, &iuml;&icirc;&divide;&agrave;&ograve;&ecirc;&icirc;&acirc;&agrave; &oslash;&acirc;&egrave;&auml;&ecirc;i&ntilde;&ograve;&uuml;
&auml;&icirc;&eth;i&acirc;&iacute;&thorn;&sup1; &iacute;&oacute;&euml;&thorn;, &agrave; &iuml;&icirc;&divide;&agrave;&ograve;&ecirc;&icirc;&acirc;&aring; &ccedil;&igrave;i&ugrave;&aring;&iacute;&iacute;&yuml; &ccedil;&agrave;&auml;&agrave;&iacute;&aring;: u(x, 0) = A sin(πx/l)
&iuml;&eth;&egrave; x ∈ [0, l].
2.8. &ETH;&icirc;&ccedil;&acirc;'&yuml;&ccedil;&agrave;&ograve;&egrave; &ccedil;&agrave;&auml;&agrave;&divide;&oacute; 2.5 &auml;&euml;&yuml; &iacute;&agrave;&iuml;i&acirc;&icirc;&aacute;&igrave;&aring;&aelig;&aring;&iacute;&icirc;&uml; &ntilde;&ograve;&eth;&oacute;&iacute;&egrave; &ccedil; &ccedil;&agrave;&ecirc;&eth;i&iuml;&euml;&aring;&iacute;&egrave;&igrave; &ecirc;i&iacute;&ouml;&aring;&igrave;.
15
3. &Igrave;&aring;&ograve;&icirc;&auml; &eth;&icirc;&ccedil;&auml;i&euml;&aring;&iacute;&iacute;&yuml; &ccedil;&igrave;i&iacute;&iacute;&egrave;&otilde;
&Igrave;&aring;&ograve;&icirc;&auml; &eth;&icirc;&ccedil;&auml;i&euml;&aring;&iacute;&iacute;&yuml; &ccedil;&igrave;i&iacute;&iacute;&egrave;&otilde;, &agrave;&aacute;&icirc; &igrave;&aring;&ograve;&icirc;&auml; &Ocirc;&oacute;&eth;'&sup1;, &sup1; &iuml;&icirc;&ograve;&oacute;&aelig;&iacute;&egrave;&igrave; i &iacute;&agrave;&eacute;&aacute;i&euml;&uuml;&oslash; &iuml;&icirc;&oslash;&egrave;&eth;&aring;&iacute;&egrave;&igrave; &ccedil;&agrave;&ntilde;&icirc;&aacute;&icirc;&igrave; &eth;&icirc;&ccedil;&acirc;'&yuml;&ccedil;&oacute;&acirc;&agrave;&iacute;&iacute;&yuml; &ccedil;&agrave;&auml;&agrave;&divide; &igrave;&agrave;&ograve;&aring;&igrave;&agrave;&ograve;&egrave;&divide;&iacute;&icirc;&uml; &ocirc;i&ccedil;&egrave;&ecirc;&egrave;. &Icirc;&iuml;&agrave;&iacute;&oacute;&acirc;&agrave;&iacute;&iacute;&yuml; &auml;&agrave;&iacute;&icirc;&atilde;&icirc; &igrave;&aring;&ograve;&icirc;&auml;&oacute; &sup1; &iuml;&aring;&eth;&aring;&auml;&oacute;&igrave;&icirc;&acirc;&icirc;&thorn; &ccedil;&agrave;&ntilde;&acirc;&icirc;&sup1;&iacute;&iacute;&yuml; &eth;i&ccedil;&iacute;&icirc;&igrave;&agrave;&iacute;i&ograve;&iacute;&egrave;&otilde; &iuml;&egrave;&ograve;&agrave;&iacute;&uuml; &ograve;&aring;&icirc;&eth;&aring;&ograve;&egrave;&divide;&iacute;&icirc;&uml; i &igrave;&agrave;&ograve;&aring;&igrave;&agrave;&ograve;&egrave;&divide;&iacute;&icirc;&uml; &ocirc;i&ccedil;&egrave;&ecirc;&egrave;. &Ccedil;&icirc;&ecirc;&eth;&aring;&igrave;&agrave;, &ccedil;&agrave;&auml;&agrave;&divide;&agrave; &iacute;&agrave; &acirc;&euml;&agrave;&ntilde;&iacute;i &ccedil;&iacute;&agrave;&divide;&aring;&iacute;&iacute;&yuml;, &agrave;&aacute;&icirc;
&ccedil;&agrave;&auml;&agrave;&divide;&agrave; &Oslash;&ograve;&oacute;&eth;&igrave;&agrave;-&Euml;i&oacute;&acirc;i&euml;&euml;&yuml;, &ugrave;&icirc; &sup1; &icirc;&aacute;&icirc;&acirc;'&yuml;&ccedil;&ecirc;&icirc;&acirc;&icirc;&thorn; &ntilde;&ecirc;&euml;&agrave;&auml;&icirc;&acirc;&icirc;&thorn; &igrave;&aring;&ograve;&icirc;&auml;&oacute; &eth;&icirc;&ccedil;&auml;i&euml;&aring;&iacute;&iacute;&yuml; &ccedil;&igrave;i&iacute;&iacute;&egrave;&otilde;, &igrave;i&ntilde;&ograve;&egrave;&ograve;&uuml; &acirc; &ntilde;&icirc;&aacute;i &ecirc;&euml;&thorn;&divide; &auml;&icirc; &eth;&icirc;&ccedil;&oacute;&igrave;i&iacute;&iacute;&yuml; &ecirc;&acirc;&agrave;&iacute;&ograve;&oacute;&acirc;&agrave;&iacute;&iacute;&yuml; (&iacute;&agrave;&aacute;&oacute;&ograve;&ograve;&yuml;
&auml;&egrave;&ntilde;&ecirc;&eth;&aring;&ograve;&iacute;&egrave;&otilde; &igrave;&icirc;&aelig;&euml;&egrave;&acirc;&egrave;&otilde; &ccedil;&iacute;&agrave;&divide;&aring;&iacute;&uuml;) &ocirc;i&ccedil;&egrave;&divide;&iacute;&egrave;&otilde; &acirc;&aring;&euml;&egrave;&divide;&egrave;&iacute; &acirc; &ecirc;&acirc;&agrave;&iacute;&ograve;&icirc;&acirc;i&eacute; &igrave;&aring;&otilde;&agrave;&iacute;&egrave;&ouml;i.
&Ograve;&egrave;&iuml;&icirc;&acirc;&egrave;&igrave;&egrave; &ccedil;&agrave;&auml;&agrave;&divide;&agrave;&igrave;&egrave;, &auml;&icirc; &eth;&icirc;&ccedil;&acirc;'&yuml;&ccedil;&oacute;&acirc;&agrave;&iacute;&iacute;&yuml; &yuml;&ecirc;&egrave;&otilde; &ccedil;&agrave;&ntilde;&ograve;&icirc;&ntilde;&icirc;&acirc;&oacute;&sup1;&ograve;&uuml;&ntilde;&yuml; &igrave;&aring;&ograve;&icirc;&auml; &eth;&icirc;&ccedil;&auml;i&euml;&aring;&iacute;&iacute;&yuml; &ccedil;&igrave;i&iacute;&iacute;&egrave;&otilde;, &sup1; &ecirc;&eth;&agrave;&eacute;&icirc;&acirc;i &ccedil;&agrave;&auml;&agrave;&divide;i &acirc; &icirc;&aacute;&igrave;&aring;&aelig;&aring;&iacute;&egrave;&otilde; &icirc;&aacute;&euml;&agrave;&ntilde;&ograve;&yuml;&otilde; &ccedil; &eth;i&acirc;&iacute;&yuml;&iacute;&iacute;&yuml;&igrave;&egrave;
&atilde;i&iuml;&aring;&eth;&aacute;&icirc;&euml;i&divide;&iacute;&icirc;&atilde;&icirc;, &iuml;&agrave;&eth;&agrave;&aacute;&icirc;&euml;i&divide;&iacute;&icirc;&atilde;&icirc; i &aring;&euml;i&iuml;&ograve;&egrave;&divide;&iacute;&icirc;&atilde;&icirc; &ograve;&egrave;&iuml;i&acirc;. &Aring;&ocirc;&aring;&ecirc;&ograve;&egrave;&acirc;&iacute;i&ntilde;&ograve;&uuml; &igrave;&aring;&ograve;&icirc;&auml;&oacute;
&iuml;&icirc;&euml;&yuml;&atilde;&agrave;&sup1; &oacute; &ntilde;&oacute;&ograve;&ograve;&sup1;&acirc;&icirc;&igrave;&oacute; &ntilde;&iuml;&eth;&icirc;&ugrave;&aring;&iacute;&iacute;i &ccedil;&agrave;&auml;&agrave;&divide;i &oslash;&euml;&yuml;&otilde;&icirc;&igrave; &ccedil;&acirc;&aring;&auml;&aring;&iacute;&iacute;&yuml; &iuml;&eth;&icirc;&aacute;&euml;&aring;&igrave;&egrave; &eth;&icirc;&ccedil;&acirc;'&yuml;&ccedil;&oacute;&acirc;&agrave;&iacute;&iacute;&yuml; &auml;&egrave;&ocirc;&aring;&eth;&aring;&iacute;&ouml;i&agrave;&euml;&uuml;&iacute;&icirc;&atilde;&icirc; &eth;i&acirc;&iacute;&yuml;&iacute;&iacute;&yuml; &ccedil; &divide;&agrave;&ntilde;&ograve;&egrave;&iacute;&iacute;&egrave;&igrave;&egrave; &iuml;&icirc;&otilde;i&auml;&iacute;&egrave;&igrave;&egrave; &auml;&icirc; &eth;&icirc;&ccedil;&acirc;'&yuml;&ccedil;&oacute;&acirc;&agrave;&iacute;&iacute;&yuml; &ccedil;&acirc;&egrave;&divide;&agrave;&eacute;&iacute;&icirc;&atilde;&icirc; &auml;&egrave;&ocirc;&aring;&eth;&aring;&iacute;&ouml;i&agrave;&euml;&uuml;&iacute;&icirc;&atilde;&icirc; &eth;i&acirc;&iacute;&yuml;&iacute;&iacute;&yuml;. &Aacute;&aring;&ccedil;&iuml;&icirc;&ntilde;&aring;&eth;&aring;&auml;&iacute;&uuml;&icirc; &ntilde;&agrave;&igrave; &igrave;&aring;&ograve;&icirc;&auml; &yuml;&acirc;&euml;&yuml;&sup1; &ntilde;&icirc;&aacute;&icirc;&thorn; &iuml;&aring;&acirc;&iacute;&oacute; &iuml;&icirc;&ntilde;&euml;i&auml;&icirc;&acirc;&iacute;i&ntilde;&ograve;&uuml; &auml;i&eacute;, &yuml;&ecirc;i &ccedil;&agrave; &ntilde;&acirc;&icirc;&sup1;&thorn; &ntilde;&oacute;&ograve;&ograve;&thorn; &icirc;&auml;&iacute;&agrave;&ecirc;&icirc;&acirc;i
&auml;&euml;&yuml; &eth;i&ccedil;&iacute;&egrave;&otilde; &ccedil;&agrave;&auml;&agrave;&divide;, &iacute;&aring;&ccedil;&agrave;&euml;&aring;&aelig;&iacute;&icirc; &acirc;i&auml; &uml;&otilde; &ocirc;i&ccedil;&egrave;&divide;&iacute;&icirc;&atilde;&icirc; &ccedil;&igrave;i&ntilde;&ograve;&oacute; i &ntilde;&ecirc;&euml;&agrave;&auml;&iacute;&icirc;&ntilde;&ograve;i. &Icirc;&ograve;&aelig;&aring;,
&ccedil;&igrave;i&ntilde;&ograve; &igrave;&aring;&ograve;&icirc;&auml;&oacute; &eth;&icirc;&ccedil;&auml;i&euml;&aring;&iacute;&iacute;&yuml; &ccedil;&igrave;i&iacute;&iacute;&egrave;&otilde; &igrave;&icirc;&aelig;&iacute;&agrave; &iuml;&icirc;&yuml;&ntilde;&iacute;&egrave;&ograve;&egrave; &iacute;&agrave; &iuml;&eth;&egrave;&ecirc;&euml;&agrave;&auml;i &auml;&icirc;&ntilde;&egrave;&ograve;&uuml;
&iuml;&eth;&icirc;&ntilde;&ograve;&icirc;&uml; &ocirc;i&ccedil;&egrave;&divide;&iacute;&icirc;&uml; &ccedil;&agrave;&auml;&agrave;&divide;i.
&Ccedil;&agrave;&auml;&agrave;&divide;&agrave; &iuml;&eth;&icirc; &acirc;i&euml;&uuml;&iacute;i &ecirc;&icirc;&euml;&egrave;&acirc;&agrave;&iacute;&iacute;&yuml; &icirc;&aacute;&igrave;&aring;&aelig;&aring;&iacute;&icirc;&uml; &ntilde;&ograve;&eth;&oacute;&iacute;&egrave;.
&Icirc;&aacute;&atilde;&icirc;&acirc;&icirc;&eth;&egrave;&igrave;&icirc; &ocirc;i&ccedil;&egrave;&divide;&iacute;&egrave;&eacute; &ccedil;&igrave;i&ntilde;&ograve; &ccedil;&agrave;&auml;&agrave;&divide;i, &ograve;&agrave; &ntilde;&ocirc;&icirc;&eth;&igrave;&oacute;&euml;&thorn;&sup1;&igrave;&icirc; &uml;&uml; &iuml;&icirc;&acirc;&iacute;&oacute; &igrave;&agrave;&ograve;&aring;&igrave;&agrave;&ograve;&egrave;&divide;&iacute;&oacute; &iuml;&icirc;&ntilde;&ograve;&agrave;&iacute;&icirc;&acirc;&ecirc;&oacute;. &Iacute;&agrave;&ograve;&yuml;&atilde;&iacute;&oacute;&ograve;&agrave; &ntilde;&ograve;&eth;&oacute;&iacute;&agrave; &auml;&icirc;&acirc;&aelig;&egrave;&iacute;&icirc;&thorn; l, &ugrave;&icirc; &ccedil;&agrave;&ecirc;&eth;i&iuml;&euml;&aring;&iacute;&agrave; &aelig;&icirc;&eth;&ntilde;&ograve;&ecirc;&icirc;
&iacute;&agrave; &ecirc;i&iacute;&ouml;&yuml;&otilde;, &igrave;&icirc;&aelig;&aring; &ecirc;&icirc;&euml;&egrave;&acirc;&agrave;&ograve;&egrave;&ntilde;&uuml; &acirc; &iuml;&icirc;&iuml;&aring;&eth;&aring;&divide;&iacute;&icirc;&igrave;&oacute; &acirc;i&auml;&iacute;&icirc;&ntilde;&iacute;&icirc; &ntilde;&ograve;&eth;&oacute;&iacute;&egrave; &iacute;&agrave;&iuml;&eth;&yuml;&igrave;&ecirc;&oacute;.
&Iacute;&aring;&otilde;&agrave;&eacute; &acirc; &iuml;&icirc;&divide;&agrave;&ograve;&ecirc;&icirc;&acirc;&egrave;&eacute; &igrave;&icirc;&igrave;&aring;&iacute;&ograve; t0 = 0 &ecirc;&icirc;&aelig;&iacute;&agrave; &ograve;&icirc;&divide;&ecirc;&agrave; &ntilde;&ograve;&eth;&oacute;&iacute;&egrave; &ccedil;&igrave;i&ugrave;&aring;&iacute;&agrave; &acirc; &iuml;&icirc;&iuml;&aring;&eth;&aring;&divide;&iacute;&icirc;&igrave;&oacute; &iacute;&agrave;&iuml;&eth;&yuml;&igrave;&ecirc;&oacute; &iacute;&agrave; &acirc;&aring;&euml;&egrave;&divide;&egrave;&iacute;&oacute; u(x, 0) = ϕ(x) i &eth;&oacute;&otilde;&agrave;&sup1;&ograve;&uuml;&ntilde;&yuml; i&ccedil; &oslash;&acirc;&egrave;&auml;&ecirc;i&ntilde;&ograve;&thorn;
ut (x, 0) = ψ(x), &auml;&aring; x &ecirc;&icirc;&icirc;&eth;&auml;&egrave;&iacute;&agrave;&ograve;&agrave; &ograve;&icirc;&divide;&ecirc;&egrave; &acirc; &ntilde;&ograve;&agrave;&iacute;i &ntilde;&iuml;&icirc;&ecirc;&icirc;&thorn; &ntilde;&ograve;&eth;&oacute;&iacute;&egrave; &acirc; &iuml;&icirc;&euml;&icirc;&aelig;&aring;&iacute;&iacute;i &eth;i&acirc;&iacute;&icirc;&acirc;&agrave;&atilde;&egrave;. &Iuml;&icirc;&divide;&agrave;&ograve;&ecirc;&icirc;&acirc;i &acirc;i&auml;&otilde;&egrave;&euml;&aring;&iacute;&iacute;&yuml; &acirc;&ntilde;i&otilde; &ograve;&icirc;&divide;&icirc;&ecirc; &ntilde;&ograve;&eth;&oacute;&iacute;&egrave; i &iuml;&icirc;&divide;&agrave;&ograve;&ecirc;&icirc;&acirc;i
&oslash;&acirc;&egrave;&auml;&ecirc;&icirc;&ntilde;&ograve;i &euml;&aring;&aelig;&agrave;&ograve;&uuml; &acirc; &icirc;&auml;&iacute;i&eacute; &iuml;&euml;&icirc;&ugrave;&egrave;&iacute;i (&iuml;&euml;&icirc;&ugrave;&egrave;&iacute;&agrave; (u, x)). &Icirc;&ograve;&aelig;&aring;, &ccedil;&eth;&icirc;&ccedil;&oacute;&igrave;i&euml;&icirc;
&ugrave;&icirc; &ecirc;&icirc;&euml;&egrave;&acirc;&agrave;&iacute;&iacute;&yuml; &ntilde;&ograve;&eth;&oacute;&iacute;&egrave; &aacute;&oacute;&auml;&aring; &acirc;i&auml;&aacute;&oacute;&acirc;&agrave;&ograve;&egrave;&ntilde;&uuml; &acirc; &ograve;i&eacute; &ntilde;&agrave;&igrave;i&eacute; &iuml;&euml;&icirc;&ugrave;&egrave;&iacute;i. &Acirc;&acirc;&agrave;&aelig;&agrave;&sup1;&igrave;&icirc;,
&ugrave;&icirc; &iacute;&agrave; &ntilde;&ograve;&eth;&oacute;&iacute;&oacute; &acirc; &iuml;&icirc;&iuml;&aring;&eth;&aring;&divide;&iacute;&icirc;&igrave;&oacute; &iacute;&agrave;&iuml;&eth;&yuml;&igrave;&ecirc;&oacute; &iacute;&aring; &auml;i&thorn;&ograve;&uuml; &iacute;i&yuml;&ecirc;i &ccedil;&icirc;&acirc;&iacute;i&oslash;&iacute;i &ntilde;&egrave;&euml;&egrave;. &Iuml;i&auml;
16
&auml;i&sup1;&thorn; &acirc;&iacute;&oacute;&ograve;&eth;i&oslash;&iacute;i&otilde; &ntilde;&egrave;&euml;, &acirc;&iacute;&agrave;&ntilde;&euml;i&auml;&icirc;&ecirc; &iuml;&icirc;&divide;&agrave;&ograve;&ecirc;&icirc;&acirc;&icirc;&atilde;&icirc; &ccedil;&igrave;i&ugrave;&aring;&iacute;&iacute;&yuml; &acirc;i&auml; &iuml;&icirc;&euml;&icirc;&aelig;&aring;&iacute;&iacute;&yuml; &eth;i&acirc;&iacute;&icirc;&acirc;&agrave;&atilde;&egrave; i &acirc;&iacute;&agrave;&ntilde;&euml;i&auml;&icirc;&ecirc; &iacute;&agrave;&yuml;&acirc;&iacute;&icirc;&ntilde;&ograve;i &iuml;&icirc;&divide;&agrave;&ograve;&ecirc;&icirc;&acirc;&icirc;&uml; &oslash;&acirc;&egrave;&auml;&ecirc;&icirc;&ntilde;&ograve;i, &ograve;&icirc;&divide;&ecirc;&egrave; &ntilde;&ograve;&eth;&oacute;&iacute;&egrave; &aacute;&oacute;&auml;&oacute;&ograve;&uuml;
&iuml;&aring;&acirc;&iacute;&egrave;&igrave; &divide;&egrave;&iacute;&icirc;&igrave; &ecirc;&icirc;&euml;&egrave;&acirc;&agrave;&ograve;&egrave;&ntilde;&uuml; &iacute;&agrave;&acirc;&ecirc;&icirc;&euml;&icirc; &iuml;&icirc;&euml;&icirc;&aelig;&aring;&iacute;&iacute;&yuml; &eth;i&acirc;&iacute;&icirc;&acirc;&agrave;&atilde;&egrave;. &Ccedil;&agrave;&auml;&agrave;&divide;&agrave; &iuml;&icirc;&euml;&yuml;&atilde;&agrave;&sup1;
&acirc; &ograve;&icirc;&igrave;&oacute;, &ugrave;&icirc;&aacute; &ccedil;&iacute;&agrave;&eacute;&ograve;&egrave; &ccedil;&igrave;i&ugrave;&aring;&iacute;&iacute;&yuml; u(x, t) &aacute;&oacute;&auml;&uuml;-&yuml;&ecirc;&icirc;&uml; &ograve;&icirc;&divide;&ecirc;&egrave; &ntilde;&ograve;&eth;&oacute;&iacute;&egrave; 0 &lt; x &lt; l &acirc;
&auml;&icirc;&acirc;i&euml;&uuml;&iacute;&egrave;&eacute; &igrave;&icirc;&igrave;&aring;&iacute;&ograve; &divide;&agrave;&ntilde;&oacute; t &gt; 0.
&Igrave;&agrave;&ograve;&aring;&igrave;&agrave;&ograve;&egrave;&divide;&iacute;&aring; &ocirc;&icirc;&eth;&igrave;&oacute;&euml;&thorn;&acirc;&agrave;&iacute;&iacute;&yuml; &auml;&agrave;&iacute;&icirc;&uml; &ccedil;&agrave;&auml;&agrave;&divide;i &ntilde;&ograve;&egrave;&ntilde;&euml;&icirc; &ccedil;&agrave;&iuml;&egrave;&ntilde;&oacute;&sup1;&ograve;&uuml;&ntilde;&yuml; &oacute; &acirc;&egrave;&atilde;&euml;&yuml;&auml;i:
utt = a2 uxx , 0 &lt; x &lt; l, t &gt; 0,
(3.1)
&frac12;
&frac12;
u(0, t) = 0,
u(l, t) = 0;
(3.2)
u(x, 0) = ϕ(x),
ut (x, 0) = ψ(x).
(3.3)
&Oslash;&oacute;&ecirc;&agrave;&iacute;&agrave; &ocirc;&oacute;&iacute;&ecirc;&ouml;i&yuml; u(x, t) &ccedil;&agrave;&euml;&aring;&aelig;&egrave;&ograve;&uuml; &acirc;i&auml; &auml;&acirc;&icirc;&otilde; &ccedil;&igrave;i&iacute;&iacute;&egrave;&otilde;: &ecirc;&icirc;&icirc;&eth;&auml;&egrave;&iacute;&agrave;&ograve;&egrave; x i &divide;&agrave;&ntilde;&oacute; t.
&Auml;&egrave;&ocirc;&aring;&eth;&aring;&iacute;&ouml;i&agrave;&euml;&uuml;&iacute;&aring; &eth;i&acirc;&iacute;&yuml;&iacute;&iacute;&yuml; &auml;&eth;&oacute;&atilde;&icirc;&atilde;&icirc; &iuml;&icirc;&eth;&yuml;&auml;&ecirc;&oacute; &ccedil; &divide;&agrave;&ntilde;&ograve;&egrave;&iacute;&iacute;&egrave;&igrave;&egrave; &iuml;&icirc;&otilde;i&auml;&iacute;&egrave;&igrave;&egrave; (3.1)
&icirc;&iuml;&egrave;&ntilde;&oacute;&sup1; &auml;&egrave;&iacute;&agrave;&igrave;i&ecirc;&oacute; &iuml;&eth;&icirc;&ouml;&aring;&ntilde;&oacute; &ecirc;&icirc;&euml;&egrave;&acirc;&agrave;&iacute;&uuml; &ntilde;&oacute;&ecirc;&oacute;&iuml;&iacute;&icirc;&ntilde;&ograve;i &oacute;&ntilde;i&otilde; &ograve;&icirc;&divide;&icirc;&ecirc; &ntilde;&ograve;&eth;&oacute;&iacute;&egrave;. &Iuml;&eth;&egrave;
&acirc;&egrave;&acirc;&aring;&auml;&aring;&iacute;&iacute;i &auml;&agrave;&iacute;&icirc;&atilde;&icirc; &eth;i&acirc;&iacute;&yuml;&iacute;&iacute;&yuml; &acirc;&eth;&agrave;&otilde;&icirc;&acirc;&agrave;&iacute;&icirc; &auml;&eth;&oacute;&atilde;&egrave;&eacute; &ccedil;&agrave;&ecirc;&icirc;&iacute; &Iacute;&uuml;&thorn;&ograve;&icirc;&iacute;&agrave; i &ccedil;&agrave;&ecirc;&icirc;&iacute; &Atilde;&oacute;&ecirc;&agrave;,
i &ecirc;&eth;i&igrave; &ograve;&icirc;&atilde;&icirc;, &acirc;&acirc;&agrave;&aelig;&agrave;&euml;&icirc;&ntilde;&uuml;, &ugrave;&icirc; &acirc;&iacute;&oacute;&ograve;&eth;i&oslash;&iacute;&sup1; &ograve;&aring;&eth;&ograve;&yuml; &acirc;i&auml;&ntilde;&oacute;&ograve;&iacute;&sup1; i, &yuml;&ecirc; &ccedil;&agrave;&ccedil;&iacute;&agrave;&divide;&aring;&iacute;&icirc; &acirc;&egrave;&ugrave;&aring;,
&acirc;i&auml;&ntilde;&oacute;&ograve;&iacute;i &ccedil;&icirc;&acirc;&iacute;i&oslash;&iacute;i &ntilde;&egrave;&euml;&egrave;. &Iuml;&agrave;&eth;&agrave;&igrave;&aring;&ograve;&eth; a, &ugrave;&icirc; &acirc;&otilde;&icirc;&auml;&egrave;&ograve;&uuml; &acirc; &eth;i&acirc;&iacute;&yuml;&iacute;&iacute;&yuml;, &ccedil;&agrave;&euml;&aring;&aelig;&egrave;&ograve;&uuml; &acirc;i&auml;
&ntilde;&egrave;&euml;&egrave; &iacute;&agrave;&ograve;&yuml;&atilde;&oacute;
&ntilde;&ograve;&eth;&oacute;&iacute;&egrave; T i &euml;i&iacute;i&eacute;&iacute;&icirc;&uml; &atilde;&oacute;&ntilde;&ograve;&egrave;&iacute;&egrave; (&igrave;&agrave;&ntilde;&egrave; &icirc;&auml;&egrave;&iacute;&egrave;&ouml;i &auml;&icirc;&acirc;&aelig;&egrave;&iacute;&egrave;) &ntilde;&ograve;&eth;&oacute;&iacute;&egrave;
p
ρ: a = T /ρ. &Ecirc;&eth;&agrave;&eacute;&icirc;&acirc;i &oacute;&igrave;&icirc;&acirc;&egrave; (3.2) &acirc;i&auml;&icirc;&aacute;&eth;&agrave;&aelig;&agrave;&thorn;&ograve;&uuml; &ocirc;i&ccedil;&egrave;&divide;&iacute;i &oacute;&igrave;&icirc;&acirc;&egrave; &iacute;&agrave; &ecirc;i&iacute;&ouml;&yuml;&otilde;
&ntilde;&ograve;&eth;&oacute;&iacute;&egrave;, &agrave; &ntilde;&agrave;&igrave;&aring;: &ecirc;i&iacute;&ouml;i &ntilde;&ograve;&eth;&oacute;&iacute;&egrave; &aelig;&icirc;&eth;&ntilde;&ograve;&ecirc;&icirc; &ccedil;&agrave;&ecirc;&eth;i&iuml;&euml;&aring;&iacute;i, &icirc;&ograve;&aelig;&aring;, &ccedil;&igrave;i&ugrave;&aring;&iacute;&iacute;&yuml; &ograve;&icirc;&divide;&icirc;&ecirc;,
&ugrave;&icirc; &igrave;&agrave;&thorn;&ograve;&uuml; &ecirc;&icirc;&icirc;&eth;&auml;&egrave;&iacute;&agrave;&ograve;&egrave; x = 0, x = l, &auml;&icirc;&eth;i&acirc;&iacute;&thorn;&thorn;&ograve;&uuml; &iacute;&oacute;&euml;&thorn; &acirc; &oacute;&ntilde;i &igrave;&icirc;&igrave;&aring;&iacute;&ograve;&egrave; &divide;&agrave;&ntilde;&oacute;
t &gt; 0.
&Ccedil;&igrave;i&ntilde;&ograve; &iuml;&icirc;&divide;&agrave;&ograve;&ecirc;&icirc;&acirc;&egrave;&otilde; &oacute;&igrave;&icirc;&acirc; (3.3) &ccedil;'&yuml;&ntilde;&icirc;&acirc;&agrave;&iacute;&icirc; &acirc;&egrave;&ugrave;&aring;. &Iuml;&eth;&egrave; &igrave;&agrave;&ograve;&aring;&igrave;&agrave;&ograve;&egrave;&divide;&iacute;&icirc;&igrave;&oacute; &ocirc;&icirc;&eth;&igrave;&oacute;&euml;&thorn;&acirc;&agrave;&iacute;&iacute;i &ccedil;&agrave;&auml;&agrave;&divide;i &ntilde;&euml;i&auml; &icirc;&aacute;&icirc;&acirc;'&yuml;&ccedil;&ecirc;&icirc;&acirc;&icirc; &acirc;&ecirc;&agrave;&ccedil;&agrave;&ograve;&egrave; &ograve;&agrave;&ecirc;&icirc;&aelig; &igrave;&aring;&aelig;i, &acirc; &yuml;&ecirc;&egrave;&otilde; &iacute;&agrave;&aacute;&oacute;&acirc;&agrave;&thorn;&ograve;&uuml;
&ccedil;&iacute;&agrave;&divide;&aring;&iacute;&uuml; &iacute;&aring;&ccedil;&agrave;&euml;&aring;&aelig;&iacute;i &ccedil;&igrave;i&iacute;&iacute;i x i t.
&Iuml;i&auml; &ecirc;&icirc;&eth;&aring;&ecirc;&ograve;&iacute;&icirc;&thorn; &iuml;&icirc;&ntilde;&ograve;&agrave;&iacute;&icirc;&acirc;&ecirc;&icirc;&thorn; &ccedil;&agrave;&auml;&agrave;&divide;i &eth;&icirc;&ccedil;&oacute;&igrave;i&thorn;&ograve;&uuml; &iacute;&agrave;&ntilde;&ograve;&oacute;&iuml;&iacute;&aring;: &ecirc;i&euml;&uuml;&ecirc;i&ntilde;&ograve;&uuml;
&oacute;&igrave;&icirc;&acirc;, &ugrave;&icirc; &auml;&icirc;&auml;&agrave;&thorn;&ograve;&uuml;&ntilde;&yuml; &auml;&icirc; &eth;i&acirc;&iacute;&yuml;&iacute;&iacute;&yuml; (3.1), &iuml;&icirc;&acirc;&egrave;&iacute;&agrave; &aacute;&oacute;&ograve;&egrave; &ograve;&agrave;&ecirc;&icirc;&thorn;, &ugrave;&icirc;&aacute;, &ccedil; &icirc;&auml;&iacute;&icirc;&atilde;&icirc; &aacute;&icirc;&ecirc;&oacute;, &iacute;&aring; &aacute;&oacute;&euml;&icirc; &ccedil;&agrave;&eacute;&acirc;&egrave;&otilde;, i&iacute;&agrave;&ecirc;&oslash;&aring; &eth;&icirc;&ccedil;&acirc;'&yuml;&ccedil;&ecirc;&oacute; &igrave;&icirc;&aelig;&aring; &iacute;&aring; i&ntilde;&iacute;&oacute;&acirc;&agrave;&ograve;&egrave; &acirc;&ccedil;&agrave;&atilde;&agrave;&euml;i, &agrave;
&ccedil; i&iacute;&oslash;&icirc;&atilde;&icirc; &aacute;&icirc;&ecirc;&oacute;, &ouml;&yuml; &ecirc;i&euml;&uuml;&ecirc;i&ntilde;&ograve;&uuml; &iuml;&icirc;&acirc;&egrave;&iacute;&agrave; &aacute;&oacute;&ograve;&egrave; &auml;&icirc;&ntilde;&ograve;&agrave;&ograve;&iacute;&uuml;&icirc;&thorn;, &ugrave;&icirc;&aacute; &eth;&icirc;&ccedil;&acirc;'&yuml;&ccedil;&icirc;&ecirc; &aacute;&oacute;&acirc;
&icirc;&auml;&iacute;&icirc;&ccedil;&iacute;&agrave;&divide;&iacute;&egrave;&igrave;. &Icirc;&ograve;&aelig;&aring;, &iuml;&eth;&egrave; &ecirc;&icirc;&eth;&aring;&ecirc;&ograve;&iacute;i&eacute; &iuml;&icirc;&ntilde;&ograve;&agrave;&iacute;&icirc;&acirc;&ouml;i &eth;&icirc;&ccedil;&acirc;'&yuml;&ccedil;&icirc;&ecirc; &ccedil;&agrave;&auml;&agrave;&divide;i i&ntilde;&iacute;&oacute;&sup1; i &acirc;i&iacute;
&euml;&egrave;&oslash;&aring; &icirc;&auml;&egrave;&iacute;. &Iuml;&eth;&egrave; &igrave;&agrave;&ograve;&aring;&igrave;&agrave;&ograve;&egrave;&divide;&iacute;&icirc;&igrave;&oacute; &ocirc;&icirc;&eth;&igrave;&oacute;&euml;&thorn;&acirc;&agrave;&iacute;&iacute;i &ocirc;i&ccedil;&egrave;&divide;&iacute;&icirc;&uml; &ccedil;&agrave;&auml;&agrave;&divide;i &auml;&icirc;&auml;&agrave;&ograve;&ecirc;&icirc;&acirc;i &oacute;&igrave;&icirc;&acirc;&egrave; &acirc;i&auml;&icirc;&aacute;&eth;&agrave;&aelig;&agrave;&thorn;&ograve;&uuml; &eth;&aring;&agrave;&euml;&uuml;&iacute;i &ocirc;i&ccedil;&egrave;&divide;&iacute;i &oacute;&igrave;&icirc;&acirc;&egrave; i i&iacute;&ograve;&oacute;&uml;&ouml;i&yuml; &ocirc;i&ccedil;&egrave;&ecirc;&agrave; &auml;&icirc;&iuml;&icirc;&igrave;&agrave;&atilde;&agrave;&sup1;
&iuml;&icirc;&ntilde;&ograve;&agrave;&acirc;&egrave;&ograve;&egrave; &ccedil;&agrave;&auml;&agrave;&divide;&oacute; &ecirc;&icirc;&eth;&aring;&ecirc;&ograve;&iacute;&icirc;.
&Acirc; &igrave;&agrave;&ograve;&aring;&igrave;&agrave;&ograve;&egrave;&divide;&iacute;i&eacute; &ocirc;i&ccedil;&egrave;&ouml;i &eth;&icirc;&ccedil;&acirc;'&yuml;&ccedil;&oacute;&sup1;&ograve;&uuml;&ntilde;&yuml; &iacute;&aring; &eth;i&acirc;&iacute;&yuml;&iacute;&iacute;&yuml;, &agrave; &ccedil;&agrave;&auml;&agrave;&divide;&agrave; &acirc; &ouml;i&euml;&icirc;&igrave;&oacute;.
&Icirc;&ograve;&aelig;&aring;, &eth;&icirc;&ccedil;&acirc;'&yuml;&ccedil;&oacute;&acirc;&agrave;&ograve;&egrave; &ccedil;&agrave;&auml;&agrave;&divide;&oacute; &ntilde;&euml;i&auml; &iuml;&icirc;&divide;&egrave;&iacute;&agrave;&ograve;&egrave; &euml;&egrave;&oslash;&aring; &iuml;i&ntilde;&euml;&yuml; &uml;&uml; &iuml;&icirc;&acirc;&iacute;&icirc;&atilde;&icirc; &ecirc;&icirc;&eth;&aring;&ecirc;&ograve;&iacute;&icirc;&atilde;&icirc; &igrave;&agrave;&ograve;&aring;&igrave;&agrave;&ograve;&egrave;&divide;&iacute;&icirc;&atilde;&icirc; &ocirc;&icirc;&eth;&igrave;&oacute;&euml;&thorn;&acirc;&agrave;&iacute;&iacute;&yuml;. &Icirc;&ntilde;&ecirc;i&euml;&uuml;&ecirc;&egrave; &ccedil;&agrave;&auml;&agrave;&divide;&oacute; (3.1)-(3.3) &ntilde;&ocirc;&icirc;&eth;17
&igrave;&oacute;&euml;&uuml;&icirc;&acirc;&agrave;&iacute;&icirc; &iuml;&icirc;&acirc;&iacute;i&ntilde;&ograve;&thorn; i &ecirc;&icirc;&eth;&aring;&ecirc;&ograve;&iacute;&icirc;, &ograve;&icirc; &iacute;&agrave;&ntilde;&ograve;&oacute;&iuml;&iacute;&egrave;&igrave; &aring;&ograve;&agrave;&iuml;&icirc;&igrave; &sup1; &eth;&icirc;&ccedil;&acirc;'&yuml;&ccedil;&agrave;&iacute;&iacute;&yuml; &ccedil;&agrave;&auml;&agrave;&divide;i, i &auml;&euml;&yuml; &ouml;&uuml;&icirc;&atilde;&icirc; &ccedil;&agrave;&ntilde;&ograve;&icirc;&ntilde;&oacute;&sup1;&igrave;&icirc; &ntilde;&agrave;&igrave;&aring; &igrave;&aring;&ograve;&icirc;&auml; &eth;&icirc;&ccedil;&auml;i&euml;&aring;&iacute;&iacute;&yuml; &ccedil;&igrave;i&iacute;&iacute;&egrave;&otilde;.
&Iuml;&eth;&icirc;&ouml;&aring;&ntilde; &eth;&icirc;&ccedil;&acirc;'&yuml;&ccedil;&oacute;&acirc;&agrave;&iacute;&iacute;&yuml; &eth;&icirc;&ccedil;i&aacute;'&sup1;&igrave;&icirc; &iacute;&agrave; &auml;&aring;&ecirc;i&euml;&uuml;&ecirc;&agrave; &iuml;&icirc;&ntilde;&euml;i&auml;&icirc;&acirc;&iacute;&egrave;&otilde; &ecirc;&eth;&icirc;&ecirc;i&acirc;. &Ntilde;&iuml;&icirc;&divide;&agrave;&ograve;&ecirc;&oacute; &iuml;&icirc;&ntilde;&ograve;&agrave;&acirc;&egrave;&igrave;&icirc; &ccedil;&agrave;&acirc;&auml;&agrave;&iacute;&iacute;&yuml; &iacute;&aring; &acirc; &iuml;&icirc;&acirc;&iacute;&icirc;&igrave;&oacute; &icirc;&aacute;&ntilde;&yuml;&ccedil;i &ccedil;&agrave;&auml;&agrave;&divide;i, &agrave; &ntilde;&ocirc;&icirc;&eth;&igrave;&oacute;&euml;&thorn;&sup1;&igrave;&icirc;
&ograve;&agrave;&ecirc; &ccedil;&acirc;&agrave;&iacute;&oacute; &icirc;&ntilde;&iacute;&icirc;&acirc;&iacute;&oacute; &auml;&icirc;&iuml;&icirc;&igrave;i&aelig;&iacute;&oacute; &ccedil;&agrave;&auml;&agrave;&divide;&oacute; :
&Ccedil;&iacute;&agrave;&eacute;&ograve;&egrave; &iacute;&aring;&ograve;&eth;&egrave;&acirc;i&agrave;&euml;&uuml;&iacute;i &eth;&icirc;&ccedil;&acirc;'&yuml;&ccedil;&ecirc;&egrave; &eth;i&acirc;&iacute;&yuml;&iacute;&iacute;&yuml; (3.1), &ugrave;&icirc; &ccedil;&agrave;&auml;&icirc;&acirc;&icirc;&euml;&uuml;&iacute;&yuml;&thorn;&ograve;&uuml;
&ecirc;&eth;&agrave;&eacute;&icirc;&acirc;&egrave;&igrave; &oacute;&igrave;&icirc;&acirc;&agrave;&igrave; (3.2) i &igrave;&agrave;&thorn;&ograve;&uuml; &acirc;&egrave;&atilde;&euml;&yuml;&auml; &auml;&icirc;&aacute;&oacute;&ograve;&ecirc;&oacute; &auml;&acirc;&icirc;&otilde; &ocirc;&oacute;&iacute;&ecirc;&ouml;i&eacute;:
u(x, t) = X(x) &middot; T (t),
(3.4)
&ecirc;&icirc;&aelig;&iacute;&agrave; &ccedil; &yuml;&ecirc;&egrave;&otilde; (X(x) &ograve;&agrave; T (t)) &ccedil;&agrave;&euml;&aring;&aelig;&egrave;&ograve;&uuml; &euml;&egrave;&oslash;&aring; &acirc;i&auml; &icirc;&auml;&iacute;i&sup1;&uml; &iacute;&aring;&ccedil;&agrave;&euml;&aring;&aelig;&iacute;&icirc;&uml;
&ccedil;&igrave;i&iacute;&iacute;&icirc;&uml; (&euml;&egrave;&oslash;&aring; &acirc;i&auml; x &agrave;&aacute;&icirc; &acirc;i&auml; t).
&Ograve;&oacute;&ograve; &iuml;i&auml; &acirc;&egrave;&eth;&agrave;&ccedil;&icirc;&igrave; &iacute;&aring;&ograve;&eth;&egrave;&acirc;i&agrave;&euml;&uuml;&iacute;i &eth;&icirc;&ccedil;&acirc;'&yuml;&ccedil;&ecirc;&egrave;, &yuml;&ecirc; &ouml;&aring; &ccedil;&acirc;&egrave;&divide;&agrave;&eacute;&iacute;&icirc; &iuml;&eth;&egrave;&eacute;&iacute;&yuml;&ograve;&icirc;,
&eth;&icirc;&ccedil;&oacute;&igrave;i&thorn;&ograve;&uuml; &eth;&icirc;&ccedil;&acirc;'&yuml;&ccedil;&ecirc;&egrave;, &ugrave;&icirc; &iacute;&aring; &auml;&icirc;&eth;i&acirc;&iacute;&thorn;&thorn;&ograve;&uuml; &iacute;&oacute;&euml;&thorn; &ograve;&icirc;&ograve;&icirc;&aelig;&iacute;&icirc;: u(x, t) ≡
/ 0. &Ccedil;&agrave;&oacute;&acirc;&agrave;&aelig;&egrave;&igrave;&icirc;, &ugrave;&icirc; &acirc; &icirc;&ntilde;&iacute;&icirc;&acirc;&iacute;i&eacute; &auml;&icirc;&iuml;&icirc;&igrave;&icirc;&aelig;&iacute;i&eacute; &ccedil;&agrave;&auml;&agrave;&divide;i &iuml;&icirc;&divide;&agrave;&ograve;&ecirc;&icirc;&acirc;i &oacute;&igrave;&icirc;&acirc;&egrave; &auml;&icirc; &oacute;&acirc;&agrave;&atilde;&egrave; &iacute;&aring;
&aacute;&aring;&eth;&oacute;&ograve;&uuml;&ntilde;&yuml;. &Auml;&agrave;&euml;i &acirc;&egrave;&igrave;&agrave;&atilde;&agrave;&sup1;&igrave;&icirc;, &ugrave;&icirc;&aacute; &ocirc;&oacute;&iacute;&ecirc;&ouml;i&yuml; u(x, t) &ccedil;&agrave;&auml;&icirc;&acirc;&icirc;&euml;&uuml;&iacute;&yuml;&euml;&agrave; &eth;i&acirc;&iacute;&yuml;&iacute;&iacute;&thorn;
(3.1), &auml;&euml;&yuml; &ouml;&uuml;&icirc;&atilde;&icirc; &iuml;i&auml;&ntilde;&ograve;&agrave;&acirc;&egrave;&igrave;&icirc; &acirc; &eth;i&acirc;&iacute;&yuml;&iacute;&iacute;&yuml; (3.1) &ocirc;&oacute;&iacute;&ecirc;&ouml;i&thorn; u(x, t) &oacute; &acirc;&egrave;&atilde;&euml;&yuml;&auml;i
(3.4). &Iuml;&icirc;&otilde;i&auml;&iacute;i &iuml;&icirc; &divide;&agrave;&ntilde;&oacute; t &iuml;&icirc;&ccedil;&iacute;&agrave;&divide;&egrave;&igrave;&icirc; &ecirc;&eth;&agrave;&iuml;&ecirc;&agrave;&igrave;&egrave;, &agrave; &iuml;&icirc;&otilde;i&auml;&iacute;i &iuml;&icirc; &ecirc;&icirc;&icirc;&eth;&auml;&egrave;&iacute;&agrave;&ograve;i x
&oslash;&ograve;&eth;&egrave;&otilde;&agrave;&igrave;&egrave;: &auml;&acirc;i &ecirc;&eth;&agrave;&iuml;&ecirc;&egrave; i &auml;&acirc;&agrave; &oslash;&ograve;&eth;&egrave;&otilde;&egrave; &icirc;&ccedil;&iacute;&agrave;&divide;&agrave;&ograve;&egrave;&igrave;&oacute;&ograve;&uuml; &acirc;i&auml;&iuml;&icirc;&acirc;i&auml;&iacute;i &iuml;&icirc;&otilde;i&auml;&iacute;i
&auml;&eth;&oacute;&atilde;&icirc;&atilde;&icirc; &iuml;&icirc;&eth;&yuml;&auml;&ecirc;&oacute;:
X(x) &middot; T̈ (t) = a2 X 00 (x) &middot; T (t).
&ETH;&icirc;&ccedil;&auml;i&euml;&egrave;&igrave;&icirc; &euml;i&acirc;&oacute; i &iuml;&eth;&agrave;&acirc;&oacute; &divide;&agrave;&ntilde;&ograve;&egrave;&iacute;&egrave; &icirc;&ograve;&eth;&egrave;&igrave;&agrave;&iacute;&icirc;&uml; &eth;i&acirc;&iacute;&icirc;&ntilde;&ograve;i &iacute;&agrave; &acirc;&egrave;&eth;&agrave;&ccedil; a2 X(x)T (t).
&Ograve;&icirc;&auml;i &igrave;&agrave;&sup1;&igrave;&icirc;:
T̈ (t)
X 00 (x)
=
.
a2 T (t)
X(x)
(3.5)
&Ccedil;&agrave;&oacute;&acirc;&agrave;&aelig;&egrave;&igrave;&icirc;, i &iacute;&agrave; &ouml;&uuml;&icirc;&igrave;&oacute; &ccedil;&eth;&icirc;&aacute;&egrave;&igrave;&icirc; &iacute;&agrave;&atilde;&icirc;&euml;&icirc;&ntilde;, &ugrave;&icirc; &eth;i&acirc;&iacute;i&ntilde;&ograve;&uuml; (3.5) &igrave;&agrave;&sup1; &acirc;&egrave;&ecirc;&icirc;&iacute;&oacute;&acirc;&agrave;&ograve;&egrave;&ntilde;&uuml; &iuml;&eth;&egrave; &acirc;&ntilde;i&otilde; &ccedil;&iacute;&agrave;&divide;&aring;&iacute;&iacute;&yuml;&otilde; 0 &lt; x &lt; l, t &gt; 0. &Acirc;&iacute;&agrave;&ntilde;&euml;i&auml;&icirc;&ecirc; &iacute;&agrave;&acirc;&aring;&auml;&aring;&iacute;&egrave;&otilde; &auml;i&eacute;
&ccedil;&igrave;i&iacute;&iacute;i &acirc; &eth;i&acirc;&iacute;&icirc;&ntilde;&ograve;i (3.5) &eth;&icirc;&ccedil;&auml;i&euml;&egrave;&euml;&egrave;&ntilde;&uuml; (&ccedil;&acirc;i&auml;&ntilde;&egrave; &iacute;&agrave;&ccedil;&acirc;&agrave; &igrave;&aring;&ograve;&icirc;&auml;&oacute; &eth;&icirc;&ccedil;&auml;i&euml;&aring;&iacute;&iacute;&yuml; &ccedil;&igrave;i&iacute;&iacute;&egrave;&otilde;): &euml;i&acirc;&agrave; &divide;&agrave;&ntilde;&ograve;&egrave;&iacute;&agrave; &eth;i&acirc;&iacute;&icirc;&ntilde;&ograve;i &ccedil;&agrave;&euml;&aring;&aelig;&egrave;&ograve;&uuml; &ograve;i&euml;&uuml;&ecirc;&egrave; &acirc;i&auml; t, &iuml;&eth;&agrave;&acirc;&agrave; &divide;&agrave;&ntilde;&ograve;&egrave;&iacute;&agrave; &ograve;i&euml;&uuml;&ecirc;&egrave;
&acirc;i&auml; x.
&Iacute;&agrave;&atilde;&agrave;&auml;&agrave;&sup1;&igrave;&icirc;, &ugrave;&icirc; &ccedil;&igrave;i&iacute;&iacute;i x i t &sup1; &iacute;&aring;&ccedil;&agrave;&euml;&aring;&aelig;&iacute;&egrave;&igrave;&egrave; &icirc;&auml;&iacute;&agrave; &acirc;i&auml; &icirc;&auml;&iacute;&icirc;&uml;. &szlig;&ecirc;&ugrave;&icirc;, &iacute;&agrave;&iuml;&eth;&egrave;&ecirc;&euml;&agrave;&auml;, &ccedil;&agrave;&ocirc;i&ecirc;&ntilde;&oacute;&acirc;&agrave;&ograve;&egrave; &auml;&aring;&yuml;&ecirc;&aring; &ccedil;&iacute;&agrave;&divide;&aring;&iacute;&iacute;&yuml; x = x0 , &agrave; t &ccedil;&igrave;i&iacute;&thorn;&acirc;&agrave;&ograve;&egrave;, &ograve;&icirc; &eth;i&acirc;&iacute;i&ntilde;&ograve;&uuml;
(3.5) &iacute;&aring; &iuml;&icirc;&eth;&oacute;&oslash;&oacute;&sup1;&ograve;&uuml;&ntilde;&yuml;:
X 00 (x0 )
T̈ (t)
=
.
a2 T (t)
X(x0 )
&Iuml;&eth;&egrave; &ocirc;i&ecirc;&ntilde;&icirc;&acirc;&agrave;&iacute;&icirc;&igrave;&oacute; &ccedil;&iacute;&agrave;&divide;&aring;&iacute;&iacute;i x = x0 &iuml;&eth;&agrave;&acirc;&agrave; &divide;&agrave;&ntilde;&ograve;&egrave;&iacute;&agrave; &eth;i&acirc;&iacute;&icirc;&ntilde;&ograve;i &sup1; &auml;&aring;&yuml;&ecirc;&icirc;&thorn; &ecirc;&icirc;&iacute;&ntilde;&ograve;&agrave;&iacute;&ograve;&icirc;&thorn;. &Iuml;&icirc;&ccedil;&iacute;&agrave;&divide;&egrave;&igrave;&icirc; &uml;&uml; &divide;&aring;&eth;&aring;&ccedil; (−λ). &Ccedil;&iacute;&agrave;&ecirc; &quot;−&quot; &ograve;&oacute;&ograve; &iacute;&aring; &igrave;&agrave;&sup1; &iuml;&eth;&egrave;&iacute;&ouml;&egrave;&iuml;&icirc;&acirc;&icirc;&atilde;&icirc;
18
&ccedil;&iacute;&agrave;&divide;&aring;&iacute;&iacute;&yuml;, &icirc;&ntilde;&ecirc;i&euml;&uuml;&ecirc;&egrave; λ &igrave;&icirc;&aelig;&aring; &aacute;&oacute;&ograve;&egrave; &yuml;&ecirc; &auml;&icirc;&auml;&agrave;&ograve;&iacute;i&igrave;, &ograve;&agrave;&ecirc; i &acirc;i&auml;'&sup1;&igrave;&iacute;&egrave;&igrave;. &Ograve;&icirc;&auml;i &auml;&euml;&yuml;
&acirc;&ntilde;i&otilde; t &gt; 0 &igrave;&agrave;&sup1;&igrave;&icirc;:
T̈ (t)
= −λ,
a2 T (t)
&agrave; &ccedil; &oacute;&eth;&agrave;&otilde;&oacute;&acirc;&agrave;&iacute;&iacute;&yuml;&igrave; (3.5) &igrave;&agrave;&sup1;&igrave;&icirc; &ograve;&agrave;&ecirc;&icirc;&aelig;:
X 00 (x)
= −λ.
X(x)
&Icirc;&ograve;&aelig;&aring;, &iacute;&agrave;&ntilde;&euml;i&auml;&ecirc;&icirc;&igrave; &eth;&icirc;&ccedil;&auml;i&euml;&aring;&iacute;&iacute;&yuml; &ccedil;&igrave;i&iacute;&iacute;&egrave;&otilde; &sup1; &eth;i&acirc;&iacute;i&ntilde;&ograve;&uuml;:
T̈ (t)
X 00 (x)
=
= −λ,
a2 T (t)
X(x)
&ugrave;&icirc; &aring;&ecirc;&acirc;i&acirc;&agrave;&euml;&aring;&iacute;&ograve;&iacute;&icirc; &ntilde;&oacute;&ecirc;&oacute;&iuml;&iacute;&icirc;&ntilde;&ograve;i &auml;&acirc;&icirc;&otilde; &eth;i&acirc;&iacute;&yuml;&iacute;&uuml;:
T̈ (t) + λa2 T (t) = 0,
(3.6)
X 00 (x) + λX(x) = 0.
(3.7)
&Acirc;i&auml;&ograve;&aring;&iuml;&aring;&eth; &ccedil;&agrave;&igrave;i&ntilde;&ograve;&uuml; &eth;i&acirc;&iacute;&yuml;&iacute;&iacute;&yuml; (3.1) &ccedil; &divide;&agrave;&ntilde;&ograve;&egrave;&iacute;&iacute;&egrave;&igrave;&egrave; &iuml;&icirc;&otilde;i&auml;&iacute;&egrave;&igrave;&egrave; &igrave;&agrave;&sup1;&igrave;&icirc; &ntilde;&iuml;&eth;&agrave;&acirc;&oacute;
i&ccedil; &ccedil;&acirc;&egrave;&divide;&agrave;&eacute;&iacute;&egrave;&igrave;&egrave; &auml;&egrave;&ocirc;&aring;&eth;&aring;&iacute;&ouml;i&agrave;&euml;&uuml;&iacute;&egrave;&igrave;&egrave; &eth;i&acirc;&iacute;&yuml;&iacute;&iacute;&yuml;&igrave;&egrave; (3.6) &ograve;&agrave; (3.7), &iuml;&icirc;&acirc;'&yuml;&ccedil;&agrave;&iacute;&egrave;&igrave;&egrave;
&igrave;i&aelig; &ntilde;&icirc;&aacute;&icirc;&thorn; &euml;&egrave;&oslash;&aring; &ntilde;&iuml;i&euml;&uuml;&iacute;&egrave;&igrave; &iuml;&agrave;&eth;&agrave;&igrave;&aring;&ograve;&eth;&icirc;&igrave; &eth;&icirc;&ccedil;&auml;i&euml;&aring;&iacute;&iacute;&yuml; λ. &Auml;&icirc;&aacute;&oacute;&ograve;&icirc;&ecirc; &auml;&icirc;&acirc;i&euml;&uuml;&iacute;&icirc;&atilde;&icirc; &eth;&icirc;&ccedil;&acirc;'&yuml;&ccedil;&ecirc;&oacute; &eth;i&acirc;&iacute;&yuml;&iacute;&iacute;&yuml; (3.6) &iacute;&agrave; &auml;&icirc;&acirc;i&euml;&uuml;&iacute;&egrave;&eacute; &eth;&icirc;&ccedil;&acirc;'&yuml;&ccedil;&icirc;&ecirc; &eth;i&acirc;&iacute;&yuml;&iacute;&iacute;&yuml; (3.7) &icirc;&aacute;&icirc;&acirc;'&yuml;&ccedil;&ecirc;&icirc;&acirc;&icirc; &aacute;&oacute;&auml;&aring; &eth;&icirc;&ccedil;&acirc;'&yuml;&ccedil;&ecirc;&icirc;&igrave; &eth;i&acirc;&iacute;&yuml;&iacute;&iacute;&yuml; (3.1). &Acirc; &icirc;&ntilde;&iacute;&icirc;&acirc;&iacute;i&eacute; &auml;&icirc;&iuml;&icirc;&igrave;i&aelig;&iacute;i&eacute; &ccedil;&agrave;&auml;&agrave;&divide;i
&acirc;&egrave;&igrave;&agrave;&atilde;&agrave;&sup1;&ograve;&uuml;&ntilde;&yuml; &ograve;&agrave;&ecirc;&icirc;&aelig;, &ugrave;&icirc;&aacute; &ouml;&aring;&eacute; &eth;&icirc;&ccedil;&acirc;'&yuml;&ccedil;&icirc;&ecirc; &ccedil;&agrave;&auml;&icirc;&acirc;&icirc;&euml;&uuml;&iacute;&yuml;&acirc; &ecirc;&eth;&agrave;&eacute;&icirc;&acirc;i &oacute;&igrave;&icirc;&acirc;&egrave; (3.2):
u(0, t) = X(0) &middot; T (t) = 0;
u(l, t) = X(l) &middot; T (t) = 0
(3.8)
&iuml;&eth;&egrave; &acirc;&ntilde;i&otilde; &ccedil;&iacute;&agrave;&divide;&aring;&iacute;&iacute;&yuml;&otilde; t &gt; 0. &szlig;&ecirc;&ugrave;&icirc; &iuml;&icirc;&ecirc;&euml;&agrave;&ntilde;&ograve;&egrave; T (t) ≡ 0, &ograve;&icirc; &ecirc;&eth;&agrave;&eacute;&icirc;&acirc;i &oacute;&igrave;&icirc;&acirc;&egrave;
(3.2) &ccedil;&agrave;&auml;&icirc;&acirc;&icirc;&euml;&uuml;&iacute;&yuml;&thorn;&ograve;&uuml;&ntilde;&yuml;, &agrave;&euml;&aring; &iuml;&eth;&egrave; &ouml;&uuml;&icirc;&igrave;&oacute; &eth;&icirc;&ccedil;&acirc;'&yuml;&ccedil;&icirc;&ecirc; u(x, t) = X(x) &middot; T (t) &auml;&icirc;&eth;i&acirc;&iacute;&thorn;&sup1; &iacute;&oacute;&euml;&thorn; &ograve;&icirc;&ograve;&icirc;&aelig;&iacute;&icirc;, &ograve;&icirc;&aacute;&ograve;&icirc; &eth;&icirc;&ccedil;&acirc;'&yuml;&ccedil;&icirc;&ecirc; &sup1; &ograve;&eth;&egrave;&acirc;i&agrave;&euml;&uuml;&iacute;&egrave;&igrave;, &ugrave;&icirc; &iuml;&eth;&icirc;&ograve;&egrave;&eth;i&divide;&egrave;&ograve;&uuml;
&icirc;&auml;&iacute;i&eacute; &ccedil; &oacute;&igrave;&icirc;&acirc; &icirc;&ntilde;&iacute;&icirc;&acirc;&iacute;&icirc;&uml; &auml;&icirc;&iuml;&icirc;&igrave;i&aelig;&iacute;&icirc;&uml; &ccedil;&agrave;&auml;&agrave;&divide;i. &Icirc;&ograve;&aelig;&aring;, &auml;&euml;&yuml; &ograve;&icirc;&atilde;&icirc;, &ugrave;&icirc;&aacute; &eth;&icirc;&ccedil;&acirc;'&yuml;&ccedil;&icirc;&ecirc; &aacute;&oacute;&acirc; &iacute;&aring;&ograve;&eth;&egrave;&acirc;i&agrave;&euml;&uuml;&iacute;&egrave;&igrave; i &icirc;&auml;&iacute;&icirc;&divide;&agrave;&ntilde;&iacute;&icirc; &ugrave;&icirc;&aacute; &oacute;&igrave;&icirc;&acirc;&egrave; (3.2) &acirc;&egrave;&ecirc;&icirc;&iacute;&oacute;&acirc;&agrave;&euml;&egrave;&ntilde;&uuml;, &ograve;&eth;&aring;&aacute;&agrave;
&iuml;&icirc;&ecirc;&euml;&agrave;&ntilde;&ograve;&egrave;: X(0) = 0 i X(l) = 0.
&ETH;&icirc;&ccedil;&atilde;&euml;&yuml;&iacute;&aring;&igrave;&icirc; &ograve;&aring;&iuml;&aring;&eth; &eth;i&acirc;&iacute;&yuml;&iacute;&iacute;&yuml; (3.7) &eth;&agrave;&ccedil;&icirc;&igrave; &ccedil; &oacute;&igrave;&icirc;&acirc;&agrave;&igrave;&egrave;, &ugrave;&icirc; &iacute;&agrave;&ecirc;&euml;&agrave;&auml;&agrave;&thorn;&ograve;&uuml;&ntilde;&yuml;
&iacute;&agrave; &ocirc;&oacute;&iacute;&ecirc;&ouml;i&thorn; X(x):
X 00 (x) + λX(x) = 0, 0 &lt; x &lt; l.
&frac12;
X(0) = 0,
X(l) = 0.
19
(3.9)
(3.10)
&Igrave;&egrave; &iuml;&eth;&egrave;&eacute;&oslash;&euml;&egrave; &auml;&icirc; &icirc;&ecirc;&eth;&aring;&igrave;&icirc;&uml; &ccedil;&agrave;&auml;&agrave;&divide;i, &ograve;&agrave;&ecirc; &ccedil;&acirc;&agrave;&iacute;&icirc;&uml; &ccedil;&agrave;&auml;&agrave;&divide;i &iacute;&agrave; &acirc;&euml;&agrave;&ntilde;&iacute;i &ccedil;&iacute;&agrave;&divide;&aring;&iacute;&iacute;&yuml;,
&agrave;&aacute;&icirc; &ccedil;&agrave;&auml;&agrave;&divide;i &Oslash;&ograve;&oacute;&eth;&igrave;&agrave;-&Euml;i&oacute;&acirc;i&euml;&euml;&yuml;. &Ntilde;&ocirc;&icirc;&eth;&igrave;&oacute;&euml;&thorn;&sup1;&igrave;&icirc; &ccedil;&igrave;i&ntilde;&ograve; &ouml;i&sup1;&uml; &ccedil;&agrave;&auml;&agrave;&divide;i:
&Ccedil;&iacute;&agrave;&eacute;&ograve;&egrave; &ograve;&agrave;&ecirc;i &ccedil;&iacute;&agrave;&divide;&aring;&iacute;&iacute;&yuml; &iuml;&agrave;&eth;&agrave;&igrave;&aring;&ograve;&eth;&agrave; λ, &iuml;&eth;&egrave; &yuml;&ecirc;&egrave;&otilde; i&ntilde;&iacute;&oacute;&thorn;&ograve;&uuml;
&iacute;&aring;&ograve;&eth;&egrave;&acirc;i&agrave;&euml;&uuml;&iacute;i &eth;&icirc;&ccedil;&acirc;'&yuml;&ccedil;&ecirc;&egrave; &eth;i&acirc;&iacute;&yuml;&iacute;&iacute;&yuml; (3.9), &ugrave;&icirc; &ccedil;&agrave;&auml;&icirc;&acirc;&icirc;&euml;&uuml;&iacute;&yuml;&thorn;&ograve;&uuml; &oacute;&igrave;&icirc;&acirc;&agrave;&igrave; (3.10), &agrave; &ograve;&agrave;&ecirc;&icirc;&aelig; &ccedil;&iacute;&agrave;&eacute;&ograve;&egrave; &ntilde;&agrave;&igrave;i &ouml;i &eth;&icirc;&ccedil;&acirc;'&yuml;&ccedil;&ecirc;&egrave;.
&Ccedil;&agrave;&oacute;&acirc;&agrave;&aelig;&egrave;&igrave;&icirc;, &ugrave;&icirc; &eth;&icirc;&ccedil;&acirc;'&yuml;&ccedil;&ecirc;&egrave; &eth;i&acirc;&iacute;&yuml;&iacute;&iacute;&yuml; (3.9) i&ntilde;&iacute;&oacute;&thorn;&ograve;&uuml; &iuml;&eth;&egrave; &aacute;&oacute;&auml;&uuml;-&yuml;&ecirc;&egrave;&otilde; &ccedil;&iacute;&agrave;&divide;&aring;&iacute;&iacute;&yuml;&otilde; λ. &Agrave;&euml;&aring; &eth;&icirc;&ccedil;&acirc;'&yuml;&ccedil;&ecirc;&egrave;, &ugrave;&icirc; &ccedil;&agrave;&auml;&icirc;&acirc;&icirc;&euml;&uuml;&iacute;&yuml;&thorn;&ograve;&uuml; &oacute;&igrave;&icirc;&acirc;&agrave;&igrave; (3.10), i&ntilde;&iacute;&oacute;&thorn;&ograve;&uuml; &euml;&egrave;&oslash;&aring;
&iuml;&eth;&egrave; &iuml;&aring;&acirc;&iacute;&egrave;&otilde; λ. &Ccedil;&iacute;&agrave;&divide;&aring;&iacute;&iacute;&yuml; λ, &iuml;&eth;&egrave; &yuml;&ecirc;&egrave;&otilde; i&ntilde;&iacute;&oacute;&thorn;&ograve;&uuml; &iacute;&aring;&ograve;&eth;&egrave;&acirc;i&agrave;&euml;&uuml;&iacute;i &eth;&icirc;&ccedil;&acirc;'&yuml;&ccedil;&ecirc;&egrave; (3.9)(3.10), &iacute;&agrave;&ccedil;&egrave;&acirc;&agrave;&thorn;&ograve;&uuml;&ntilde;&yuml; &acirc;&euml;&agrave;&ntilde;&iacute;&egrave;&igrave;&egrave; &ccedil;&iacute;&agrave;&divide;&aring;&iacute;&iacute;&yuml;&igrave;&egrave; &ccedil;&agrave;&auml;&agrave;&divide;i &Oslash;&ograve;&oacute;&eth;&igrave;&agrave;-&Euml;i&oacute;&acirc;i&euml;&euml;&yuml;, &agrave;
&ntilde;&agrave;&igrave;i &eth;&icirc;&ccedil;&acirc;'&yuml;&ccedil;&ecirc;&egrave;, &ugrave;&icirc; &acirc;i&auml;&iuml;&icirc;&acirc;i&auml;&agrave;&thorn;&ograve;&uuml; &ouml;&egrave;&igrave; &acirc;&euml;&agrave;&ntilde;&iacute;&egrave;&igrave; &ccedil;&iacute;&agrave;&divide;&aring;&iacute;&iacute;&yuml;&igrave; &acirc;&euml;&agrave;&ntilde;&iacute;&egrave;&igrave;&egrave;
&ocirc;&oacute;&iacute;&ecirc;&ouml;i&yuml;&igrave;&egrave;.
&Ccedil;&agrave;&auml;&agrave;&divide;&agrave; &Oslash;&ograve;&oacute;&eth;&igrave;&agrave;-&Euml;i&oacute;&acirc;i&euml;&euml;&yuml; &sup1; &ntilde;&ecirc;&euml;&agrave;&auml;&icirc;&acirc;&icirc;&thorn; &divide;&agrave;&ntilde;&ograve;&egrave;&iacute;&icirc;&thorn; &icirc;&ntilde;&iacute;&icirc;&acirc;&iacute;&icirc;&uml; &auml;&icirc;&iuml;&icirc;&igrave;i&aelig;&iacute;&icirc;&uml;
&ccedil;&agrave;&auml;&agrave;&divide;i. &Icirc;&aacute;&icirc;&acirc;'&yuml;&ccedil;&ecirc;&icirc;&acirc;&egrave;&igrave;&egrave; &eth;&egrave;&ntilde;&agrave;&igrave;&egrave; &ccedil;&agrave;&auml;&agrave;&divide;i &Oslash;&ograve;&oacute;&eth;&igrave;&agrave;-&Euml;i&oacute;&acirc;i&euml;&euml;&yuml; &sup1; &iacute;&agrave;&yuml;&acirc;&iacute;i&ntilde;&ograve;&uuml; &iuml;&agrave;&eth;&agrave;&igrave;&aring;&ograve;&eth;&agrave; i &auml;&icirc;&auml;&agrave;&ograve;&ecirc;&icirc;&acirc;&egrave;&otilde; &oacute;&igrave;&icirc;&acirc;, &ugrave;&icirc; &iacute;&agrave;&ecirc;&euml;&agrave;&auml;&agrave;&thorn;&ograve;&uuml;&ntilde;&yuml; &iacute;&agrave; &eth;&icirc;&ccedil;&acirc;'&yuml;&ccedil;&ecirc;&egrave; &auml;&egrave;&ocirc;&aring;&eth;&aring;&iacute;&ouml;i&agrave;&euml;&uuml;&iacute;&icirc;&atilde;&icirc; &eth;i&acirc;&iacute;&yuml;&iacute;&iacute;&yuml;.
&Ccedil;&icirc;&ntilde;&aring;&eth;&aring;&auml;&egrave;&igrave;&icirc;&ntilde;&uuml; &ograve;&aring;&iuml;&aring;&eth; &iacute;&agrave; &eth;&icirc;&ccedil;&acirc;'&yuml;&ccedil;&agrave;&iacute;&iacute;i &ccedil;&agrave;&auml;&agrave;&divide;i &Oslash;&ograve;&oacute;&eth;&igrave;&agrave;-&Euml;i&oacute;&acirc;i&euml;&euml;&yuml; (3.9)-(3.10).
&Auml;&euml;&yuml; &ouml;&uuml;&icirc;&atilde;&icirc; &ograve;&eth;&aring;&aacute;&agrave; &ntilde;&iuml;&icirc;&divide;&agrave;&ograve;&ecirc;&oacute; &ccedil;&agrave;&iuml;&egrave;&ntilde;&agrave;&ograve;&egrave; &ccedil;&agrave;&atilde;&agrave;&euml;&uuml;&iacute;&egrave;&eacute; &eth;&icirc;&ccedil;&acirc;'&yuml;&ccedil;&icirc;&ecirc; &eth;i&acirc;&iacute;&yuml;&iacute;&iacute;&yuml; (3.9).
&Acirc;&egrave;&atilde;&euml;&yuml;&auml; &ouml;&uuml;&icirc;&atilde;&icirc; &eth;&icirc;&ccedil;&acirc;'&yuml;&ccedil;&ecirc;&oacute; &ccedil;&agrave;&euml;&aring;&aelig;&egrave;&ograve;&uuml; &acirc;i&auml; &ograve;&icirc;&atilde;&icirc;, &divide;&egrave; &sup1; &ccedil;&iacute;&agrave;&divide;&aring;&iacute;&iacute;&yuml; &iuml;&agrave;&eth;&agrave;&igrave;&aring;&ograve;&eth;&oacute; λ
&auml;&icirc;&auml;&agrave;&ograve;&iacute;i&igrave;, &acirc;i&auml;'&sup1;&igrave;&iacute;&egrave;&igrave; &divide;&egrave; &acirc;&icirc;&iacute;&icirc; &auml;&icirc;&eth;i&acirc;&iacute;&thorn;&sup1; &iacute;&oacute;&euml;&thorn;. &ETH;&icirc;&ccedil;&atilde;&euml;&yuml;&iacute;&aring;&igrave;&icirc; &iuml;&icirc;&ntilde;&euml;i&auml;&icirc;&acirc;&iacute;&icirc; &ouml;i
&ograve;&eth;&egrave; &acirc;&egrave;&iuml;&agrave;&auml;&ecirc;&egrave; &icirc;&ecirc;&eth;&aring;&igrave;&icirc;.
1. &Iacute;&aring;&otilde;&agrave;&eacute; λ &lt; 0. &Ograve;&icirc;&auml;i &ccedil;&agrave;&atilde;&agrave;&euml;&uuml;&iacute;&egrave;&eacute; &eth;&icirc;&ccedil;&acirc;'&yuml;&ccedil;&icirc;&ecirc; &eth;i&acirc;&iacute;&yuml;&iacute;&iacute;&yuml; (3.9) &igrave;&agrave;&sup1; &acirc;&egrave;&atilde;&euml;&yuml;&auml;:
√
√
|λ|x
− |λ|x
+ Be
.
X(x) = Ae
(3.11)
&Ccedil;'&yuml;&ntilde;&oacute;&sup1;&igrave;&icirc;, &divide;&egrave; i&ntilde;&iacute;&oacute;&thorn;&ograve;&uuml; &ograve;&agrave;&ecirc;i &ccedil;&iacute;&agrave;&divide;&aring;&iacute;&iacute;&yuml; &ecirc;&icirc;&aring;&ocirc;i&ouml;i&sup1;&iacute;&ograve;i&acirc; A i B , &iuml;&eth;&egrave; &yuml;&ecirc;&egrave;&otilde; &eth;&icirc;&ccedil;&acirc;'&yuml;&ccedil;&icirc;&ecirc; (3.11) &ccedil;&agrave;&auml;&icirc;&acirc;&icirc;&euml;&uuml;&iacute;&yuml;&sup1; &ecirc;&eth;&agrave;&eacute;&icirc;&acirc;&egrave;&igrave; &oacute;&igrave;&icirc;&acirc;&agrave;&igrave; (3.10). &Ccedil;&agrave;&oacute;&acirc;&agrave;&aelig;&egrave;&igrave;&icirc;, &ugrave;&icirc; A i B
&iacute;&aring; &iuml;&icirc;&acirc;&egrave;&iacute;&iacute;i &icirc;&auml;&iacute;&icirc;&divide;&agrave;&ntilde;&iacute;&icirc; &auml;&icirc;&eth;i&acirc;&iacute;&thorn;&acirc;&agrave;&ograve;&egrave; &iacute;&oacute;&euml;&thorn;, &icirc;&ntilde;&ecirc;i&euml;&uuml;&ecirc;&egrave; &iuml;&eth;&egrave; A = 0 i B = 0
&eth;&icirc;&ccedil;&acirc;'&yuml;&ccedil;&icirc;&ecirc; X(x) ≡ 0 &sup1; &ograve;&eth;&egrave;&acirc;i&agrave;&euml;&uuml;&iacute;&egrave;&igrave;, &ugrave;&icirc; &iuml;&eth;&icirc;&ograve;&egrave;&eth;i&divide;&egrave;&ograve;&uuml; &oacute;&igrave;&icirc;&acirc;i &ccedil;&agrave;&auml;&agrave;&divide;i &Oslash;&ograve;&oacute;&eth;&igrave;&agrave;&Euml;i&oacute;&acirc;i&euml;&euml;&yuml;.
&Icirc;&ograve;&aelig;&aring;, &ccedil; &ecirc;&eth;&agrave;&eacute;&icirc;&acirc;&egrave;&otilde; &oacute;&igrave;&icirc;&acirc; (3.10) &igrave;&agrave;&sup1;&igrave;&icirc;:
&frac12;
(
X(0) = 0
⇒
X(l) = 0
A+
√B = 0
Ae
|λ|l
+ Be
−
√
|λ|l
= 0.
(3.12)
&Igrave;&agrave;&sup1;&igrave;&icirc; &ntilde;&egrave;&ntilde;&ograve;&aring;&igrave;&oacute; &auml;&acirc;&icirc;&otilde; &agrave;&euml;&atilde;&aring;&aacute;&eth;&agrave;&uml;&divide;&iacute;&egrave;&otilde; &eth;i&acirc;&iacute;&yuml;&iacute;&uuml; &acirc;i&auml;&iacute;&icirc;&ntilde;&iacute;&icirc; &iacute;&aring;&acirc;i&auml;&icirc;&igrave;&egrave;&otilde; A i B .
&Icirc;&auml;&iacute;&icirc;&eth;i&auml;&iacute;&agrave; &ntilde;&egrave;&ntilde;&ograve;&aring;&igrave;&agrave; (3.12) &igrave;&agrave;&sup1; &iacute;&aring;&ograve;&eth;&egrave;&acirc;i&agrave;&euml;&uuml;&iacute;&egrave;&eacute; &eth;&icirc;&ccedil;&acirc;'&yuml;&ccedil;&icirc;&ecirc; (A i B &iacute;&aring; &auml;&icirc;&eth;i&acirc;20
&iacute;&thorn;&thorn;&ograve;&uuml; &iacute;&oacute;&euml;&thorn; &icirc;&auml;&iacute;&icirc;&divide;&agrave;&ntilde;&iacute;&icirc;), &yuml;&ecirc;&ugrave;&icirc; &acirc;&egrave;&ccedil;&iacute;&agrave;&divide;&iacute;&egrave;&ecirc; &ntilde;&egrave;&ntilde;&ograve;&aring;&igrave;&egrave;
&macr;
&macr;1
&macr;
∆ = &macr; √|λ|
&macr;e
l
1 √
e− |λ|
&macr;
&macr;
√
√
&macr;
− |λ| l
− e |λ|
&macr;=e
&macr;
l
l
&auml;&icirc;&eth;i&acirc;&iacute;&thorn;&sup1; &iacute;&oacute;&euml;&thorn;. &Aacute;&aring;&ccedil;&iuml;&icirc;&ntilde;&aring;&eth;&aring;&auml;&iacute;&uuml;&icirc; &ccedil; &acirc;&egrave;&atilde;&euml;&yuml;&auml;&oacute; &iuml;&eth;&agrave;&acirc;&icirc;&uml;√&divide;&agrave;&ntilde;&ograve;&egrave;&iacute;&egrave; &ouml;i&sup1;&uml;√
&eth;i&acirc;&iacute;&icirc;&ntilde;&ograve;i &acirc;&egrave;&auml;&iacute;&icirc;, &ugrave;&icirc; &acirc; &auml;&agrave;&iacute;&icirc;&igrave;&oacute; &acirc;&egrave;&iuml;&agrave;&auml;&ecirc;&oacute; ∆ 6= 0, &icirc;&ntilde;&ecirc;i&euml;&uuml;&ecirc;&egrave; e− |λ| l &lt; 1, a e |λ| l &gt; 1 &ccedil;&agrave;
&aacute;&oacute;&auml;&uuml;-&yuml;&ecirc;&egrave;&otilde; &ccedil;&iacute;&agrave;&divide;&aring;&iacute;&uuml; λ &lt; 0.
&Acirc;&egrave;&ntilde;&iacute;&icirc;&acirc;&icirc;&ecirc;: &iuml;&eth;&egrave; λ &lt; 0 &ccedil;&agrave;&auml;&agrave;&divide;&agrave; &Oslash;&ograve;&oacute;&eth;&igrave;&agrave;-&Euml;i&oacute;&acirc;i&euml;&euml;&yuml; (3.9), (3.10) &iacute;&aring; &igrave;&agrave;&sup1;
&eth;&icirc;&ccedil;&acirc;'&yuml;&ccedil;&ecirc;i&acirc;, &ograve;&icirc;&aacute;&ograve;&icirc; &iuml;&eth;&egrave; λ &lt; 0 &iacute;&aring; i&ntilde;&iacute;&oacute;&sup1; &acirc;&euml;&agrave;&ntilde;&iacute;&egrave;&otilde; &ccedil;&iacute;&agrave;&divide;&aring;&iacute;&uuml;, &agrave; &icirc;&ograve;&aelig;&aring; &iacute;&aring; i&ntilde;&iacute;&oacute;&sup1;
&acirc;i&auml;&iuml;&icirc;&acirc;i&auml;&iacute;&egrave;&otilde; &uml;&igrave; &acirc;&euml;&agrave;&ntilde;&iacute;&egrave;&otilde; &ocirc;&oacute;&iacute;&ecirc;&ouml;i&eacute;.
2. &Iacute;&aring;&otilde;&agrave;&eacute; λ = 0. &Ograve;&icirc;&auml;i &eth;i&acirc;&iacute;&yuml;&iacute;&iacute;&yuml; (3.9) &iacute;&agrave;&aacute;&oacute;&acirc;&agrave;&sup1; &acirc;&egrave;&atilde;&euml;&yuml;&auml;&oacute;
X 00 (x) = 0,
&agrave; &eacute;&icirc;&atilde;&icirc; &eth;&icirc;&ccedil;&acirc;'&yuml;&ccedil;&icirc;&ecirc; &sup1; &euml;i&iacute;i&eacute;&iacute;&icirc;&thorn; &ocirc;&oacute;&iacute;&ecirc;&ouml;i&sup1;&thorn; x:
(3.13)
X(x) = Ax + B.
&Ccedil; &oacute;&igrave;&icirc;&acirc; (3.10) &igrave;&agrave;&sup1;&igrave;&icirc;:
&frac12;
X(0) = 0
⇒
X(l) = 0
&frac12;
B=0
Al = 0.
&Icirc;&ntilde;&ecirc;i&euml;&uuml;&ecirc;&egrave; l 6= 0, &ograve;&icirc; A = 0 i B = 0 &icirc;&auml;&iacute;&icirc;&divide;&agrave;&ntilde;&iacute;&icirc;; &ograve;&icirc;&aacute;&ograve;&icirc;, i&ntilde;&iacute;&oacute;&sup1; &euml;&egrave;&oslash;&aring; &ograve;&eth;&egrave;&acirc;i&agrave;&euml;&uuml;&iacute;&egrave;&eacute; &eth;&icirc;&ccedil;&acirc;'&yuml;&ccedil;&icirc;&ecirc; X(x) ≡ 0. &Ccedil;&acirc;i&auml;&ntilde;&egrave; &acirc;&egrave;&ntilde;&iacute;&icirc;&acirc;&icirc;&ecirc;: λ = 0 &iacute;&aring; &sup1; &acirc;&euml;&agrave;&ntilde;&iacute;&egrave;&igrave; &ccedil;&iacute;&agrave;&divide;&aring;&iacute;&iacute;&yuml;&igrave;
&ccedil;&agrave;&auml;&agrave;&divide;i &Oslash;&ograve;&oacute;&eth;&igrave;&agrave;-&Euml;i&oacute;&acirc;i&euml;&euml;&yuml; (3.9), (3.10).
&Iacute;&agrave;&eth;&aring;&oslash;&ograve;i, &eth;&icirc;&ccedil;&atilde;&euml;&yuml;&iacute;&aring;&igrave;&icirc; &icirc;&ntilde;&ograve;&agrave;&iacute;&iacute;i&eacute; &ccedil; &igrave;&icirc;&aelig;&euml;&egrave;&acirc;&egrave;&otilde; &acirc;&egrave;&iuml;&agrave;&auml;&ecirc;i&acirc;.
3. &Iacute;&aring;&otilde;&agrave;&eacute; λ &gt; 0. &szlig;&ecirc; i &acirc; &auml;&acirc;&icirc;&otilde; &iuml;&icirc;&iuml;&aring;&eth;&aring;&auml;&iacute;i&otilde; &acirc;&egrave;&iuml;&agrave;&auml;&ecirc;&agrave;&otilde;, &ccedil;&agrave;&iuml;&egrave;&oslash;&aring;&igrave;&icirc; &ccedil;&agrave;&atilde;&agrave;&euml;&uuml;&iacute;&egrave;&eacute;
&eth;&icirc;&ccedil;&acirc;'&yuml;&ccedil;&icirc;&ecirc; &eth;i&acirc;&iacute;&yuml;&iacute;&iacute;&yuml; (3.9):
X(x) = A cos
&sup3;√
&acute;
λx + B sin
&sup3;√
&acute;
λx
i &acirc;&egrave;&igrave;&agrave;&atilde;&agrave;&sup1;&igrave;&icirc;, &ugrave;&icirc;&aacute; &acirc;i&iacute; &ccedil;&agrave;&auml;&icirc;&acirc;&icirc;&euml;&uuml;&iacute;&yuml;&acirc; &oacute;&igrave;&icirc;&acirc;&egrave; (3.10):
&frac12;
(
X(0) = 0
⇒
X(l) = 0
A = 0&sup3;
√ &acute;
B sin
λl = 0.
(3.14)
&szlig;&ecirc;&ugrave;&icirc; &iuml;&icirc;&ecirc;&euml;&agrave;&ntilde;&ograve;&egrave; B = 0, &ograve;&icirc; &oacute;&igrave;&icirc;&acirc;&egrave; (3.10) &ccedil;&agrave;&auml;&icirc;&acirc;&icirc;&euml;&uuml;&iacute;&yuml;&ograve;&uuml;&ntilde;&yuml;, &agrave;&euml;&aring; &ograve;&icirc;&auml;i A = 0
i B = 0 &icirc;&auml;&iacute;&icirc;&divide;&agrave;&ntilde;&iacute;&icirc;, &ograve;&icirc;&aacute;&ograve;&icirc; &icirc;&ograve;&eth;&egrave;&igrave;&agrave;&sup1;&igrave;&icirc; &ograve;&eth;&egrave;&acirc;i&agrave;&euml;&uuml;&iacute;&egrave;&eacute; &eth;&icirc;&ccedil;&acirc;'&yuml;&ccedil;&icirc;&ecirc; X(x) ≡ 0,
&yuml;&ecirc;&egrave;&eacute; &iacute;&aring; &acirc;&acirc;&agrave;&aelig;&agrave;&sup1;&ograve;&uuml;&ntilde;&yuml; &ccedil;&agrave; &eth;&icirc;&ccedil;&acirc;'&yuml;&ccedil;&icirc;&ecirc; &ccedil;&agrave;&auml;&agrave;&divide;i &Oslash;&ograve;&oacute;&eth;&igrave;&agrave;-&Euml;i&oacute;&acirc;i&euml;&euml;&yuml;.
21
&Iacute;&agrave; &acirc;i&auml;&igrave;i&iacute;&oacute; &acirc;i&auml; &auml;&acirc;&icirc;&otilde; &iuml;&icirc;&iuml;&aring;&eth;&aring;&auml;&iacute;i&otilde;, &acirc; &auml;&agrave;&iacute;&iacute;&icirc;&igrave;&oacute; &acirc;&egrave;&iuml;&agrave;&auml;&ecirc;&oacute; &sup1; &agrave;&euml;&uuml;&ograve;&aring;&eth;&iacute;&agrave;&ograve;&egrave;&acirc;&agrave;:
&acirc;&acirc;&agrave;&aelig;&agrave;&sup1;&igrave;&icirc;, &ugrave;&icirc; B 6= 0, &ograve;&icirc;&auml;i
&sup3;√ &acute;
sin
λl = 0.
&Ograve;&agrave;&ecirc;&agrave; &eth;i&acirc;&iacute;i&ntilde;&ograve;&uuml; &igrave;&agrave;&sup1; &igrave;i&ntilde;&ouml;&aring;, &yuml;&ecirc;&ugrave;&icirc;
√
λl = πn, &auml;&aring; n = 1, 2, 3, ...
(3.15)
√
&Acirc;i&auml;'&sup1;&igrave;&iacute;i &ouml;i&euml;i &ccedil;&iacute;&agrave;&divide;&aring;&iacute;&iacute;&yuml; n &iacute;&aring; &aacute;&aring;&eth;&oacute;&ograve;&uuml;&ntilde;&yuml; &auml;&icirc; &oacute;&acirc;&agrave;&atilde;&egrave;, &icirc;&ntilde;&ecirc;i&euml;&uuml;&ecirc;&egrave; λl &auml;&icirc;&auml;&agrave;&ograve;&iacute;&yuml;
&acirc;&aring;&euml;&egrave;&divide;&egrave;&iacute;&agrave;.
&Ccedil; (3.15) &acirc;&egrave;&iuml;&euml;&egrave;&acirc;&agrave;&sup1;, &ugrave;&icirc; &iacute;&aring;&ograve;&eth;&egrave;&acirc;i&agrave;&euml;&uuml;&iacute;i &eth;&icirc;&ccedil;&acirc;'&yuml;&ccedil;&ecirc;&egrave; &ccedil;&agrave;&auml;&agrave;&divide;i &Oslash;&ograve;&oacute;&eth;&igrave;&agrave;-&Euml;i&oacute;&acirc;i&euml;&euml;&yuml;
i&ntilde;&iacute;&oacute;&thorn;&ograve;&uuml; &euml;&egrave;&oslash;&aring; &iuml;&eth;&egrave; &iuml;&aring;&acirc;&iacute;&egrave;&otilde; &ccedil;&iacute;&agrave;&divide;&aring;&iacute;&iacute;&yuml;&otilde; &iuml;&agrave;&eth;&agrave;&igrave;&aring;&ograve;&eth;&agrave; λ. &Ecirc;&icirc;&aelig;&iacute;&icirc;&igrave;&oacute; &ccedil;&iacute;&agrave;&divide;&aring;&iacute;&iacute;&thorn; n
2
&acirc;i&auml;&iuml;&icirc;&acirc;i&auml;&agrave;&sup1; &ntilde;&acirc;&icirc;&sup1; &ccedil;&iacute;&agrave;&divide;&aring;&iacute;&iacute;&yuml; λn = (πn/l) . &Ecirc;&icirc;&aelig;&iacute;&icirc;&igrave;&oacute; &ccedil;&iacute;&agrave;&divide;&aring;&iacute;&iacute;&thorn; λn &acirc;i&auml;&iuml;&icirc;&acirc;i&auml;&agrave;&sup1;
&ocirc;&oacute;&iacute;&ecirc;&ouml;i&yuml;
&sup3;p &acute;
&sup3; πn &acute;
Xn (x) = B̃n sin
λn l ⇒ Xn (x) = B̃n sin
x .
l
&ETH;&icirc;&ccedil;&acirc;'&yuml;&ccedil;&ecirc;&icirc;&igrave; &ccedil;&agrave;&auml;&agrave;&divide;i &Oslash;&ograve;&oacute;&eth;&igrave;&agrave;-&Euml;i&oacute;&acirc;i&euml;&euml;&yuml; &iacute;&agrave;&ccedil;&egrave;&acirc;&agrave;&sup1;&ograve;&uuml;&ntilde;&yuml; &ntilde;&oacute;&ecirc;&oacute;&iuml;&iacute;i&ntilde;&ograve;&uuml; &acirc;&ntilde;i&otilde;
&acirc;&euml;&agrave;&ntilde;&iacute;&egrave;&otilde; &ccedil;&iacute;&agrave;&divide;&aring;&iacute;&uuml; λn i &acirc;i&auml;&iuml;&icirc;&acirc;i&auml;&iacute;&egrave;&otilde; &acirc;&euml;&agrave;&ntilde;&iacute;&egrave;&otilde; &ocirc;&oacute;&iacute;&ecirc;&ouml;i&eacute; Xn (x):
(
&iexcl; &cent;2
λn = πn
l
&iexcl;
&cent;
(3.16)
Xn (x) = B̃n sin πn
x
,
l
&auml;&aring; n = 1, 2, 3...; B̃n &auml;&icirc;&acirc;i&euml;&uuml;&iacute;&agrave; &ntilde;&ograve;&agrave;&euml;&agrave;. &Icirc;&ograve;&aelig;&aring;, &ccedil;&agrave;&auml;&agrave;&divide;&agrave; &Oslash;&ograve;&oacute;&eth;&igrave;&agrave;-&Euml;i&oacute;&acirc;i&euml;&euml;&yuml;
(3.9), (3.10) &igrave;&agrave;&sup1; &iacute;&aring;&ntilde;&ecirc;i&iacute;&divide;&aring;&iacute;&iacute;&oacute;, &ccedil;&euml;i&divide;&aring;&iacute;&iacute;&oacute; &igrave;&iacute;&icirc;&aelig;&egrave;&iacute;&oacute; &acirc;&euml;&agrave;&ntilde;&iacute;&egrave;&otilde; &ocirc;&oacute;&iacute;&ecirc;&ouml;i&eacute; i &acirc;&euml;&agrave;&ntilde;&iacute;&egrave;&otilde; &ccedil;&iacute;&agrave;&divide;&aring;&iacute;&uuml;. &Ccedil;&agrave;&oacute;&acirc;&agrave;&aelig;&egrave;&igrave;&icirc;, &ugrave;&icirc; &acirc;&ntilde;i &acirc;&euml;&agrave;&ntilde;&iacute;i &ccedil;&iacute;&agrave;&divide;&aring;&iacute;&iacute;&yuml; &ccedil;&agrave;&auml;&agrave;&divide;i (3.9), (3.10) &auml;&icirc;&auml;&agrave;&ograve;&iacute;i. &Acirc; &icirc;&auml;&iacute;&icirc;&igrave;&oacute; &ccedil; &iacute;&agrave;&ntilde;&ograve;&oacute;&iuml;&iacute;&egrave;&otilde; &eth;&icirc;&ccedil;&auml;i&euml;i&acirc; &aacute;&oacute;&auml;&oacute;&ograve;&uuml; &ntilde;&ocirc;&icirc;&eth;&igrave;&oacute;&euml;&uuml;&icirc;&acirc;&agrave;&iacute;i &ccedil;&agrave;&atilde;&agrave;&euml;&uuml;&iacute;i
&oacute;&igrave;&icirc;&acirc;&egrave; &ccedil;&agrave; &yuml;&ecirc;&egrave;&otilde; &acirc;&ntilde;i &acirc;&euml;&agrave;&ntilde;&iacute;i &ccedil;&iacute;&agrave;&divide;&aring;&iacute;&iacute;&yuml; &ccedil;&agrave;&auml;&agrave;&divide;i &Oslash;&ograve;&oacute;&eth;&igrave;&agrave;-&Euml;i&oacute;&acirc;i&euml;&euml;&yuml; &sup1; &auml;&icirc;&auml;&agrave;&ograve;&iacute;i&igrave;&egrave;.
&Icirc;&ntilde;&ecirc;i&euml;&uuml;&ecirc;&egrave; &eth;&icirc;&ccedil;&acirc;'&yuml;&ccedil;&icirc;&ecirc; &ccedil;&agrave;&auml;&agrave;&divide;i &Oslash;&ograve;&oacute;&eth;&igrave;&agrave;-&Euml;i&oacute;&acirc;i&euml;&euml;&yuml; &ccedil;&iacute;&agrave;&eacute;&auml;&aring;&iacute;&icirc;, &ograve;&icirc; &iuml;&eth;&icirc;&auml;&icirc;&acirc;&aelig;&oacute;&sup1;&igrave;&icirc; &auml;&agrave;&euml;i &iacute;&aring;&icirc;&aacute;&otilde;i&auml;&iacute;i &auml;i&uml; &ugrave;&icirc;&auml;&icirc; &eth;&icirc;&ccedil;&acirc;'&yuml;&ccedil;&agrave;&iacute;&iacute;&yuml; &icirc;&ntilde;&iacute;&icirc;&acirc;&iacute;&icirc;&uml; &auml;&icirc;&iuml;&icirc;&igrave;i&aelig;&iacute;&icirc;&uml; &ccedil;&agrave;&auml;&agrave;&divide;i. &Ccedil;&acirc;&aring;&eth;&ograve;&agrave;&sup1;&igrave;&icirc;&ntilde;&uuml; &ograve;&aring;&iuml;&aring;&eth; &auml;&icirc; &eth;i&acirc;&iacute;&yuml;&iacute;&iacute;&yuml; (3.6). &Iacute;&agrave;&atilde;&agrave;&auml;&agrave;&sup1;&igrave;&icirc;, &ugrave;&icirc; (3.6) i (3.7) &iuml;&icirc;&acirc;'&yuml;&ccedil;&agrave;&iacute;i &ntilde;&iuml;i&euml;&uuml;&iacute;&egrave;&igrave; &iuml;&agrave;&eth;&agrave;&igrave;&aring;&ograve;&eth;&icirc;&igrave; λ. &Ograve;&icirc;&igrave;&oacute;, &icirc;&ntilde;&ecirc;i&euml;&uuml;&ecirc;&egrave; &eth;&icirc;&ccedil;&acirc;'&yuml;&ccedil;&ecirc;&egrave; &ccedil;&agrave;&auml;&agrave;&divide;i &Oslash;&ograve;&oacute;&eth;&igrave;&agrave;&Euml;i&oacute;&acirc;i&euml;&euml;&yuml; i&ntilde;&iacute;&oacute;&thorn;&ograve;&uuml; &euml;&egrave;&oslash;&aring; &ccedil;&agrave; &iuml;&aring;&acirc;&iacute;&egrave;&otilde; &auml;&egrave;&ntilde;&ecirc;&eth;&aring;&ograve;&iacute;&egrave;&otilde; &ccedil;&iacute;&agrave;&divide;&aring;&iacute;&iacute;&uuml; λn , &ograve;&icirc;&iexcl; i &eth;i&acirc;&iacute;&yuml;&iacute;&iacute;&yuml;
&cent;2
(3.6) &ntilde;&euml;i&auml; &eth;&icirc;&ccedil;&acirc;'&yuml;&ccedil;&oacute;&acirc;&agrave;&ograve;&egrave; &ograve;i&euml;&uuml;&ecirc;&egrave; &iuml;&eth;&egrave; &ograve;&egrave;&otilde; &ntilde;&agrave;&igrave;&egrave;&otilde; &ccedil;&iacute;&agrave;&divide;&aring;&iacute;&iacute;&yuml;&otilde; λn = πn
, &icirc;&ograve;&aelig;&aring;
l
&eth;i&acirc;&iacute;&yuml;&iacute;&iacute;&yuml; (3.6) &ograve;&eth;&aring;&aacute;&agrave; &iuml;&aring;&eth;&aring;&iuml;&egrave;&ntilde;&agrave;&ograve;&egrave; &oacute; &acirc;&egrave;&atilde;&euml;&yuml;&auml;i:
T̈n (t) + λn a2 Tn (t) = 0,
&agrave;&aacute;&icirc;
T̈n (t) + ωn2 Tn (t) = 0,
22
(3.17)
&auml;&aring; ωn = πna
l . I&iacute;&auml;&aring;&ecirc;&ntilde; n &oacute; &ocirc;&oacute;&iacute;&ecirc;&ouml;i&uml; Tn (t) &icirc;&ccedil;&iacute;&agrave;&divide;&agrave;&sup1;, &ugrave;&icirc; &eth;i&ccedil;&iacute;&egrave;&igrave; &ccedil;&iacute;&agrave;&divide;&aring;&iacute;&iacute;&yuml;&igrave; λn
&acirc;i&auml;&iuml;&icirc;&acirc;i&auml;&agrave;&thorn;&ograve;&uuml; &eth;i&ccedil;&iacute;i &ocirc;&oacute;&iacute;&ecirc;&ouml;i&uml; Tn (t).
&Ccedil;&agrave;&iuml;&egrave;&oslash;&aring;&igrave;&icirc; &ccedil;&agrave;&atilde;&agrave;&euml;&uuml;&iacute;&egrave;&eacute; &eth;&icirc;&ccedil;&acirc;'&yuml;&ccedil;&icirc;&ecirc; &eth;i&acirc;&iacute;&yuml;&iacute;&iacute;&yuml; (3.17):
Tn (t) = Cn cos (ωn t) + Dn sin (ωn t) .
&Ograve;&icirc;&auml;i &ocirc;&oacute;&iacute;&ecirc;&ouml;i&uml;
&sup3; πn &acute;
un (x, t) = Xn (x) &middot; Tn (t) = (An cos ωn t + Bn sin ωn t) &middot; sin
x ,
l
(3.18)
&sup1; &divide;&agrave;&ntilde;&ograve;&egrave;&iacute;&iacute;&egrave;&igrave;&egrave; &eth;&icirc;&ccedil;&acirc;'&yuml;&ccedil;&ecirc;&agrave;&igrave;&egrave; &icirc;&ntilde;&iacute;&icirc;&acirc;&iacute;&icirc;&uml; &auml;&icirc;&iuml;&icirc;&igrave;i&aelig;&iacute;&icirc;&uml; &ccedil;&agrave;&auml;&agrave;&divide;i. &Ograve;&icirc;&aacute;&ograve;&icirc; un (x, t),
n = 1, 2, ... &sup1; &eth;&icirc;&ccedil;&acirc;'&yuml;&ccedil;&ecirc;&agrave;&igrave;&egrave; &eth;i&acirc;&iacute;&yuml;&iacute;&iacute;&yuml; (3.1), &ugrave;&icirc; &ccedil;&agrave;&auml;&icirc;&acirc;&icirc;&euml;&iacute;&yuml;&thorn;&ograve;&uuml; &ecirc;&eth;&agrave;&eacute;&icirc;&acirc;i &oacute;&igrave;&icirc;&acirc;&egrave;
(3.2) &iuml;&eth;&egrave; &auml;&icirc;&acirc;i&euml;&uuml;&iacute;&egrave;&otilde; &ccedil;&iacute;&agrave;&divide;&aring;&iacute;&iacute;&yuml;&otilde; &ecirc;&icirc;&iacute;&ntilde;&ograve;&agrave;&iacute;&ograve; An = Cn &middot; B̃n , Bn = Dn &middot; B̃n . &Ograve;&agrave;&ecirc;&egrave;&igrave;
&divide;&egrave;&iacute;&icirc;&igrave;, &icirc;&ntilde;&iacute;&icirc;&acirc;&iacute;&oacute; &auml;&icirc;&iuml;&icirc;&igrave;i&aelig;&iacute;&oacute; &ccedil;&agrave;&auml;&agrave;&divide;&oacute; &eth;&icirc;&ccedil;&acirc;'&yuml;&ccedil;&agrave;&iacute;&icirc;, &icirc;&ntilde;&ecirc;i&euml;&uuml;&ecirc;&egrave; &eth;&icirc;&ccedil;&acirc;'&yuml;&ccedil;&ecirc;&egrave; (3.18)
&ccedil;&agrave;&auml;&icirc;&acirc;&icirc;&euml;&uuml;&iacute;&yuml;&thorn;&ograve;&uuml; &acirc;&ntilde;i&igrave; &acirc;&egrave;&igrave;&icirc;&atilde;&agrave;&igrave; &auml;&agrave;&iacute;&icirc;&uml; &ccedil;&agrave;&auml;&agrave;&divide;i.
&Icirc;&ntilde;&ecirc;i&euml;&uuml;&ecirc;&egrave; &eth;i&acirc;&iacute;&yuml;&iacute;&iacute;&yuml; (3.1) &sup1; &euml;i&iacute;i&eacute;&iacute;&egrave;&igrave;, &ecirc;&eth;&agrave;&eacute;&icirc;&acirc;i &oacute;&igrave;&icirc;&acirc;&egrave; (3.2) &sup1; &euml;i&iacute;i&eacute;&iacute;&egrave;&igrave;&egrave; &ograve;&agrave; &icirc;&auml;&iacute;&icirc;&eth;i&auml;&iacute;&egrave;&igrave;&egrave;, &ograve;&icirc; &ccedil;&agrave; &iuml;&eth;&egrave;&iacute;&ouml;&egrave;&iuml;&icirc;&igrave; &ntilde;&oacute;&iuml;&aring;&eth;&iuml;&icirc;&ccedil;&egrave;&ouml;i&uml; &auml;&euml;&yuml; &euml;i&iacute;i&eacute;&iacute;&egrave;&otilde; &ccedil;&agrave;&auml;&agrave;&divide;,
&auml;&icirc;&acirc;i&euml;&uuml;&iacute;&agrave; &euml;i&iacute;i&eacute;&iacute;&agrave; &ecirc;&icirc;&igrave;&aacute;i&iacute;&agrave;&ouml;i&yuml; &divide;&agrave;&ntilde;&ograve;&egrave;&iacute;&iacute;&egrave;&otilde; &eth;&icirc;&ccedil;&acirc;'&yuml;&ccedil;&ecirc;i&acirc; un (x, t):
&sup3; πn &acute;
u(x, t) =
(An cos ωn t + Bn sin ωn t) &middot; sin
x
l
n=1
∞
X
(3.19)
&ograve;&agrave;&ecirc;&icirc;&aelig; &sup1; &eth;&icirc;&ccedil;&acirc;'&yuml;&ccedil;&ecirc;&icirc;&igrave; &eth;i&acirc;&iacute;&yuml;&iacute;&iacute;&yuml; (3.1), &ugrave;&icirc; &ccedil;&agrave;&auml;&icirc;&acirc;&icirc;&euml;&iacute;&yuml;&thorn;&ograve;&uuml; &ecirc;&eth;&agrave;&eacute;&icirc;&acirc;i &oacute;&igrave;&icirc;&acirc;&egrave; (3.2)
&iuml;&eth;&egrave; &aacute;&oacute;&auml;&uuml;-&yuml;&ecirc;&egrave;&otilde; &ccedil;&iacute;&agrave;&divide;&aring;&iacute;&iacute;&yuml;&otilde; &ecirc;&icirc;&iacute;&ntilde;&ograve;&agrave;&iacute;&ograve; An i Bn .
&Acirc;&egrave;&ecirc;&icirc;&iacute;&agrave;&sup1;&igrave;&icirc; &ograve;&aring;&iuml;&aring;&eth; &icirc;&ntilde;&ograve;&agrave;&iacute;&iacute;i &auml;i&uml; &auml;&euml;&yuml; &icirc;&ograve;&eth;&egrave;&igrave;&agrave;&iacute;&iacute;&yuml; &eth;&icirc;&ccedil;&acirc;'&yuml;&ccedil;&ecirc;&oacute; &ccedil;&agrave;&auml;&agrave;&divide;i (3.1)-(3.3)
&iuml;&eth;&icirc; &ecirc;&icirc;&euml;&egrave;&acirc;&agrave;&iacute;&iacute;&yuml; &icirc;&aacute;&igrave;&aring;&aelig;&aring;&iacute;&icirc;&uml; &ntilde;&ograve;&eth;&oacute;&iacute;&egrave;. &Iuml;&icirc;&ntilde;&ograve;&agrave;&acirc;&egrave;&igrave;&icirc; &iuml;&egrave;&ograve;&agrave;&iacute;&iacute;&yuml; &iacute;&agrave;&ntilde;&ograve;&oacute;&iuml;&iacute;&egrave;&igrave; &divide;&egrave;&iacute;&icirc;&igrave;:
&agrave; &divide;&egrave; &iacute;&aring; &igrave;&icirc;&aelig;&iacute;&agrave; &quot;&iuml;i&auml;i&aacute;&eth;&agrave;&ograve;&egrave;&quot;&ccedil;&iacute;&agrave;&divide;&aring;&iacute;&iacute;&yuml; &ecirc;&icirc;&aring;&ocirc;i&ouml;i&sup1;&iacute;&ograve;i&acirc; An i Bn &acirc; (3.19) &ograve;&agrave;&ecirc;&egrave;&igrave;&egrave;, &ugrave;&icirc;&aacute; &ocirc;&oacute;&iacute;&ecirc;&ouml;i&yuml; un (x, t) &oacute; &acirc;&egrave;&atilde;&euml;&yuml;&auml;i (3.19) &ccedil;&agrave;&auml;&icirc;&acirc;&icirc;&euml;&uuml;&iacute;&yuml;&euml;&agrave; &aacute; i &iuml;&icirc;&divide;&agrave;&ograve;&ecirc;&icirc;&acirc;&egrave;&igrave;
&oacute;&igrave;&icirc;&acirc;&agrave;&igrave; (3.3)? &Icirc;&ograve;&aelig;&aring;, &ecirc;&icirc;&aring;&ocirc;i&ouml;i&sup1;&iacute;&ograve;&egrave; An i Bn &iuml;&icirc;&acirc;&egrave;&iacute;i &igrave;&agrave;&ograve;&egrave; &ograve;&agrave;&ecirc;i &ccedil;&iacute;&agrave;&divide;&aring;&iacute;&iacute;&yuml;, &ugrave;&icirc;&aacute;
&acirc;&egrave;&ecirc;&icirc;&iacute;&oacute;&acirc;&agrave;&euml;&egrave;&ntilde;&uuml; &eth;i&acirc;&iacute;&icirc;&ntilde;&ograve;i:
&sup3; πn &acute;
x ,
u(x, 0) = ϕ(x) =
An sin
l
n=1
(3.20)
&sup3; πn &acute;
ut (x, 0) = ψ(x) =
Bn ωn sin
x .
l
n=1
(3.21)
∞
X
∞
X
&Ccedil;&agrave; &ntilde;&acirc;&icirc;&uml;&igrave; &ccedil;&igrave;i&ntilde;&ograve;&icirc;&igrave; (3.20) i (3.21) &sup1; &eth;&icirc;&ccedil;&ecirc;&euml;&agrave;&auml;&icirc;&igrave; &ccedil;&agrave;&auml;&agrave;&iacute;&egrave;&otilde; &ocirc;&oacute;&iacute;&ecirc;&ouml;i&eacute; ϕ(x) i
ψ(x) &acirc; &eth;&yuml;&auml; &Ocirc;&oacute;&eth;&sup1; &ccedil;&agrave; &ntilde;&egrave;&iacute;&oacute;&ntilde;&agrave;&igrave;&egrave;. &Icirc;&ograve;&aelig;&aring;, &auml;&euml;&yuml; &ccedil;&iacute;&agrave;&otilde;&icirc;&auml;&aelig;&aring;&iacute;&iacute;&yuml; &ecirc;&icirc;&aring;&ocirc;i&ouml;i&sup1;&iacute;&ograve;i&acirc; An
i Bn &igrave;&icirc;&aelig;&iacute;&agrave; &ntilde;&ecirc;&icirc;&eth;&egrave;&ntilde;&ograve;&agrave;&ograve;&egrave;&ntilde;&uuml; &acirc;i&auml;&icirc;&igrave;&egrave;&igrave;&egrave; &ocirc;&icirc;&eth;&igrave;&oacute;&euml;&agrave;&igrave;&egrave; &auml;&euml;&yuml; &ecirc;&icirc;&aring;&ocirc;i&ouml;i&sup1;&iacute;&ograve;i&acirc; &eth;&yuml;&auml;&oacute;
&Ocirc;&oacute;&eth;'&sup1;.
23
&Agrave;&euml;&aring;, &acirc; &ccedil;&agrave;&atilde;&agrave;&euml;&uuml;&iacute;&icirc;&igrave;&oacute; &acirc;&egrave;&iuml;&agrave;&auml;&ecirc;&oacute;, &acirc;&euml;&agrave;&ntilde;&iacute;i &ocirc;&oacute;&iacute;&ecirc;&ouml;i&uml; &ccedil;&agrave;&auml;&agrave;&divide;i &Oslash;&ograve;&oacute;&eth;&igrave;&agrave;-&Euml;i&oacute;&acirc;i&euml;&euml;&yuml;
Xn (x) &iacute;&aring; &ccedil;&acirc;&icirc;&auml;&yuml;&ograve;&uuml;&ntilde;&yuml; &auml;&icirc; &ograve;&eth;&egrave;&atilde;&icirc;&iacute;&icirc;&igrave;&aring;&ograve;&eth;&egrave;&divide;&iacute;&egrave;&otilde; &ocirc;&oacute;&iacute;&ecirc;&ouml;i&eacute;, i &ograve;&icirc;&auml;i &eth;&icirc;&ccedil;&ecirc;&euml;&agrave;&auml;&agrave;&iacute;&iacute;&yuml; &acirc;
&eth;&yuml;&auml; &ccedil;&agrave; &ocirc;&oacute;&iacute;&ecirc;&ouml;i&yuml;&igrave;&egrave; Xn (x) &iacute;&agrave;&ccedil;&egrave;&acirc;&agrave;&sup1;&ograve;&uuml;&ntilde;&yuml; &eth;&icirc;&ccedil;&ecirc;&euml;&agrave;&auml;&agrave;&iacute;&iacute;&yuml;&igrave; &acirc; &oacute;&ccedil;&agrave;&atilde;&agrave;&euml;&uuml;&iacute;&aring;&iacute;&egrave;&eacute; &eth;&yuml;&auml;
&Ocirc;&oacute;&eth;'&sup1;. &Ccedil;&iacute;&agrave;&eacute;&auml;&aring;&igrave;&icirc; &ccedil;&iacute;&agrave;&divide;&aring;&iacute;&iacute;&yuml; &ecirc;&icirc;&aring;&ocirc;i&ouml;i&sup1;&iacute;&ograve;i&acirc; An i Bn &acirc; (3.20) i (3.21) &ntilde;&ograve;&agrave;&iacute;&auml;&agrave;&eth;&ograve;&iacute;&egrave;&igrave; &igrave;&aring;&ograve;&icirc;&auml;&icirc;&igrave;, &yuml;&ecirc;&egrave;&eacute; &acirc;&egrave;&ecirc;&icirc;&eth;&egrave;&ntilde;&ograve;&icirc;&acirc;&oacute;&sup1;&ograve;&uuml;&ntilde;&yuml; &iuml;&eth;&egrave; &eth;&icirc;&ccedil;&ecirc;&euml;&agrave;&auml;&agrave;&iacute;&iacute;i &ocirc;&oacute;&iacute;&ecirc;&ouml;i&eacute; &ccedil;&agrave;
&acirc;&euml;&agrave;&ntilde;&iacute;&egrave;&igrave;&egrave; &ocirc;&oacute;&iacute;&ecirc;&ouml;i&yuml;&igrave;&egrave; &aacute;&oacute;&auml;&uuml;-&yuml;&ecirc;&icirc;&uml; &ccedil;&agrave;&auml;&agrave;&divide;i &Oslash;&ograve;&oacute;&eth;&igrave;&agrave;-&Euml;i&oacute;&acirc;i&euml;&euml;&yuml;.
&iexcl;
&cent;
&iexcl; πm &cent;
&Ntilde;&ecirc;&icirc;&eth;&egrave;&ntilde;&ograve;&agrave;&sup1;&igrave;&icirc;&ntilde;&uuml; &ograve;&egrave;&igrave;, &ugrave;&icirc; &ocirc;&oacute;&iacute;&ecirc;&ouml;i&uml; Xn (x) = sin πn
x
i
X
(x)
=
sin
m
l
l x
&icirc;&eth;&ograve;&icirc;&atilde;&icirc;&iacute;&agrave;&euml;&uuml;&iacute;i &igrave;i&aelig; &ntilde;&icirc;&aacute;&icirc;&thorn; &iacute;&agrave; &acirc;i&auml;&eth;i&ccedil;&ecirc;&oacute; x ∈ [0, l], &iuml;&eth;&egrave; m 6= n:
Z
l
0
&sup3; πm &acute;
&sup3; πn &acute;
l
x sin
x dx = δnm .
sin
l
l
2
&Auml;&icirc;&igrave;&iacute;&icirc;&aelig;&egrave;&igrave;&icirc; &euml;i&acirc;&oacute; i &iuml;&eth;&agrave;&acirc;&oacute; &divide;&agrave;&ntilde;&ograve;&egrave;&iacute;&egrave; &eth;i&acirc;&iacute;&icirc;&ntilde;&ograve;i (3.20) &iacute;&agrave; sin
&atilde;&eth;&oacute;&sup1;&igrave;&icirc; &iuml;&icirc; x &icirc;&ograve;&eth;&egrave;&igrave;&agrave;&iacute;i &acirc;&egrave;&eth;&agrave;&ccedil;&egrave; &iacute;&agrave; &acirc;i&auml;&eth;i&ccedil;&ecirc;&oacute; [0, l]:
Z
l
0
&iexcl; πm &cent;
l x i &iuml;&eth;&icirc;i&iacute;&ograve;&aring;-
Z l
∞
&sup3; πm &acute;
&sup3; πn &acute;
&sup3; πm &acute;
X
ϕ(x) sin
x dx =
An
sin
x sin
x dx =
l
l
l
0
n=1
=
∞
X
l
l
An δnm = Am .
2
2
n=1
&Ccedil;&acirc;i&auml;&ntilde;&egrave;:
Z
&sup3; πm &acute;
2 l
ϕ(x) sin
x dx, m = 1, 2, 3...
(3.22)
Am =
l 0
l
&Icirc;&ntilde;&ecirc;i&euml;&uuml;&ecirc;&egrave; m i n &iuml;&eth;&icirc;&aacute;i&atilde;&agrave;&thorn;&ograve;&uuml; &icirc;&auml;&iacute;&oacute; i &ograve;&oacute; &ntilde;&agrave;&igrave;&oacute; &igrave;&iacute;&icirc;&aelig;&egrave;&iacute;&oacute; &ccedil;&iacute;&agrave;&divide;&aring;&iacute;&uuml;: m = 1, 2, 3...;
n = 1, 2, 3..., &ograve;&icirc; &acirc; (3.22) &igrave;&icirc;&aelig;&iacute;&agrave; &ccedil;&agrave;&igrave;i&iacute;&egrave;&ograve;&egrave; m &iacute;&agrave; n:
Z
&sup3; πn &acute;
2 l
An =
ϕ(x) sin
x dx.
(3.23)
l 0
l
&Acirc;&egrave;&ecirc;&icirc;&eth;&egrave;&ntilde;&ograve;&icirc;&acirc;&oacute;&thorn;&divide;&egrave; &agrave;&iacute;&agrave;&euml;&icirc;&atilde;i&divide;&iacute;&oacute; &iuml;&eth;&icirc;&ouml;&aring;&auml;&oacute;&eth;&oacute; &iuml;&icirc; &acirc;i&auml;&iacute;&icirc;&oslash;&aring;&iacute;&iacute;&thorn; &auml;&icirc; &eth;i&acirc;&iacute;&yuml;&iacute;&iacute;&yuml; (3.21)
&icirc;&ograve;&eth;&egrave;&igrave;&agrave;&sup1;&igrave;&icirc;:
l
ω n Bn =
2
Z
l
0
&sup3; πn &acute;
ψ(x) sin
x dx,
l
&ccedil;&acirc;i&auml;&ntilde;&egrave;:
Z l
&sup3; πn &acute;
2
Bn =
ψ(x) sin
x dx.
(3.24)
ωn l 0
l
&Ograve;&agrave;&ecirc;&egrave;&igrave; &divide;&egrave;&iacute;&icirc;&igrave;, &acirc;&ntilde;i &ecirc;&icirc;&aring;&ocirc;i&ouml;i&sup1;&iacute;&ograve;&egrave; An i Bn &acirc;&egrave;&ccedil;&iacute;&agrave;&divide;&aring;&iacute;i &icirc;&auml;&iacute;&icirc;&ccedil;&iacute;&agrave;&divide;&iacute;&icirc; &ccedil;&atilde;i&auml;&iacute;&icirc;
(3.23) i (3.24) i, &icirc;&ograve;&aelig;&aring;, &ocirc;&oacute;&iacute;&ecirc;&ouml;i&yuml; (3.19) &sup1; &sup1;&auml;&egrave;&iacute;&egrave;&igrave; &eth;&icirc;&ccedil;&acirc;'&yuml;&ccedil;&ecirc;&icirc;&igrave; &ccedil;&agrave;&auml;&agrave;&divide;i (3.1)(3.3) &iuml;&eth;&icirc; &acirc;i&euml;&uuml;&iacute;i &ecirc;&icirc;&euml;&egrave;&acirc;&agrave;&iacute;&iacute;&yuml; &icirc;&aacute;&igrave;&aring;&aelig;&aring;&iacute;&icirc;&uml; &ntilde;&ograve;&eth;&oacute;&iacute;&egrave;.
24
&Ocirc;i&ccedil;&egrave;&divide;&iacute;&agrave; i&iacute;&ograve;&aring;&eth;&iuml;&eth;&aring;&ograve;&agrave;&ouml;i&yuml; &eth;&icirc;&ccedil;&acirc;'&yuml;&ccedil;&ecirc;&oacute; &ccedil;&agrave;&auml;&agrave;&divide;i.
&Icirc;&ograve;&eth;&egrave;&igrave;&agrave;&iacute;&iacute;&yuml; &igrave;&agrave;&ograve;&aring;&igrave;&agrave;&ograve;&egrave;&divide;&iacute;&icirc;&atilde;&icirc; &acirc;&egrave;&eth;&agrave;&ccedil;&oacute; &auml;&euml;&yuml; &eth;&icirc;&ccedil;&acirc;'&yuml;&ccedil;&ecirc;&oacute; &ccedil;&agrave;&auml;&agrave;&divide;i &igrave;&agrave;&ograve;&aring;&igrave;&agrave;&ograve;&egrave;&divide;&iacute;&icirc;&uml;
&ocirc;i&ccedil;&egrave;&ecirc;&egrave; &auml;&icirc;&ccedil;&acirc;&icirc;&euml;&yuml;&sup1; &iuml;&aring;&eth;&aring;&eacute;&ograve;&egrave; &auml;&icirc; &iacute;&agrave;&ntilde;&ograve;&oacute;&iuml;&iacute;&icirc;&atilde;&icirc; &aring;&ograve;&agrave;&iuml;&oacute; &ograve;&aring;&icirc;&eth;&aring;&ograve;&egrave;&divide;&iacute;&icirc;&atilde;&icirc; &auml;&icirc;&ntilde;&euml;i&auml;&aelig;&aring;&iacute;&iacute;&yuml; &ocirc;i&ccedil;&egrave;&divide;&iacute;&icirc;&atilde;&icirc; &iuml;&eth;&icirc;&ouml;&aring;&ntilde;&oacute;. &Acirc;&ntilde;&aring;&aacute;i&divide;&iacute;&egrave;&eacute; &agrave;&iacute;&agrave;&euml;i&ccedil; &ouml;&uuml;&icirc;&atilde;&icirc; &acirc;&egrave;&eth;&agrave;&ccedil;&oacute; &auml;&agrave;&sup1; &igrave;&icirc;&aelig;&euml;&egrave;&acirc;i&ntilde;&ograve;&uuml; &acirc;&egrave;&yuml;&acirc;&egrave;&ograve;&egrave; &acirc;&euml;&agrave;&ntilde;&ograve;&egrave;&acirc;&icirc;&ntilde;&ograve;i &ograve;&agrave; &otilde;&agrave;&eth;&agrave;&ecirc;&ograve;&aring;&eth;&iacute;i &icirc;&ntilde;&icirc;&aacute;&euml;&egrave;&acirc;&icirc;&ntilde;&ograve;i, &iuml;&eth;&egrave;&ograve;&agrave;&igrave;&agrave;&iacute;&iacute;i &aacute;&aring;&ccedil;&iuml;&icirc;&ntilde;&aring;&eth;&aring;&auml;&iacute;&uuml;&icirc;
&auml;&agrave;&iacute;&icirc;&igrave;&oacute; &yuml;&acirc;&egrave;&ugrave;&oacute;, &auml;&icirc;&ccedil;&acirc;&icirc;&euml;&yuml;&sup1; &iacute;&agrave;&auml;&agrave;&ograve;&egrave; &eacute;&icirc;&igrave;&oacute; &yuml;&ecirc;i&ntilde;&iacute;&aring; &ograve;&euml;&oacute;&igrave;&agrave;&divide;&aring;&iacute;&iacute;&yuml; &ograve;&agrave; &iuml;&eth;&icirc;&acirc;&aring;&ntilde;&ograve;&egrave; &agrave;&iacute;&agrave;&euml;&icirc;&atilde;i&uml; &ccedil; i&iacute;&oslash;&egrave;&igrave;&egrave; &iuml;&eth;&icirc;&ouml;&aring;&ntilde;&agrave;&igrave;&egrave; &ograve;&agrave; &yuml;&acirc;&egrave;&ugrave;&agrave;&igrave;&egrave;.
&ETH;&icirc;&ccedil;&atilde;&euml;&yuml;&iacute;&aring;&igrave;&icirc; &icirc;&ecirc;&eth;&aring;&igrave;&egrave;&eacute; &auml;&icirc;&auml;&agrave;&iacute;&icirc;&ecirc; &acirc; &eth;&icirc;&ccedil;&acirc;'&yuml;&ccedil;&ecirc;&oacute; (3.19) &iacute;&agrave;&oslash;&icirc;&uml; &ccedil;&agrave;&auml;&agrave;&divide;i:
&sup3; πn &acute;
un (x, t) = {An sin ωn t + Bn cos ωn t} sin
x ,
(3.25)
l
&ograve;&agrave; &iuml;&eth;&aring;&auml;&ntilde;&ograve;&agrave;&acirc;&egrave;&igrave;&icirc; &eacute;&icirc;&atilde;&icirc; &oacute; &acirc;&egrave;&atilde;&euml;&yuml;&auml;i:
&sup3; πn &acute;
un (x, t) = αn &middot; sin
x &middot; cos(ωn t + δn ),
(3.26)
l
p
n
&auml;&aring; αn = A2n + Bn2 , δn = −arctg B
An .
&Acirc;&egrave;&eth;&agrave;&ccedil; (3.26) &igrave;&agrave;&sup1; &ograve;&egrave;&iuml;&icirc;&acirc;&egrave;&eacute; &acirc;&egrave;&atilde;&euml;&yuml;&auml; &ocirc;&oacute;&iacute;&ecirc;&ouml;i&uml;, &ugrave;&icirc; &icirc;&iuml;&egrave;&ntilde;&oacute;&sup1; &ntilde;&ograve;&icirc;&yuml;&divide;i &otilde;&acirc;&egrave;&euml;i. &Ccedil;
&iacute;&uuml;&icirc;&atilde;&icirc; &acirc;&egrave;&iuml;&euml;&egrave;&acirc;&agrave;&sup1;, &ugrave;&icirc; &acirc;&ntilde;i &ograve;&icirc;&divide;&ecirc;&egrave; &ntilde;&ograve;&eth;&oacute;&iacute;&egrave; &ccedil;&auml;i&eacute;&ntilde;&iacute;&thorn;&thorn;&ograve;&uuml; &atilde;&agrave;&eth;&igrave;&icirc;&iacute;i&divide;&iacute;i &ecirc;&icirc;&euml;&egrave;&acirc;&agrave;&iacute;&iacute;&yuml;
&ccedil; &icirc;&auml;&iacute;i&sup1;&thorn; i &ograve;i&sup1;&thorn; &aelig; &divide;&agrave;&ntilde;&ograve;&icirc;&ograve;&icirc;&thorn; ωn . &Agrave;&igrave;&iuml;&euml;i&ograve;&oacute;&auml;&agrave; &ecirc;&icirc;&euml;&egrave;&acirc;&agrave;&iacute;&uuml; u0 = |αn sin(πnx/l)|
&ccedil;&agrave;&euml;&aring;&aelig;&egrave;&ograve;&uuml; &acirc;i&auml; &ecirc;&icirc;&icirc;&eth;&auml;&egrave;&iacute;&agrave;&ograve;&egrave; x, &icirc;&ograve;&aelig;&aring;, &acirc;&ccedil;&agrave;&atilde;&agrave;&euml;i &ecirc;&agrave;&aelig;&oacute;&divide;&egrave;, &eth;i&ccedil;&iacute;i &ograve;&icirc;&divide;&ecirc;&egrave; &ntilde;&ograve;&eth;&oacute;&iacute;&egrave;
&ecirc;&icirc;&euml;&egrave;&acirc;&agrave;&thorn;&ograve;&uuml;&ntilde;&yuml; &ccedil; &eth;i&ccedil;&iacute;&egrave;&igrave;&egrave; &agrave;&igrave;&iuml;&euml;i&ograve;&oacute;&auml;&agrave;&igrave;&egrave;.
&Ograve;&icirc;&divide;&ecirc;&egrave; &ccedil; &ecirc;&icirc;&icirc;&eth;&auml;&egrave;&iacute;&agrave;&ograve;&agrave;&igrave;&egrave; xk = k nl , (k = 0, 1, 2, ..., n), &auml;&euml;&yuml; &yuml;&ecirc;&egrave;&otilde; sin(πnx/l) =
0, &ccedil;&agrave;&euml;&egrave;&oslash;&agrave;&thorn;&ograve;&uuml;&ntilde;&yuml; &iacute;&aring;&eth;&oacute;&otilde;&icirc;&igrave;&egrave;&igrave;&egrave; &acirc;&iuml;&eth;&icirc;&auml;&icirc;&acirc;&aelig; &acirc;&ntilde;&uuml;&icirc;&atilde;&icirc; &iuml;&eth;&icirc;&ouml;&aring;&ntilde;&oacute;, &ograve;&icirc;&aacute;&ograve;&icirc; &agrave;&igrave;&iuml;&euml;i&ograve;&oacute;&auml;&agrave;
&ecirc;&icirc;&euml;&egrave;&acirc;&agrave;&iacute;&uuml; &auml;&agrave;&iacute;&egrave;&otilde; &ograve;&icirc;&divide;&icirc;&ecirc; &auml;&icirc;&eth;i&acirc;&iacute;&thorn;&sup1; &iacute;&oacute;&euml;&thorn;. &Ograve;&agrave;&ecirc;i &ograve;&icirc;&divide;&ecirc;&egrave; &iacute;&agrave;&ccedil;&egrave;&acirc;&agrave;&thorn;&ograve;&uuml; &acirc;&oacute;&ccedil;&euml;&agrave;&igrave;&egrave;
&ntilde;&ograve;&icirc;&yuml;&divide;&icirc;&uml; &otilde;&acirc;&egrave;&euml;i.
(2m+1)l
&Ograve;&icirc;&divide;&ecirc;&egrave; &ccedil; &ecirc;&icirc;&icirc;&eth;&auml;&egrave;&iacute;&agrave;&ograve;&agrave;&igrave;&egrave; xm = 2n (m = 0, 1, 2, ..., n − 1), &auml;&euml;&yuml; &yuml;&ecirc;&egrave;&otilde;
| sin(πnx/l)| = 1, &ecirc;&icirc;&euml;&egrave;&acirc;&agrave;&thorn;&ograve;&uuml;&ntilde;&yuml; &ccedil; &igrave;&agrave;&ecirc;&ntilde;&egrave;&igrave;&agrave;&euml;&uuml;&iacute;&icirc;&thorn; &agrave;&igrave;&iuml;&euml;i&ograve;&oacute;&auml;&icirc;&thorn;, &ugrave;&icirc; &auml;&icirc;&eth;i&acirc;&iacute;&thorn;&sup1;
αn . &Ograve;&agrave;&ecirc;i &ograve;&icirc;&divide;&ecirc;&egrave; &iacute;&agrave;&ccedil;&egrave;&acirc;&agrave;&thorn;&ograve;&uuml; &iuml;&oacute;&divide;&iacute;&icirc;&ntilde;&ograve;&yuml;&igrave;&egrave; &ntilde;&ograve;&icirc;&yuml;&divide;&icirc;&uml; &otilde;&acirc;&egrave;&euml;i.
&Acirc;i&auml;&ntilde;&ograve;&agrave;&iacute;&uuml; &igrave;i&aelig; &auml;&acirc;&icirc;&igrave;&agrave; &ntilde;&oacute;&ntilde;i&auml;&iacute;i&igrave;&egrave; &acirc;&oacute;&ccedil;&euml;&agrave;&igrave;&egrave;, &ograve;&agrave;&ecirc; &ntilde;&agrave;&igrave;&icirc;, &yuml;&ecirc; &acirc;i&auml;&ntilde;&ograve;&agrave;&iacute;&uuml; &igrave;i&aelig; &auml;&acirc;&icirc;&igrave;&agrave; &ntilde;&oacute;&ntilde;i&auml;&iacute;i&igrave;&egrave; &iuml;&oacute;&divide;&iacute;&icirc;&ntilde;&ograve;&yuml;&igrave;&egrave;, &auml;&icirc;&eth;i&acirc;&iacute;&thorn;&sup1; ∆x = l/n, i &ecirc;&icirc;&aelig;&iacute;&agrave; &iuml;&oacute;&divide;&iacute;i&ntilde;&ograve;&uuml; &ccedil;&iacute;&agrave;&otilde;&icirc;&auml;&egrave;&ograve;&uuml;&ntilde;&yuml; &iuml;&icirc;&ntilde;&aring;&eth;&aring;&auml;&egrave;&iacute;i &igrave;i&aelig; &auml;&acirc;&icirc;&igrave;&agrave; &ntilde;&oacute;&ntilde;i&auml;&iacute;i&igrave;&egrave; &acirc;&oacute;&ccedil;&euml;&agrave;&igrave;&egrave;. &Ocirc;&agrave;&ccedil;&agrave; &ecirc;&icirc;&euml;&egrave;&acirc;&agrave;&iacute;&uuml; &ograve;&icirc;&divide;&icirc;&ecirc;
&ntilde;&ograve;&eth;&oacute;&iacute;&egrave; &ccedil;&agrave;&euml;&aring;&aelig;&egrave;&ograve;&uuml; &acirc;i&auml; &ccedil;&iacute;&agrave;&ecirc;&oacute; sin(πnx/l):
&frac12;
ωn t + δn ,
&yuml;&ecirc;&ugrave;&icirc; sin(πnx/l) &gt; 0,
(3.27)
ϕn (t) =
ωn t + δn &plusmn; π, &yuml;&ecirc;&ugrave;&icirc; sin(πnx/l) &lt; 0.
&Icirc;&ntilde;&ecirc;i&euml;&uuml;&ecirc;&egrave; &ocirc;&oacute;&iacute;&ecirc;&ouml;i&yuml; sin(πnx/l) &ccedil;&igrave;i&iacute;&thorn;&sup1; &ccedil;&iacute;&agrave;&ecirc; (&quot;+&quot; &iacute;&agrave; &quot;−&quot; &agrave;&aacute;&icirc; &iacute;&agrave;&acirc;&iuml;&agrave;&ecirc;&egrave;) &euml;&egrave;&oslash;&aring; &oacute; &acirc;&oacute;&ccedil;&euml;&agrave;&otilde;, &ograve;&icirc; &ccedil;&atilde;i&auml;&iacute;&icirc; (3.27) &acirc;&ntilde;i &ograve;&icirc;&divide;&ecirc;&egrave; &ntilde;&ograve;&eth;&oacute;&iacute;&egrave; &igrave;i&aelig; &ntilde;&oacute;&ntilde;i&auml;&iacute;i&igrave;&egrave; &acirc;&oacute;&ccedil;&euml;&agrave;&igrave;&egrave;
xk , xk+1 &ecirc;&icirc;&euml;&egrave;&acirc;&agrave;&thorn;&ograve;&uuml;&ntilde;&yuml; &oacute; &ocirc;&agrave;&ccedil;i: &acirc; &iuml;&eth;&icirc;&ouml;&aring;&ntilde;i &ecirc;&icirc;&euml;&egrave;&acirc;&agrave;&iacute;&iacute;&yuml; &acirc;&icirc;&iacute;&egrave; &icirc;&auml;&iacute;&icirc;&divide;&agrave;&ntilde;&iacute;&icirc; &iuml;&eth;&icirc;&otilde;&icirc;&auml;&yuml;&ograve;&uuml; &iuml;&icirc;&euml;&icirc;&aelig;&aring;&iacute;&iacute;&yuml; &eth;i&acirc;&iacute;&icirc;&acirc;&agrave;&atilde;&egrave; i &icirc;&auml;&iacute;&icirc;&divide;&agrave;&ntilde;&iacute;&icirc; &auml;&icirc;&ntilde;&yuml;&atilde;&agrave;&thorn;&ograve;&uuml; (&ecirc;&icirc;&aelig;&iacute;&agrave; &ograve;&icirc;&divide;&ecirc;&agrave; &ntilde;&acirc;&icirc;&atilde;&icirc;)
&igrave;&agrave;&ecirc;&ntilde;&egrave;&igrave;&agrave;&euml;&uuml;&iacute;&icirc;&atilde;&icirc; &acirc;i&auml;&otilde;&egrave;&euml;&aring;&iacute;&iacute;&yuml; &acirc;i&auml; &iuml;&icirc;&euml;&icirc;&aelig;&aring;&iacute;&iacute;&yuml; &eth;i&acirc;&iacute;&icirc;&acirc;&agrave;&atilde;&egrave;.
25
&Ocirc;&agrave;&ccedil;&egrave; &ecirc;&icirc;&euml;&egrave;&acirc;&agrave;&iacute;&uuml; &ograve;&icirc;&divide;&icirc;&ecirc;, &ugrave;&icirc; &eth;&icirc;&ccedil;&ograve;&agrave;&oslash;&icirc;&acirc;&agrave;&iacute;i &iuml;&icirc; &eth;i&ccedil;&iacute;i &aacute;&icirc;&ecirc;&egrave; &acirc;&oacute;&ccedil;&euml;&agrave;, &acirc;i&auml;&eth;i&ccedil;&iacute;&yuml;&thorn;&ograve;&uuml;&ntilde;&yuml;, &yuml;&ecirc; &acirc;&egrave;&auml;&iacute;&icirc; &ccedil; (3.27), &iacute;&agrave; ∆ϕ = &plusmn;π , &ograve;&icirc;&aacute;&ograve;&icirc; &ograve;&agrave;&ecirc;i &ograve;&icirc;&divide;&ecirc;&egrave; &ecirc;&icirc;&euml;&egrave;&acirc;&agrave;&thorn;&ograve;&uuml;&ntilde;&yuml;
&oacute; &iuml;&eth;&icirc;&ograve;&egrave;&ocirc;&agrave;&ccedil;i. &Acirc;&icirc;&iacute;&egrave; &icirc;&auml;&iacute;&icirc;&divide;&agrave;&ntilde;&iacute;&icirc; &iuml;&eth;&icirc;&otilde;&icirc;&auml;&yuml;&ograve;&uuml; &iuml;&icirc;&euml;&icirc;&aelig;&aring;&iacute;&iacute;&yuml; &eth;i&acirc;&iacute;&icirc;&acirc;&agrave;&atilde;&egrave; (&ccedil; &iuml;&eth;&icirc;&ograve;&egrave;&euml;&aring;&aelig;&iacute;&egrave;&igrave;&egrave; &iacute;&agrave;&iuml;&eth;&yuml;&igrave;&agrave;&igrave;&egrave; &oslash;&acirc;&egrave;&auml;&ecirc;&icirc;&ntilde;&ograve;&aring;&eacute;) i &icirc;&auml;&iacute;&icirc;&divide;&agrave;&ntilde;&iacute;&icirc; &auml;&icirc;&ntilde;&yuml;&atilde;&agrave;&thorn;&ograve;&uuml; &igrave;&agrave;&ecirc;&ntilde;&egrave;&igrave;&agrave;&euml;&uuml;&iacute;&icirc;&atilde;&icirc;
&acirc;i&auml;&otilde;&egrave;&euml;&aring;&iacute;&iacute;&yuml; (&agrave;&euml;&aring; &iuml;&icirc; &eth;i&ccedil;&iacute;i &aacute;&icirc;&ecirc;&egrave;) &acirc;i&auml; &iuml;&icirc;&euml;&icirc;&aelig;&aring;&iacute;&iacute;&yuml; &eth;i&acirc;&iacute;&icirc;&acirc;&agrave;&atilde;&egrave;. &Iuml;&eth;&icirc;&ocirc;i&euml;&uuml; &ntilde;&ograve;&eth;&oacute;&iacute;&egrave;,
&ugrave;&icirc; &ecirc;&icirc;&euml;&egrave;&acirc;&agrave;&sup1;&ograve;&uuml;&ntilde;&yuml; &ccedil;&agrave; &ccedil;&agrave;&ecirc;&icirc;&iacute;&icirc;&igrave; (3.25), &acirc; &aacute;&oacute;&auml;&uuml;-&yuml;&ecirc;&egrave;&eacute; &igrave;&icirc;&igrave;&aring;&iacute;&ograve; &divide;&agrave;&ntilde;&oacute; &yuml;&acirc;&euml;&yuml;&sup1; &ntilde;&icirc;&aacute;&icirc;&thorn;
&ntilde;&egrave;&iacute;&oacute;&ntilde;&icirc;&uml;&auml;&oacute;:
&sup3; πn &acute;
un (x, t) = γn (t) &middot; sin
x ,
(3.28)
l
&auml;&aring; γn (t) = αn &middot; cos(ωn t + δn ). &Iuml;i&auml; &divide;&agrave;&ntilde; &ecirc;&icirc;&euml;&egrave;&acirc;&agrave;&iacute;&iacute;&yuml; &acirc;i&auml;&aacute;&oacute;&acirc;&agrave;&sup1;&ograve;&uuml;&ntilde;&yuml; &iuml;&aring;&eth;i&icirc;&auml;&egrave;&divide;&iacute;&aring;
&iuml;&aring;&eth;&aring;&ograve;&acirc;&icirc;&eth;&aring;&iacute;&iacute;&yuml; &iuml;&icirc;&ograve;&aring;&iacute;&ouml;i&agrave;&euml;&uuml;&iacute;&icirc;&uml; &aring;&iacute;&aring;&eth;&atilde;i&uml; &ntilde;&ograve;&eth;&oacute;&iacute;&egrave; &acirc; &ecirc;i&iacute;&aring;&ograve;&egrave;&divide;&iacute;&oacute; i &iacute;&agrave;&acirc;&iuml;&agrave;&ecirc;&egrave;. &Acirc; &igrave;&icirc;&igrave;&aring;&iacute;&ograve; &iacute;&agrave;&eacute;&aacute;i&euml;&uuml;&oslash;&icirc;&atilde;&icirc; &acirc;i&auml;&otilde;&egrave;&euml;&aring;&iacute;&iacute;&yuml; &ntilde;&ograve;&eth;&oacute;&iacute;&egrave; &acirc;i&auml; &iuml;&icirc;&euml;&icirc;&aelig;&aring;&iacute;&iacute;&yuml; &eth;i&acirc;&iacute;&icirc;&acirc;&agrave;&atilde;&egrave; &ecirc;i&iacute;&aring;&ograve;&egrave;&divide;&iacute;&agrave;
&aring;&iacute;&aring;&eth;&atilde;i&yuml; &auml;&icirc;&eth;i&acirc;&iacute;&thorn;&sup1; &iacute;&oacute;&euml;&thorn;, &agrave; &iuml;&icirc;&ograve;&aring;&iacute;&ouml;i&agrave;&euml;&uuml;&iacute;&agrave; &auml;&icirc;&ntilde;&yuml;&atilde;&agrave;&sup1; &ntilde;&acirc;&icirc;&atilde;&icirc; &iacute;&agrave;&eacute;&aacute;i&euml;&uuml;&oslash;&icirc;&atilde;&icirc; &ccedil;&iacute;&agrave;&divide;&aring;&iacute;&iacute;&yuml;. &Oslash;&acirc;&egrave;&auml;&ecirc;&icirc;&ntilde;&ograve;i &acirc;&ntilde;i&otilde; &ograve;&icirc;&divide;&icirc;&ecirc; &ntilde;&ograve;&eth;&oacute;&iacute;&egrave; &acirc; &ograve;&agrave;&ecirc;&egrave;&eacute; &igrave;&icirc;&igrave;&aring;&iacute;&ograve; &auml;&icirc;&eth;i&acirc;&iacute;&thorn;&thorn;&ograve;&uuml; &iacute;&oacute;&euml;&thorn;.
&Iuml;&eth;&egrave; &iuml;&eth;&icirc;&otilde;&icirc;&auml;&aelig;&aring;&iacute;&iacute;i &ograve;&icirc;&divide;&icirc;&ecirc; &ntilde;&ograve;&eth;&oacute;&iacute;&egrave; &divide;&aring;&eth;&aring;&ccedil; &iuml;&icirc;&euml;&icirc;&aelig;&aring;&iacute;&iacute;&yuml; &eth;i&acirc;&iacute;&icirc;&acirc;&agrave;&atilde;&egrave; &ecirc;i&iacute;&aring;&ograve;&egrave;&divide;&iacute;&agrave;
&aring;&iacute;&aring;&eth;&atilde;i&yuml; &auml;&icirc;&ntilde;&yuml;&atilde;&agrave;&sup1; &igrave;&agrave;&ecirc;&ntilde;&egrave;&igrave;&agrave;&euml;&uuml;&iacute;&icirc;&atilde;&icirc; &ccedil;&iacute;&agrave;&divide;&aring;&iacute;&iacute;&yuml;, &agrave; &iuml;&icirc;&ograve;&aring;&iacute;&ouml;i&agrave;&euml;&uuml;&iacute;&agrave; &aring;&iacute;&aring;&eth;&atilde;i&yuml; &ntilde;&ograve;&agrave;&sup1; &igrave;i&iacute;i&igrave;&agrave;&euml;&uuml;&iacute;&icirc;&thorn;.
&Ccedil;&iacute;&agrave;&eacute;&auml;&aring;&igrave;&icirc; &iuml;&icirc;&acirc;&iacute;&oacute; &aring;&iacute;&aring;&eth;&atilde;i&thorn; &ntilde;&ograve;&eth;&oacute;&iacute;&egrave;, &ugrave;&icirc; &ecirc;&icirc;&euml;&egrave;&acirc;&agrave;&sup1;&ograve;&uuml;&ntilde;&yuml; &ccedil;&agrave; &ccedil;&agrave;&ecirc;&icirc;&iacute;&icirc;&igrave; (3.25). &Ccedil;&agrave;
&iacute;&oacute;&euml;&uuml;&icirc;&acirc;&egrave;&eacute; &eth;i&acirc;&aring;&iacute;&uuml; &iuml;&icirc;&ograve;&aring;&iacute;&ouml;i&agrave;&euml;&uuml;&iacute;&icirc;&uml; &aring;&iacute;&aring;&eth;&atilde;i&uml; &acirc;i&ccedil;&uuml;&igrave;&aring;&igrave;&icirc; &uml;&uml; &igrave;i&iacute;i&igrave;&agrave;&euml;&uuml;&iacute;&aring; &ccedil;&iacute;&agrave;&divide;&aring;&iacute;&iacute;&yuml;.
&Ograve;&icirc;&auml;i &iuml;&icirc;&acirc;&iacute;&agrave; &aring;&iacute;&aring;&eth;&atilde;i&yuml; &ecirc;&icirc;&euml;&egrave;&acirc;&agrave;&iacute;&uuml; &ntilde;&ograve;&eth;&oacute;&iacute;&egrave; &auml;&icirc;&eth;i&acirc;&iacute;&thorn;&sup1; &igrave;&agrave;&ecirc;&ntilde;&egrave;&igrave;&agrave;&euml;&uuml;&iacute;&icirc;&igrave;&oacute; &ccedil;&iacute;&agrave;&divide;&aring;&iacute;&iacute;&thorn;
&ecirc;i&iacute;&aring;&ograve;&egrave;&divide;&iacute;&icirc;&uml; &aring;&iacute;&aring;&eth;&atilde;i&uml;: E = Ekmax .
&Ecirc;i&iacute;&aring;&ograve;&egrave;&divide;&iacute;&agrave; &aring;&iacute;&aring;&eth;&atilde;i&yuml; &ntilde;&ograve;&eth;&oacute;&iacute;&egrave; &icirc;&aacute;&divide;&egrave;&ntilde;&euml;&thorn;&sup1;&ograve;&uuml;&ntilde;&yuml; &yuml;&ecirc; &ntilde;&oacute;&igrave;&agrave; (i&iacute;&ograve;&aring;&atilde;&eth;&agrave;&euml;) &ecirc;i&iacute;&aring;&ograve;&egrave;&divide;&iacute;&egrave;&otilde;
&aring;&iacute;&aring;&eth;&atilde;i&eacute; &icirc;&ecirc;&eth;&aring;&igrave;&egrave;&otilde; &iacute;&aring;&ntilde;&ecirc;i&iacute;&divide;&aring;&iacute;&iacute;&icirc;-&igrave;&agrave;&euml;&egrave;&otilde; &aring;&euml;&aring;&igrave;&aring;&iacute;&ograve;i&acirc; &ntilde;&ograve;&eth;&oacute;&iacute;&egrave;
Z l 2
v (x)
Ek =
ρ
dx,
2
0
&auml;&aring; ρ &euml;i&iacute;i&eacute;&iacute;&agrave; &atilde;&oacute;&ntilde;&ograve;&egrave;&iacute;&agrave; &ntilde;&ograve;&eth;&oacute;&iacute;&egrave; (ρdx &igrave;&agrave;&ntilde;&agrave; &aring;&euml;&aring;&igrave;&aring;&iacute;&ograve;&agrave; &ntilde;&ograve;&eth;&oacute;&iacute;&egrave; &auml;&icirc;&acirc;&aelig;&egrave;&iacute;&icirc;&thorn;
dx).
&Oslash;&acirc;&egrave;&auml;&ecirc;i&ntilde;&ograve;&uuml; &ograve;&icirc;&divide;&icirc;&ecirc; &ntilde;&ograve;&eth;&oacute;&iacute;&egrave;
&sup3; πn &acute;
∂un
v=
= −αn ωn sin
x sin(ωn t + δn ),
∂t
l
&agrave; &igrave;&agrave;&ecirc;&ntilde;&egrave;&igrave;&agrave;&euml;&uuml;&iacute;&aring; &ccedil;&iacute;&agrave;&divide;&aring;&iacute;&iacute;&yuml; &oslash;&acirc;&egrave;&auml;&ecirc;&icirc;&ntilde;&ograve;i (&ccedil;&agrave; &agrave;&aacute;&ntilde;&icirc;&euml;&thorn;&ograve;&iacute;&icirc;&thorn; &acirc;&aring;&euml;&egrave;&divide;&egrave;&iacute;&icirc;&thorn;) &auml;&icirc;&ntilde;&yuml;&atilde;&agrave;&sup1;&iexcl;
&cent;
&ograve;&uuml;&ntilde;&yuml; &acirc; &igrave;&icirc;&igrave;&aring;&iacute;&ograve; &divide;&agrave;&ntilde;&oacute;, &ecirc;&icirc;&euml;&egrave; | sin(ωn t+δn )| = 1. &Ograve;&icirc;&auml;i |vmax (x)| = αn ωn | sin πn
x
|,
l
&agrave; &igrave;&agrave;&ecirc;&ntilde;&egrave;&igrave;&agrave;&euml;&uuml;&iacute;&aring; &ccedil;&iacute;&agrave;&divide;&aring;&iacute;&iacute;&yuml; &ecirc;i&iacute;&aring;&ograve;&egrave;&divide;&iacute;&icirc;&uml; &aring;&iacute;&aring;&eth;&atilde;i&uml;
Z
Z l 2
ραn2 ωn2 l 2 &sup3; πn &acute;
vmax (x)
max
dx =
sin
x dx =
Ek =
ρ
2
2
l
0
0
ραn2 ωn2 l
M αn2 ωn2
M (A2n + Bn2 )ωn2
=
=
=
,
2 2
4
4
26
&auml;&aring; M = ρl &igrave;&agrave;&ntilde;&agrave; &ntilde;&ograve;&eth;&oacute;&iacute;&egrave;.
&Ograve;&agrave;&ecirc;&egrave;&igrave; &divide;&egrave;&iacute;&icirc;&igrave;, &iuml;&icirc;&acirc;&iacute;&agrave; &aring;&iacute;&aring;&eth;&atilde;i&yuml; &ecirc;&icirc;&euml;&egrave;&acirc;&agrave;&iacute;&uuml; &ntilde;&ograve;&eth;&oacute;&iacute;&egrave; E = Ekmax = 41 M (A2n +
Bn2 )ωn2 &iuml;&eth;&icirc;&iuml;&icirc;&eth;&ouml;i&eacute;&iacute;&agrave; &uml;&uml; &igrave;&agrave;&ntilde;i, &ecirc;&acirc;&agrave;&auml;&eth;&agrave;&ograve;&oacute; &agrave;&igrave;&iuml;&euml;i&ograve;&oacute;&auml;&egrave; &ograve;&agrave; &ecirc;&acirc;&agrave;&auml;&eth;&agrave;&ograve;&oacute; &divide;&agrave;&ntilde;&ograve;&icirc;&ograve;&egrave;.
&Icirc;&ecirc;&eth;&aring;&igrave;&oacute; &ntilde;&ograve;&icirc;&yuml;&divide;&oacute; &otilde;&acirc;&egrave;&euml;&thorn; &ccedil; &iuml;&aring;&acirc;&iacute;&icirc;&thorn; &divide;&agrave;&ntilde;&ograve;&icirc;&ograve;&icirc;&thorn; ωn &iacute;&agrave;&ccedil;&egrave;&acirc;&agrave;&thorn;&ograve;&uuml; &atilde;&agrave;&eth;&igrave;&icirc;&iacute;i&ecirc;&icirc;&thorn;.
&Icirc;&ograve;&aelig;&aring;, &eth;&icirc;&ccedil;&acirc;'&yuml;&ccedil;&icirc;&ecirc; &oacute; &acirc;&egrave;&atilde;&euml;&yuml;&auml;i (3.19) &ccedil;&agrave;&auml;&agrave;&divide;i &iuml;&eth;&icirc; &ecirc;&icirc;&euml;&egrave;&acirc;&agrave;&iacute;&iacute;&yuml; &ntilde;&ograve;&eth;&oacute;&iacute;&egrave; &yuml;&acirc;&euml;&yuml;&sup1; &ntilde;&icirc;&aacute;&icirc;&thorn; &ntilde;&oacute;&iuml;&aring;&eth;&iuml;&icirc;&ccedil;&egrave;&ouml;i&thorn; &ntilde;&ograve;&icirc;&yuml;&divide;&egrave;&otilde; &otilde;&acirc;&egrave;&euml;&uuml;, &agrave;&aacute;&icirc;, i&iacute;&agrave;&ecirc;&oslash;&aring; &ecirc;&agrave;&aelig;&oacute;&divide;&egrave;, &ntilde;&oacute;&iuml;&aring;&eth;&iuml;&icirc;&ccedil;&egrave;&ouml;i&thorn; &atilde;&agrave;&eth;&igrave;&icirc;&iacute;i&ecirc;. &times;&agrave;&ntilde;&ograve;&icirc;&ograve;&egrave; ωn = πna
l , &ugrave;&icirc; &acirc;i&auml;&iuml;&icirc;&acirc;i&auml;&agrave;&thorn;&ograve;&uuml; &icirc;&ecirc;&eth;&aring;&igrave;&egrave;&igrave; &atilde;&agrave;&eth;&igrave;&icirc;&iacute;i&ecirc;&agrave;&igrave;, &iacute;&agrave;&ccedil;&egrave;&acirc;&agrave;&thorn;&ograve;&uuml;&ntilde;&yuml; &acirc;&euml;&agrave;&ntilde;&iacute;&egrave;&igrave;&egrave; &divide;&agrave;&ntilde;&ograve;&icirc;&ograve;&agrave;&igrave;&egrave; &ecirc;&icirc;&euml;&egrave;&acirc;&agrave;&iacute;&uuml;
p &ntilde;&ograve;&eth;&oacute;&iacute;&egrave;. &Icirc;&ntilde;&ecirc;i&euml;&uuml;&ecirc;&egrave; &iuml;&agrave;&eth;&agrave;&igrave;&aring;&ograve;&eth; a, &ugrave;&icirc;
&acirc;&otilde;&icirc;&auml;&egrave;&ograve;&uuml; &acirc; &eth;i&acirc;&iacute;&yuml;&iacute;&iacute;&yuml; (3.1), &auml;&icirc;&eth;i&acirc;&iacute;&thorn;&sup1; T /ρ, &ograve;&icirc;
s
ωn =
πn
l
T
, (n = 1, 2, 3, ...).
ρ
(3.29)
&Ecirc;&icirc;&euml;&egrave;&acirc;&agrave;&iacute;&iacute;&yuml; &ntilde;&ograve;&eth;&oacute;&iacute;&egrave; &ntilde;&iuml;&eth;&egrave;&eacute;&igrave;&agrave;&sup1;&ograve;&uuml;&ntilde;&yuml; &iacute;&agrave;&igrave;&egrave; &ccedil;&agrave;&acirc;&auml;&yuml;&ecirc;&egrave; &ccedil;&acirc;&oacute;&ecirc;&oacute;, &yuml;&ecirc;&egrave;&eacute; &acirc;&egrave;&auml;&agrave;&sup1; &ntilde;&ograve;&eth;&oacute;&iacute;&agrave;
&oacute; &acirc;&egrave;&atilde;&euml;&yuml;&auml;i &iacute;&agrave;&ecirc;&euml;&agrave;&auml;&agrave;&iacute;&iacute;&yuml; &iuml;&eth;&icirc;&ntilde;&ograve;&egrave;&otilde; &ograve;&icirc;&iacute;i&acirc;, &ugrave;&icirc; &acirc;i&auml;&iuml;&icirc;&acirc;i&auml;&agrave;&thorn;&ograve;&uuml; &icirc;&ecirc;&eth;&aring;&igrave;&egrave;&igrave; &ntilde;&ograve;&icirc;&yuml;&divide;&egrave;&igrave;
&otilde;&acirc;&egrave;&euml;&yuml;&igrave;. &Acirc;&egrave;&ntilde;&icirc;&ograve;&agrave; &ograve;&icirc;&iacute;&oacute; &ccedil;&agrave;&euml;&aring;&aelig;&egrave;&ograve;&uuml; &acirc;i&auml; &divide;&agrave;&ntilde;&ograve;&icirc;&ograve;&egrave; &ecirc;&icirc;&euml;&egrave;&acirc;&agrave;&iacute;&uuml;, &agrave; &ntilde;&egrave;&euml;&agrave; &ograve;&icirc;&iacute;&oacute; &acirc;&egrave;&ccedil;&iacute;&agrave;&divide;&agrave;&sup1;&ograve;&uuml;&ntilde;&yuml; &aring;&iacute;&aring;&eth;&atilde;i&sup1;&thorn; &ntilde;&ograve;&icirc;&yuml;&divide;&icirc;&uml; &otilde;&acirc;&egrave;&euml;i, &agrave; &icirc;&ograve;&aelig;&aring;, &agrave;&igrave;&iuml;&euml;i&ograve;&oacute;&auml;&icirc;&thorn; &ecirc;&icirc;&euml;&egrave;&acirc;&agrave;&iacute;&uuml;.
&Ccedil;&atilde;i&auml;&iacute;&icirc; (3.29) &ntilde;&agrave;&igrave;&egrave;&eacute; &iacute;&egrave;&ccedil;&uuml;&ecirc;&egrave;&eacute; &ograve;&icirc;&iacute;, &yuml;&ecirc;&egrave;&eacute; &igrave;&icirc;&aelig;&aring; &oacute;&ograve;&acirc;&icirc;&eth;&thorn;&acirc;&agrave;&ograve;&egrave; &ntilde;&ograve;&eth;&oacute;&iacute;&agrave;, &acirc;i&auml;&iuml;&icirc;&acirc;i&auml;&agrave;&sup1; &iacute;&agrave;&eacute;&igrave;&aring;&iacute;&oslash;i&eacute; &ccedil; &oacute;&ntilde;i&otilde; &igrave;&icirc;&aelig;&euml;&egrave;&acirc;&egrave;&otilde; &acirc;&euml;&agrave;&ntilde;&iacute;&egrave;&otilde; &divide;&agrave;&ntilde;&ograve;&icirc;&ograve; &ntilde;&ograve;&eth;&oacute;&iacute;&egrave;:
s
ω1 =
π
l
T
,
ρ
(3.30)
i &iacute;&agrave;&ccedil;&egrave;&acirc;&agrave;&sup1;&ograve;&uuml;&ntilde;&yuml; &icirc;&ntilde;&iacute;&icirc;&acirc;&iacute;&egrave;&igrave; &ograve;&icirc;&iacute;&icirc;&igrave; &ntilde;&ograve;&eth;&oacute;&iacute;&egrave;. &ETH;&aring;&oslash;&ograve;&agrave; &ograve;&icirc;&iacute;i&acirc;, &divide;&agrave;&ntilde;&ograve;&icirc;&ograve;&egrave; &yuml;&ecirc;&egrave;&otilde; &sup1; &ecirc;&eth;&agrave;&ograve;&iacute;&egrave;&igrave;&egrave; ω1 , &iacute;&agrave;&ccedil;&egrave;&acirc;&agrave;&thorn;&ograve;&uuml;&ntilde;&yuml; &icirc;&aacute;&aring;&eth;&ograve;&icirc;&iacute;&agrave;&igrave;&egrave;. &Ograve;&aring;&igrave;&aacute;&eth; &ccedil;&acirc;&oacute;&ecirc;&oacute; &ccedil;&agrave;&euml;&aring;&aelig;&egrave;&ograve;&uuml; &acirc;i&auml; &iacute;&agrave;&yuml;&acirc;&iacute;&icirc;&ntilde;&ograve;i, &iuml;&icirc;&eth;&yuml;&auml; &ccedil; &icirc;&ntilde;&iacute;&icirc;&acirc;&iacute;&egrave;&igrave; &ograve;&icirc;&iacute;&icirc;&igrave;, &icirc;&aacute;&aring;&eth;&ograve;&icirc;&iacute;i&acirc;, &agrave; &ograve;&agrave;&ecirc;&icirc;&aelig; &acirc;i&auml; &eth;&icirc;&ccedil;&iuml;&icirc;&auml;i&euml;&oacute; &aring;&iacute;&aring;&eth;&atilde;i&uml; &iuml;&icirc;
&atilde;&agrave;&eth;&igrave;&icirc;&iacute;i&ecirc;&agrave;&otilde;. &szlig;&ecirc; &acirc;&egrave;&auml;&iacute;&icirc; &ccedil; (3.30), &divide;&agrave;&ntilde;&ograve;&icirc;&ograve;&agrave; &icirc;&ntilde;&iacute;&icirc;&acirc;&iacute;&icirc;&atilde;&icirc; &ograve;&icirc;&iacute;&oacute; (&agrave; &ograve;&agrave;&ecirc;&icirc;&aelig; &icirc;&aacute;&aring;&eth;&ograve;&icirc;&iacute;i&acirc;) &ccedil;&agrave;&euml;&aring;&aelig;&egrave;&ograve;&uuml; &acirc;i&auml; &auml;&icirc;&acirc;&aelig;&egrave;&iacute;&egrave; &ntilde;&ograve;&eth;&oacute;&iacute;&egrave;, &euml;i&iacute;i&eacute;&iacute;&icirc;&uml; &atilde;&oacute;&ntilde;&ograve;&egrave;&iacute;&egrave; (&agrave;&aacute;&icirc; &igrave;&agrave;&ntilde;&egrave;) &ograve;&agrave; &acirc;i&auml;
&ntilde;&egrave;&euml;&egrave; &iacute;&agrave;&ograve;&yuml;&atilde;&oacute; &ntilde;&ograve;&eth;&oacute;&iacute;&egrave;.
&Iacute;&agrave;&iuml;&eth;&egrave;&ecirc;i&iacute;&ouml;i &ccedil;&agrave;&oacute;&acirc;&agrave;&aelig;&egrave;&igrave;&icirc;, &ugrave;&icirc; &ccedil;&agrave;&auml;&agrave;&divide;i &iuml;&eth;&icirc; &acirc;i&euml;&uuml;&iacute;i &igrave;&aring;&otilde;&agrave;&iacute;i&divide;&iacute;i &ecirc;&icirc;&euml;&egrave;&acirc;&agrave;&iacute;&iacute;&yuml; (&iuml;&icirc;&acirc;&ccedil;&auml;&icirc;&acirc;&aelig;&iacute;i) &acirc; &iuml;&eth;&oacute;&aelig;&iacute;&uuml;&icirc;&igrave;&oacute; &ntilde;&ograve;&aring;&eth;&aelig;&iacute;i, &ecirc;&icirc;&euml;&egrave;&acirc;&agrave;&iacute;&iacute;&yuml; &iuml;&icirc;&acirc;i&ograve;&eth;&yuml; &acirc; &ograve;&eth;&oacute;&aacute;&ouml;i, &ecirc;&icirc;&euml;&egrave;&acirc;&agrave;&iacute;&iacute;&yuml;
&ntilde;&ograve;&eth;&oacute;&igrave;&oacute; i &iacute;&agrave;&iuml;&eth;&oacute;&atilde;&egrave; &acirc; &iuml;&eth;&icirc;&acirc;i&auml;&iacute;&egrave;&ecirc;&agrave;&otilde;, &agrave;&aacute;&icirc; &iuml;&eth;&icirc; &aring;&euml;&aring;&ecirc;&ograve;&eth;&icirc;&igrave;&agrave;&atilde;&iacute;i&ograve;&iacute;i &ecirc;&icirc;&euml;&egrave;&acirc;&agrave;&iacute;&iacute;&yuml; &acirc;
&eth;&aring;&ccedil;&icirc;&iacute;&agrave;&ograve;&icirc;&eth;&agrave;&otilde;, &agrave;&aacute;&ntilde;&icirc;&euml;&thorn;&ograve;&iacute;&icirc; i&auml;&aring;&iacute;&ograve;&egrave;&divide;&iacute;i &ugrave;&icirc;&eacute;&iacute;&icirc; &eth;&icirc;&ccedil;&atilde;&euml;&yuml;&iacute;&oacute;&ograve;i&eacute; &ccedil;&agrave;&auml;&agrave;&divide;i &iuml;&eth;&icirc; &iuml;&icirc;&iuml;&aring;&eth;&aring;&divide;&iacute;i &ecirc;&icirc;&euml;&egrave;&acirc;&agrave;&iacute;&iacute;&yuml; &ntilde;&ograve;&eth;&oacute;&iacute;&egrave;. &Icirc;&ecirc;&eth;&aring;&igrave;i &ocirc;&eth;&agrave;&atilde;&igrave;&aring;&iacute;&ograve;&egrave; &auml;&agrave;&iacute;&icirc;&uml; &ccedil;&agrave;&auml;&agrave;&divide;i &iuml;&icirc;&acirc;&ograve;&icirc;&eth;&thorn;&thorn;&ograve;&uuml;&ntilde;&yuml;
&oacute; &ntilde;&agrave;&igrave;&egrave;&otilde; &eth;i&ccedil;&iacute;&icirc;&igrave;&agrave;&iacute;i&ograve;&iacute;&egrave;&otilde; &ccedil;&agrave;&auml;&agrave;&divide;&agrave;&otilde; &ograve;&aring;&icirc;&eth;&aring;&ograve;&egrave;&divide;&iacute;&icirc;&uml; &ocirc;i&ccedil;&egrave;&ecirc;&egrave;. &Ccedil;&icirc;&ecirc;&eth;&aring;&igrave;&agrave;, &acirc; &ecirc;&acirc;&agrave;&iacute;&ograve;&icirc;&acirc;i&eacute; &igrave;&aring;&otilde;&agrave;&iacute;i&ouml;i &auml;&egrave;&ntilde;&ecirc;&eth;&aring;&ograve;&iacute;i &ccedil;&iacute;&agrave;&divide;&aring;&iacute;&iacute;&yuml; &aring;&iacute;&aring;&eth;&atilde;i&uml;, &igrave;&icirc;&igrave;&aring;&iacute;&ograve;&oacute; &ecirc;i&euml;&uuml;&ecirc;&icirc;&ntilde;&ograve;i &eth;&oacute;&otilde;&oacute;, &ograve;&icirc;&ugrave;&icirc;
&acirc;&egrave;&iacute;&egrave;&ecirc;&agrave;&thorn;&ograve;&uuml; &acirc; &ecirc;&acirc;&agrave;&iacute;&ograve;&icirc;&acirc;i&eacute; &igrave;&aring;&otilde;&agrave;&iacute;i&ouml;i &yuml;&ecirc; &acirc;&euml;&agrave;&ntilde;&iacute;i &ccedil;&iacute;&agrave;&divide;&aring;&iacute;&iacute;&yuml; &acirc;i&auml;&iuml;&icirc;&acirc;i&auml;&iacute;&egrave;&otilde; &ccedil;&agrave;&auml;&agrave;&divide;
&Oslash;&ograve;&oacute;&eth;&igrave;&agrave;-&Euml;i&oacute;&acirc;i&euml;&euml;&yuml;.
27
&Ccedil;&agrave;&euml;&aring;&aelig;&iacute;i&ntilde;&ograve;&uuml; &eth;&icirc;&ccedil;&acirc;'&yuml;&ccedil;&ecirc;&oacute; &ccedil;&agrave;&auml;&agrave;&divide;i &iuml;&eth;&icirc; &ecirc;&icirc;&euml;&egrave;&acirc;&agrave;&iacute;&iacute;&yuml; &ntilde;&ograve;&eth;&oacute;&iacute;&egrave; &acirc;i&auml; &acirc;&egrave;&aacute;&icirc;&eth;&oacute; &ecirc;&eth;&agrave;&eacute;&icirc;&acirc;&egrave;&otilde; &oacute;&igrave;&icirc;&acirc;.
&Ccedil;&igrave;i&iacute;&egrave;&igrave;&icirc; &ograve;&aring;&iuml;&aring;&eth; &ecirc;&eth;&agrave;&eacute;&icirc;&acirc;&oacute; &oacute;&igrave;&icirc;&acirc;&oacute; &iacute;&agrave; &icirc;&auml;&iacute;&icirc;&igrave;&oacute; &ccedil; &ecirc;i&iacute;&ouml;i&acirc; &ntilde;&ograve;&eth;&oacute;&iacute;&egrave;: &acirc;&acirc;&agrave;&aelig;&agrave;&sup1;&igrave;&icirc;,
&ugrave;&icirc; &euml;i&acirc;&egrave;&eacute; &ecirc;i&iacute;&aring;&ouml;&uuml; &ntilde;&ograve;&eth;&oacute;&iacute;&egrave; &ccedil;&agrave;&euml;&egrave;&oslash;&agrave;&sup1;&ograve;&uuml;&ntilde;&yuml; &ccedil;&agrave;&ecirc;&eth;i&iuml;&euml;&aring;&iacute;&egrave;&igrave;, &agrave; &iuml;&eth;&agrave;&acirc;&egrave;&eacute; &ecirc;i&iacute;&aring;&ouml;&uuml; &acirc;i&euml;&uuml;&iacute;&egrave;&igrave;. &Ntilde;&euml;&icirc;&acirc;&icirc; &quot;&acirc;i&euml;&uuml;&iacute;&egrave;&eacute;&quot; &iacute;&aring; &ntilde;&euml;i&auml; &ograve;&oacute;&ograve; &eth;&icirc;&ccedil;&oacute;&igrave;i&ograve;&egrave; &aacute;&oacute;&ecirc;&acirc;&agrave;&euml;&uuml;&iacute;&icirc;. &Auml;i&eacute;&ntilde;&iacute;&icirc;, &auml;&euml;&yuml; &ograve;&icirc;&atilde;&icirc;,
&ugrave;&icirc;&aacute; &ntilde;&ograve;&eth;&oacute;&iacute;&agrave; &igrave;&icirc;&atilde;&euml;&agrave; &ecirc;&icirc;&euml;&egrave;&acirc;&agrave;&ograve;&egrave;&ntilde;&uuml;, &ograve;&eth;&aring;&aacute;&agrave; &ccedil;&agrave;&aacute;&aring;&ccedil;&iuml;&aring;&divide;&egrave;&ograve;&egrave; &iacute;&agrave;&ograve;&yuml;&atilde; &ntilde;&ograve;&eth;&oacute;&iacute;&egrave; &auml;&aring;&yuml;&ecirc;&icirc;&thorn;
&ntilde;&egrave;&euml;&icirc;&thorn; T , &ugrave;&icirc; &iuml;&eth;&egrave;&ecirc;&euml;&agrave;&auml;&aring;&iacute;&agrave; &auml;&icirc; &ecirc;i&iacute;&ouml;&yuml; &ntilde;&ograve;&eth;&oacute;&iacute;&egrave; &acirc; &iuml;&icirc;&acirc;&ccedil;&auml;&icirc;&acirc;&aelig;&iacute;&icirc;&igrave;&oacute; &iacute;&agrave;&iuml;&eth;&yuml;&igrave;&ecirc;&oacute;. &Icirc;&ograve;&aelig;&aring;, &quot;&acirc;i&euml;&uuml;&iacute;&egrave;&eacute;&quot; &acirc; &auml;&agrave;&iacute;&icirc;&igrave;&oacute; &acirc;&egrave;&iuml;&agrave;&auml;&ecirc;&oacute; &icirc;&ccedil;&iacute;&agrave;&divide;&agrave;&sup1;, &ugrave;&icirc; &ecirc;i&iacute;&aring;&ouml;&uuml; &ntilde;&ograve;&eth;&oacute;&iacute;&egrave; &igrave;&icirc;&aelig;&aring; &acirc;i&euml;&uuml;&iacute;&icirc;
&ccedil;&igrave;i&ugrave;&oacute;&acirc;&agrave;&ograve;&egrave;&ntilde;&uuml; &acirc; &iuml;&icirc;&iuml;&aring;&eth;&aring;&divide;&iacute;&icirc;&igrave;&oacute; &iuml;&icirc; &acirc;i&auml;&iacute;&icirc;&oslash;&aring;&iacute;&iacute;&thorn; &auml;&icirc; &iacute;&aring;&eacute;&ograve;&eth;&agrave;&euml;&uuml;&iacute;&icirc;&atilde;&icirc; &iuml;&icirc;&euml;&icirc;&aelig;&aring;&iacute;&iacute;&yuml;
&ntilde;&ograve;&eth;&oacute;&iacute;&egrave; &iacute;&agrave;&iuml;&eth;&yuml;&igrave;&ecirc;&oacute;, i &iacute;&agrave; &ecirc;i&iacute;&aring;&ouml;&uuml; &ntilde;&ograve;&eth;&oacute;&iacute;&egrave; &acirc; &iuml;&icirc;&iuml;&aring;&eth;&aring;&divide;&iacute;&icirc;&igrave;&oacute; &iacute;&agrave;&iuml;&eth;&yuml;&igrave;&ecirc;&oacute; &iacute;&aring; &auml;i&sup1; &iacute;i&yuml;&ecirc;&agrave;
&ntilde;&egrave;&euml;&agrave;. &Ntilde;&otilde;&aring;&igrave;&oacute; &eth;&aring;&agrave;&euml;i&ccedil;&agrave;&ouml;i&uml; &ecirc;&eth;&agrave;&eacute;&icirc;&acirc;&egrave;&otilde; &oacute;&igrave;&icirc;&acirc; &ograve;&agrave;&ecirc;&icirc;&atilde;&icirc; &ograve;&egrave;&iuml;&oacute; &acirc;&ecirc;&agrave;&ccedil;&agrave;&iacute;&icirc; &iacute;&agrave; &eth;&egrave;&ntilde;&oacute;&iacute;&ecirc;&oacute;:
&euml;i&acirc;&egrave;&eacute; &ecirc;i&iacute;&aring;&ouml;&uuml; &ntilde;&ograve;&eth;&oacute;&iacute;&egrave; (x = 0) &ccedil;&agrave;&ecirc;&eth;i&iuml;&euml;&aring;&iacute;&egrave;&eacute; &aelig;&icirc;&eth;&ntilde;&ograve;&ecirc;&icirc;, &agrave; &iuml;&eth;&agrave;&acirc;&egrave;&eacute; &ccedil;&agrave; &auml;&icirc;&iuml;&icirc;&igrave;&icirc;&atilde;&icirc;&thorn; &ecirc;i&euml;&uuml;&ouml;&yuml; (&igrave;&agrave;&ntilde;&icirc;&thorn; &yuml;&ecirc;&icirc;&atilde;&icirc; &iacute;&aring;&otilde;&ograve;&oacute;&sup1;&igrave;&icirc;), &ugrave;&icirc; &igrave;&icirc;&aelig;&aring; &aacute;&aring;&ccedil; &ograve;&aring;&eth;&ograve;&yuml; &ecirc;&icirc;&acirc;&ccedil;&agrave;&ograve;&egrave; &acirc;&ccedil;&auml;&icirc;&acirc;&aelig;
&iacute;&agrave;&iuml;&eth;&agrave;&acirc;&euml;&yuml;&thorn;&divide;&icirc;&atilde;&icirc; &ntilde;&ograve;&aring;&eth;&aelig;&iacute;&yuml; &oacute; &acirc;&aring;&eth;&ograve;&egrave;&ecirc;&agrave;&euml;&uuml;&iacute;&icirc;&igrave;&oacute; &iacute;&agrave;&iuml;&eth;&yuml;&igrave;&ecirc;&oacute;. &Igrave;&agrave;&ograve;&aring;&igrave;&agrave;&ograve;&egrave;&divide;&iacute;&aring; &ocirc;&icirc;&eth;&igrave;&oacute;&euml;&thorn;&acirc;&agrave;&iacute;&iacute;&yuml; &auml;&agrave;&iacute;&icirc;&uml; &ccedil;&agrave;&auml;&agrave;&divide;i &acirc;i&auml;&eth;i&ccedil;&iacute;&yuml;&sup1;&ograve;&uuml;&ntilde;&yuml; &acirc;i&auml; (3.1)-(3.3) &ograve;i&euml;&uuml;&ecirc;&egrave; &icirc;&auml;&iacute;i&sup1;&thorn; &ecirc;&eth;&agrave;&eacute;&icirc;&acirc;&icirc;&thorn; &oacute;&igrave;&icirc;&acirc;&icirc;&thorn;: u(l, t) = 0 &ntilde;&euml;i&auml; &ccedil;&agrave;&igrave;i&iacute;&egrave;&ograve;&egrave; &iacute;&agrave; ux (l, t) = 0, &ugrave;&icirc; &iuml;&eth;&egrave;&ccedil;&acirc;&icirc;&auml;&egrave;&ograve;&uuml;
&auml;&icirc; &ccedil;&agrave;&igrave;i&iacute;&egrave; &acirc;i&auml;&iuml;&icirc;&acirc;i&auml;&iacute;&icirc;&uml; &oacute;&igrave;&icirc;&acirc;&egrave; X(l) = 0 &iacute;&agrave; i&iacute;&oslash;&oacute; &oacute;&igrave;&icirc;&acirc;&oacute; X 0 (l) = 0 &acirc; &ccedil;&agrave;&auml;&agrave;&divide;i
&Oslash;&ograve;&oacute;&eth;&igrave;&agrave;-&Euml;i&oacute;&acirc;i&euml;&euml;&yuml; (3.9), (3.10). &Auml;&icirc;&ograve;&eth;&egrave;&igrave;&oacute;&thorn;&divide;&egrave;&ntilde;&uuml; &eth;&icirc;&ccedil;&atilde;&euml;&yuml;&iacute;&oacute;&ograve;&icirc;&uml; &acirc;&egrave;&ugrave;&aring; &ntilde;&otilde;&aring;&igrave;&egrave;
&eth;&icirc;&ccedil;&acirc;'&yuml;&ccedil;&oacute;&acirc;&agrave;&iacute;&iacute;&yuml; &ccedil;&agrave;&auml;&agrave;&divide;i &Oslash;&ograve;&oacute;&eth;&igrave;&agrave;-&Euml;i&oacute;&acirc;i&euml;&euml;&yuml;, &iuml;&eth;&egrave;&otilde;&icirc;&auml;&egrave;&igrave;&icirc; &auml;&icirc; &acirc;&egrave;&ntilde;&iacute;&icirc;&acirc;&ecirc;&oacute;, &ugrave;&icirc; &iuml;&eth;&egrave;
λ &lt; 0 i λ = 0 i &acirc; &auml;&agrave;&iacute;&icirc;&igrave;&oacute; &acirc;&egrave;&iuml;&agrave;&auml;&ecirc;&oacute; &iacute;&aring; i&ntilde;&iacute;&oacute;&sup1; &acirc;&euml;&agrave;&ntilde;&iacute;&egrave;&otilde; &ccedil;&iacute;&agrave;&divide;&aring;&iacute;&uuml; (&agrave; &icirc;&ograve;&aelig;&aring;,
i &acirc;&euml;&agrave;&ntilde;&iacute;&egrave;&otilde; &ocirc;&oacute;&iacute;&ecirc;&ouml;i&eacute;). &Iuml;&eth;&egrave; λ &gt; 0 &ccedil;&agrave;&ntilde;&ograve;&icirc;&ntilde;&icirc;&acirc;&oacute;&sup1;&igrave;&icirc; &auml;&icirc; &ccedil;&agrave;&atilde;&agrave;&euml;&uuml;&iacute;&icirc;&atilde;&icirc; &eth;&icirc;&ccedil;&acirc;'&yuml;&ccedil;&ecirc;&oacute;
&eth;i&acirc;&iacute;&yuml;&iacute;&iacute;&yuml; (3.9)
√
√
X(x) = A cos( λx) + B sin( λx)
&ecirc;&eth;&agrave;&eacute;&icirc;&acirc;i &oacute;&igrave;&icirc;&acirc;&egrave; X(0) = 0, X 0 (l) = 0 i &iuml;&eth;&egrave;&otilde;&icirc;&auml;&egrave;&igrave;&icirc; &auml;&icirc; &ntilde;&egrave;&ntilde;&ograve;&aring;&igrave;&egrave; &eth;i&acirc;&iacute;&yuml;&iacute;&uuml;
&acirc;i&auml;&iacute;&icirc;&ntilde;&iacute;&icirc; &ecirc;&icirc;&aring;&ocirc;i&ouml;i&sup1;&iacute;&ograve;i&acirc; A i B :
&frac12;
A√
= 0, √
B λ cos( λ l) = 0.
&Icirc;&ntilde;&ecirc;i&euml;&uuml;&ecirc;&egrave; λ &gt; 0 i B 6= 0√(&eth;&icirc;&ccedil;&acirc;'&yuml;&ccedil;&icirc;&ecirc; &iacute;&aring;&ograve;&eth;&egrave;&acirc;i&agrave;&euml;&uuml;&iacute;&egrave;&eacute;), &ograve;&icirc; &ccedil; &auml;&eth;&oacute;&atilde;&icirc;&atilde;&icirc; &eth;i&acirc;&iacute;&yuml;&iacute;&iacute;&yuml; &ntilde;&egrave;&ntilde;&ograve;&aring;&igrave;&egrave; &igrave;&agrave;&sup1;&igrave;&icirc;: cos( λ l) = 0 i &ccedil;&acirc;i&auml;&ntilde;&egrave; &ccedil;&iacute;&agrave;&otilde;&icirc;&auml;&egrave;&igrave;&icirc; &acirc;&euml;&agrave;&ntilde;&iacute;i &ccedil;&iacute;&agrave;&divide;&aring;&iacute;&iacute;&yuml;
28
i &acirc;&euml;&agrave;&ntilde;&iacute;i &ocirc;&oacute;&iacute;&ecirc;&ouml;i&uml; &ccedil;&agrave;&auml;&agrave;&divide;i &Oslash;&ograve;&oacute;&eth;&igrave;&agrave;-&Euml;i&oacute;&acirc;i&euml;&euml;&yuml;:
&micro;
&para;
π(2n + 1)
λn =
, Xn (x) = B̃n sin
x ,
(3.31)
2l
√
(n = 0, 1, 2, ...). &Acirc;&euml;&agrave;&ntilde;&iacute;i &divide;&agrave;&ntilde;&ograve;&icirc;&ograve;&egrave; &ntilde;&ograve;&eth;&oacute;&iacute;&egrave; ωn = λn &middot; a = π(2n+1)a
&acirc;&egrave;&ccedil;&iacute;&agrave;&divide;&agrave;&thorn;2l
&ograve;&uuml;&ntilde;&yuml; &acirc;&egrave;&atilde;&euml;&yuml;&auml;&icirc;&igrave; &eth;i&acirc;&iacute;&yuml;&iacute;&iacute;&yuml; (3.17) &ccedil; &acirc;&eth;&agrave;&otilde;&oacute;&acirc;&agrave;&iacute;&iacute;&yuml;&igrave; &iacute;&icirc;&acirc;&egrave;&otilde; &ccedil;&iacute;&agrave;&divide;&aring;&iacute;&uuml; λn .
&Ograve;&agrave;&ecirc;&egrave;&igrave; &divide;&egrave;&iacute;&icirc;&igrave;, &eth;&icirc;&ccedil;&acirc;'&yuml;&ccedil;&icirc;&ecirc; &ccedil;&agrave;&auml;&agrave;&divide;i &iuml;&eth;&icirc; &ecirc;&icirc;&euml;&egrave;&acirc;&agrave;&iacute;&iacute;&yuml; &ntilde;&ograve;&eth;&oacute;&iacute;&egrave; &oacute; &acirc;&egrave;&iuml;&agrave;&auml;&ecirc;&oacute;, &ecirc;&icirc;&euml;&egrave;
&icirc;&auml;&egrave;&iacute; (&euml;i&acirc;&egrave;&eacute;) &ecirc;i&iacute;&aring;&ouml;&uuml; &ntilde;&ograve;&eth;&oacute;&iacute;&egrave; &ccedil;&agrave;&ecirc;&eth;i&iuml;&euml;&aring;&iacute;&egrave;&eacute;, &agrave; &auml;&eth;&oacute;&atilde;&egrave;&eacute; (&iuml;&eth;&agrave;&acirc;&egrave;&eacute;) &acirc;i&euml;&uuml;&iacute;&egrave;&eacute;, &igrave;&agrave;&sup1;
&acirc;&egrave;&atilde;&euml;&yuml;&auml;
&micro;
&para;
∞
X
π(2n + 1)
u(x, t) =
(An cos ωn t + Bn sin ωn t) sin
x ,
(3.32)
2l
n=0
π(2n + 1)
2l
&para;2
&micro;
&auml;&aring; &ecirc;&icirc;&aring;&ocirc;i&ouml;i&sup1;&iacute;&ograve;&egrave; &eth;&yuml;&auml;&oacute; &icirc;&aacute;&divide;&egrave;&ntilde;&euml;&thorn;&thorn;&ograve;&uuml;&ntilde;&yuml; &ccedil;&agrave; &ocirc;&icirc;&eth;&igrave;&oacute;&euml;&agrave;&igrave;&egrave;:
&para;
π(2n + 1)
ϕ(x) sin
x dx,
2l
0
&micro;
&para;
Z l
π(2n + 1)
2
ψ(x) sin
Bn =
x dx.
ωn l 0
2l
2
An =
l
Z
l
&micro;
&szlig;&ecirc; i &acirc; &iuml;&icirc;&iuml;&aring;&eth;&aring;&auml;&iacute;&uuml;&icirc;&igrave;&oacute; &acirc;&egrave;&iuml;&agrave;&auml;&ecirc;&oacute; (&ecirc;i&iacute;&ouml;i &ntilde;&ograve;&eth;&oacute;&iacute;&egrave; &ccedil;&agrave;&ecirc;&eth;i&iuml;&euml;&aring;&iacute;i), &eth;&icirc;&ccedil;&acirc;'&yuml;&ccedil;&icirc;&ecirc; (3.31)
&yuml;&acirc;&euml;&yuml;&sup1; &ntilde;&icirc;&aacute;&icirc;&thorn; &ntilde;&oacute;&iuml;&aring;&eth;&iuml;&icirc;&ccedil;&egrave;&ouml;i&thorn; &ntilde;&ograve;&icirc;&yuml;&divide;&egrave;&otilde; &otilde;&acirc;&egrave;&euml;&uuml; &agrave;&aacute;&icirc; &atilde;&agrave;&eth;&igrave;&icirc;&iacute;i&ecirc;. &ETH;i&ccedil;&iacute;&egrave;&ouml;&yuml; &iuml;&icirc;&euml;&yuml;&atilde;&agrave;&sup1;
&euml;&egrave;&oslash;&aring; &acirc; &eth;&icirc;&ccedil;&ograve;&agrave;&oslash;&oacute;&acirc;&agrave;&iacute;&iacute;i &acirc;&oacute;&ccedil;&euml;i&acirc; i &iuml;&oacute;&divide;&iacute;&icirc;&ntilde;&ograve;&aring;&eacute;, &agrave; &ograve;&agrave;&ecirc;&icirc;&aelig; &acirc; &ccedil;&iacute;&agrave;&divide;&aring;&iacute;&iacute;&yuml;&otilde; &acirc;&euml;&agrave;&ntilde;&iacute;&egrave;&otilde;
&divide;&agrave;&ntilde;&ograve;&icirc;&ograve; &ntilde;&ograve;&eth;&oacute;&iacute;&egrave;.
&Ccedil;&atilde;i&auml;&iacute;&icirc; (3.31) &acirc;&oacute;&ccedil;&euml;&egrave; &ntilde;&ograve;&icirc;&yuml;&divide;&icirc;&uml; &otilde;&acirc;&egrave;&euml;i &ccedil; &acirc;&euml;&agrave;&ntilde;&iacute;&icirc;&thorn; &divide;&agrave;&ntilde;&ograve;&icirc;&ograve;&icirc;&thorn; ωn &eth;&icirc;&ccedil;&ograve;&agrave;&oslash;&icirc;&acirc;&agrave;&iacute;i
2k
&acirc; &ograve;&icirc;&divide;&ecirc;&agrave;&otilde; xk = 2n+1
&middot;l, (k = 0, 1...n), &agrave; &iuml;&oacute;&divide;&iacute;&icirc;&ntilde;&ograve;i &acirc; &ograve;&icirc;&divide;&ecirc;&agrave;&otilde; xm = 2m+1
2n+1 &middot;l , (m =
0, 1...n). &Icirc;&ograve;&aelig;&aring;, &iuml;&eth;&egrave; &auml;&icirc;&acirc;i&euml;&uuml;&iacute;&icirc;&igrave;&oacute; &ccedil;&iacute;&agrave;&divide;&aring;&iacute;&iacute;i n &iacute;&agrave; &euml;i&acirc;&icirc;&igrave;&oacute; &ecirc;i&iacute;&ouml;i &ntilde;&ograve;&eth;&oacute;&iacute;&egrave; &ccedil;&agrave;&acirc;&aelig;&auml;&egrave;
&aacute;&oacute;&auml;&aring; &acirc;&oacute;&ccedil;&icirc;&euml;, &agrave; &iacute;&agrave; &iuml;&eth;&agrave;&acirc;&icirc;&igrave;&oacute; &iuml;&oacute;&divide;&iacute;i&ntilde;&ograve;&uuml;. &Iacute;&agrave;&eacute;&igrave;&aring;&iacute;&oslash;&agrave; &divide;&agrave;&ntilde;&ograve;&icirc;&ograve;&agrave; &atilde;&agrave;&eth;&igrave;&icirc;&iacute;i&ecirc;&egrave;, &ugrave;&icirc;
&ccedil;&agrave;&auml;&agrave;&sup1; &ograve;&icirc;&iacute; &ccedil;&acirc;&oacute;&divide;&agrave;&iacute;&iacute;&yuml; &ntilde;&ograve;&eth;&oacute;&iacute;&egrave;, ω0 = πa
2l . &Icirc;&ntilde;&ecirc;i&euml;&uuml;&ecirc;&egrave; &iuml;&agrave;&eth;&agrave;&igrave;&aring;&ograve;&eth; a &igrave;&agrave;&sup1; &ccedil;&igrave;i&ntilde;&ograve;
&oslash;&acirc;&egrave;&auml;&ecirc;&icirc;&ntilde;&ograve;i &iuml;&icirc;&oslash;&egrave;&eth;&aring;&iacute;&iacute;&yuml; &otilde;&acirc;&egrave;&euml;&uuml; &acirc; &ntilde;&ograve;&eth;&oacute;&iacute;i, &ograve;&icirc; &divide;&agrave;&ntilde;&ograve;&icirc;&ograve;i ω0 &acirc;i&auml;&iuml;&icirc;&acirc;i&auml;&agrave;&sup1; &auml;&icirc;&acirc;&aelig;&egrave;&iacute;&agrave;
2πa
&otilde;&acirc;&egrave;&euml;i λ̃0 = 2πa
ω0 = πa/(2l) = 4l . &Icirc;&ograve;&aelig;&aring;, &iuml;&eth;&egrave; &ecirc;&icirc;&euml;&egrave;&acirc;&agrave;&iacute;&iacute;i &ccedil; &iacute;&agrave;&eacute;&igrave;&aring;&iacute;&oslash;&icirc;&thorn; &acirc;&euml;&agrave;&ntilde;&iacute;&icirc;&thorn;
&divide;&agrave;&ntilde;&ograve;&icirc;&ograve;&icirc;&thorn; ω0 &iacute;&agrave; &auml;&icirc;&acirc;&aelig;&egrave;&iacute;i &ntilde;&ograve;&eth;&oacute;&iacute;&egrave; &acirc;&ecirc;&euml;&agrave;&auml;&agrave;&sup1;&ograve;&uuml;&ntilde;&yuml; &euml;&egrave;&oslash;&aring; &divide;&acirc;&aring;&eth;&ograve;&uuml; &auml;&icirc;&acirc;&aelig;&egrave;&iacute;&egrave; &otilde;&acirc;&egrave;&euml;i
l = λ̃0 /4, &acirc; &ograve;&icirc;&eacute; &divide;&agrave;&ntilde;, &yuml;&ecirc; &oacute; &acirc;&egrave;&iuml;&agrave;&auml;&ecirc;&oacute; &icirc;&aacute;&icirc;&otilde; &ccedil;&agrave;&ecirc;&eth;i&iuml;&euml;&aring;&iacute;&egrave;&otilde; &ecirc;i&iacute;&ouml;i&acirc; &iuml;&icirc;&euml;&icirc;&acirc;&egrave;&iacute;&agrave;
&auml;&icirc;&acirc;&aelig;&egrave;&iacute;&egrave; &otilde;&acirc;&egrave;&euml;i. &Ccedil; &eth;i&acirc;&iacute;&icirc;&ntilde;&ograve;i (3.31) &acirc;&egrave;&iuml;&euml;&egrave;&acirc;&agrave;&sup1;, &ugrave;&icirc; &acirc;&ntilde;i &acirc;&euml;&agrave;&ntilde;&iacute;i &divide;&agrave;&ntilde;&ograve;&icirc;&ograve;&egrave; &ntilde;&ograve;&eth;&oacute;&iacute;&egrave;
&ecirc;&eth;&agrave;&ograve;&iacute;i &iacute;&agrave;&eacute;&igrave;&aring;&iacute;&oslash;i&eacute; &divide;&agrave;&ntilde;&ograve;&icirc;&ograve;i
ωn = (2n + 1)ω0 ,
i &ecirc;&icirc;&aring;&ocirc;i&ouml;i&sup1;&iacute;&ograve;&egrave; &ecirc;&eth;&agrave;&ograve;&iacute;&icirc;&ntilde;&ograve;i &sup1; &ouml;i&euml;&egrave;&igrave;&egrave; &iacute;&aring;&iuml;&agrave;&eth;&iacute;&egrave;&igrave;&egrave; &divide;&egrave;&ntilde;&euml;&agrave;&igrave;&egrave;.
29
&Ecirc;&icirc;&eth;&icirc;&ograve;&ecirc;&icirc; &icirc;&aacute;&atilde;&icirc;&acirc;&icirc;&eth;&egrave;&igrave;&icirc; &ograve;&aring;&iuml;&aring;&eth; &ntilde;&iuml;&aring;&ouml;&egrave;&ocirc;i&ecirc;&oacute; &ccedil;&agrave;&auml;&agrave;&divide;i &ccedil; &ecirc;&eth;&agrave;&eacute;&icirc;&acirc;&egrave;&igrave;&egrave; &oacute;&igrave;&icirc;&acirc;&agrave;&igrave;&egrave;,
&ugrave;&icirc; &acirc;i&auml;&iuml;&icirc;&acirc;i&auml;&agrave;&thorn;&ograve;&uuml; &ntilde;&egrave;&ograve;&oacute;&agrave;&ouml;i&uml;, &ecirc;&icirc;&euml;&egrave; &icirc;&aacute;&egrave;&auml;&acirc;&agrave; &ecirc;i&iacute;&ouml;i &ntilde;&ograve;&eth;&oacute;&iacute;&egrave; &acirc;i&euml;&uuml;&iacute;i.
&Iuml;&icirc;&acirc;&ograve;&icirc;&eth;&thorn;&thorn;&divide;&egrave; &auml;i&uml;, &eth;&icirc;&ccedil;&atilde;&euml;&yuml;&iacute;&oacute;&ograve;i &eth;&agrave;&iacute;i&oslash;&aring;, &iuml;&eth;&egrave;&otilde;&icirc;&auml;&egrave;&igrave;&icirc; &auml;&icirc; &ccedil;&agrave;&auml;&agrave;&divide;i &Oslash;&ograve;&oacute;&eth;&igrave;&agrave;&Euml;i&oacute;&acirc;i&euml;&euml;&yuml;

 X 00 (x) + λX(x) = 0, 0 &lt; x &lt; l
X 0 (0) = 0,
 0
X (l) = 0.
(3.33)
&Euml;&aring;&atilde;&ecirc;&icirc; &iuml;&aring;&eth;&aring;&ecirc;&icirc;&iacute;&agrave;&ograve;&egrave;&ntilde;&uuml;, &ugrave;&icirc; i &acirc; &auml;&agrave;&iacute;&icirc;&igrave;&oacute; &acirc;&egrave;&iuml;&agrave;&auml;&ecirc;&oacute; &iacute;&aring; i&ntilde;&iacute;&oacute;&sup1; &acirc;&euml;&agrave;&ntilde;&iacute;&egrave;&otilde; &ccedil;&iacute;&agrave;&divide;&aring;&iacute;&uuml;
i &acirc;&euml;&agrave;&ntilde;&iacute;&egrave;&otilde; &ocirc;&oacute;&iacute;&ecirc;&ouml;i&eacute; &iuml;&eth;&egrave; λ &lt; 0. &ETH;&icirc;&ccedil;&acirc;'&yuml;&ccedil;&ecirc;&agrave;&igrave;&egrave; &ccedil;&agrave;&auml;&agrave;&divide;i (3.33) λ &gt; 0 &sup1; &acirc;&euml;&agrave;2
&ntilde;&iacute;i &ccedil;&iacute;&agrave;&divide;&aring;&iacute;&iacute;&yuml;
&iexcl; πn &cent; λn = (πn/l) , &yuml;&ecirc;&egrave;&igrave; &acirc;i&auml;&iuml;&icirc;&acirc;i&auml;&agrave;&thorn;&ograve;&uuml; &acirc;&euml;&agrave;&ntilde;&iacute;i &ocirc;&oacute;&iacute;&ecirc;&ouml;i&uml; Xn (x) =
An cos l x .
&Iacute;&agrave; &acirc;i&auml;&igrave;i&iacute;&oacute; &acirc;i&auml; &auml;&acirc;&icirc;&otilde; &iuml;&icirc;&iuml;&aring;&eth;&aring;&auml;&iacute;i&otilde; &acirc;&agrave;&eth;i&agrave;&iacute;&ograve;i&acirc; &ecirc;&eth;&agrave;&eacute;&icirc;&acirc;&egrave;&otilde; &oacute;&igrave;&icirc;&acirc;, &acirc;&egrave;&yuml;&acirc;&euml;&yuml;&sup1;&ograve;&uuml;&ntilde;&yuml;,
&ugrave;&icirc; &ecirc;&icirc;&euml;&egrave; &icirc;&aacute;&egrave;&auml;&acirc;&agrave; &ecirc;i&iacute;&ouml;i &ntilde;&ograve;&eth;&oacute;&iacute;&egrave; &acirc;i&euml;&uuml;&iacute;i, &ograve;&icirc; λ = 0 &ograve;&agrave;&ecirc;&icirc;&aelig; &sup1; &acirc;&euml;&agrave;&ntilde;&iacute;&egrave;&igrave; &ccedil;&iacute;&agrave;&divide;&aring;&iacute;&iacute;&yuml;&igrave;.
&Auml;i&eacute;&ntilde;&iacute;&icirc;, &iuml;&eth;&egrave; λ = 0 &ccedil;&agrave;&auml;&agrave;&divide;&agrave; &Oslash;&ograve;&oacute;&eth;&igrave;&agrave;-&Euml;i&oacute;&acirc;i&euml;&euml;&yuml; &igrave;&agrave;&sup1; &acirc;&egrave;&atilde;&euml;&yuml;&auml;:
 00
 X (x) = 0, 0 &lt; x &lt; l
X 0 (0) = 0,
 0
X (l) = 0.
&Ccedil;&agrave;&atilde;&agrave;&euml;&uuml;&iacute;&egrave;&igrave; &eth;&icirc;&ccedil;&acirc;'&yuml;&ccedil;&ecirc;&icirc;&igrave; &auml;&egrave;&ocirc;&aring;&eth;&aring;&iacute;&ouml;i&agrave;&euml;&uuml;&iacute;&icirc;&atilde;&icirc; &eth;i&acirc;&iacute;&yuml;&iacute;&iacute;&yuml; &sup1; &euml;i&iacute;i&eacute;&iacute;&agrave; &ocirc;&oacute;&iacute;&ecirc;&ouml;i&yuml; X(x) =
C + Dx, &agrave; &ccedil; &ecirc;&eth;&agrave;&eacute;&icirc;&acirc;&egrave;&otilde; &oacute;&igrave;&icirc;&acirc; &acirc;&egrave;&iuml;&euml;&egrave;&acirc;&agrave;&sup1;: X 0 (0) = D = 0, X 0 (l) = D = 0. &Icirc;&ograve;&aelig;&aring;, &acirc;&euml;&agrave;&ntilde;&iacute;&icirc;&igrave;&oacute; &ccedil;&iacute;&agrave;&divide;&aring;&iacute;&iacute;&thorn; λ = λ0 = 0 &acirc;i&auml;&iuml;&icirc;&acirc;i&auml;&agrave;&sup1; &acirc;&euml;&agrave;&ntilde;&iacute;&agrave; &ocirc;&oacute;&iacute;&ecirc;&ouml;i&yuml; X0 (x) = C ,
&yuml;&ecirc;&agrave; &iacute;&aring; &auml;&icirc;&eth;i&acirc;&iacute;&thorn;&sup1; &ograve;&icirc;&ograve;&icirc;&aelig;&iacute;&uuml;&icirc; &iacute;&oacute;&euml;&thorn;. &Iuml;&eth;&egrave; λ = 0 &eth;i&acirc;&iacute;&yuml;&iacute;&iacute;&yuml; (3.6) &iacute;&agrave;&aacute;&oacute;&acirc;&agrave;&sup1; &acirc;&egrave;&atilde;&euml;&yuml;&auml;&oacute; T̈ = 0, &eth;&icirc;&ccedil;&acirc;'&yuml;&ccedil;&ecirc;&agrave;&igrave;&egrave; &yuml;&ecirc;&icirc;&atilde;&icirc; &sup1; &ocirc;&oacute;&iacute;&ecirc;&ouml;i&yuml;
T0 (t) = Ã0 + B̃0 t.
&Ograve;&agrave;&ecirc;&egrave;&igrave; &divide;&egrave;&iacute;&icirc;&igrave;, &eth;&icirc;&ccedil;&acirc;'&yuml;&ccedil;&icirc;&ecirc; &ccedil;&agrave;&auml;&agrave;&divide;i &iuml;&eth;&icirc; &ecirc;&icirc;&euml;&egrave;&acirc;&agrave;&iacute;&iacute;&yuml; &icirc;&aacute;&igrave;&aring;&aelig;&aring;&iacute;&icirc;&uml; &ntilde;&ograve;&eth;&oacute;&iacute;&egrave; &ccedil; &acirc;i&euml;&uuml;&iacute;&egrave;&igrave;&egrave; &ecirc;i&iacute;&ouml;&yuml;&igrave;&egrave; &igrave;&agrave;&sup1; &acirc;&egrave;&atilde;&euml;&yuml;&auml;
u(x, t) = A0 + B0 t +
&micro;
&para;
∞
X
π(2n + 1)
+
(An cos ωn t + Bn sin ωn t) sin
x , (3.34)
2l
n=1
30
&auml;&aring; &ecirc;&icirc;&aring;&ocirc;i&ouml;i&sup1;&iacute;&ograve;&egrave; &eth;&yuml;&auml;&oacute; &icirc;&aacute;&divide;&egrave;&ntilde;&euml;&thorn;&thorn;&ograve;&uuml;&ntilde;&yuml; &ccedil;&agrave; &ocirc;&icirc;&eth;&igrave;&oacute;&euml;&agrave;&igrave;&egrave;
1
A0 =
l
Z
l
0
1
ϕ(x)dx, Bn =
l
Z
l
ψ(x),
0
Z
Z l
&sup3; πn &acute;
&sup3; πn &acute;
2
2 l
ϕ(x) cos
x dx, Bn =
ψ(x) cos
x dx.
An =
l 0
l
ωn l 0
l
&Acirc;&euml;&agrave;&ntilde;&iacute;i &divide;&agrave;&ntilde;&ograve;&icirc;&ograve;&egrave; &ecirc;&icirc;&euml;&egrave;&acirc;&agrave;&iacute;&uuml; &ntilde;&ograve;&eth;&oacute;&iacute;&egrave; ωn = πna
l .
&Iuml;&icirc;&yuml;&ntilde;&iacute;&egrave;&igrave;&icirc; &ocirc;i&ccedil;&egrave;&divide;&iacute;&egrave;&eacute; &ccedil;&igrave;i&ntilde;&ograve; &iuml;&aring;&eth;&oslash;&egrave;&otilde; &auml;&acirc;&icirc;&otilde; &auml;&icirc;&auml;&agrave;&iacute;&ecirc;i&acirc; &acirc; &iuml;&eth;&agrave;&acirc;i&eacute; &divide;&agrave;&ntilde;&ograve;&egrave;&iacute;i (3.34).
Rl
&szlig;&ecirc;&ugrave;&icirc; 0 ϕ(x)dx 6= 0, &ograve;&icirc; &iuml;&eth;&egrave; t = 0 &ouml;&aring;&iacute;&ograve;&eth; &igrave;&agrave;&ntilde; &ntilde;&ograve;&eth;&oacute;&iacute;&egrave; &aacute;&oacute;&auml;&aring; &ccedil;&igrave;i&ugrave;&aring;&iacute;&egrave;&eacute; &oacute; &acirc;&aring;&eth;&ograve;&egrave;&ecirc;&agrave;&euml;&uuml;&iacute;&icirc;&igrave;&oacute; &iacute;&agrave;&iuml;&eth;&yuml;&igrave;&ecirc;&oacute; &iacute;&agrave; &acirc;&aring;&euml;&egrave;&divide;&egrave;&iacute;&oacute;
Rl
Z l
ρϕ(x)dx
1
ũ0 = 0 R l
=
ϕ(x)dx = A0 .
l
0
0 ρdx
&Icirc;&ntilde;&ecirc;i&euml;&uuml;&ecirc;&egrave; &iacute;&agrave; &ntilde;&ograve;&eth;&oacute;&iacute;&oacute; &ccedil; &acirc;i&euml;&uuml;&iacute;&egrave;&igrave;&egrave; &ecirc;i&iacute;&ouml;&yuml;&igrave;&egrave; &acirc; &iuml;&icirc;&iuml;&aring;&eth;&aring;&divide;&iacute;&icirc;&igrave;&oacute; &iacute;&agrave;&iuml;&eth;&yuml;&igrave;&ecirc;&oacute; &iacute;&aring; &auml;i&thorn;&ograve;&uuml; &ccedil;&icirc;&acirc;&iacute;i&oslash;&iacute;i &ntilde;&egrave;&euml;&egrave;, &ograve;&icirc; &ccedil;&igrave;i&ugrave;&aring;&iacute;&iacute;&yuml; &ouml;&aring;&iacute;&ograve;&eth;&oacute; &igrave;&agrave;&ntilde; &ccedil;&agrave;&euml;&egrave;&oslash;&agrave;&sup1;&ograve;&uuml;&ntilde;&yuml; &iuml;&icirc;&ntilde;&ograve;i&eacute;&iacute;&egrave;&igrave; &acirc; &divide;&agrave;&ntilde;i
i &ecirc;&icirc;&euml;&egrave;&acirc;&agrave;&iacute;&iacute;&yuml; &ntilde;&ograve;&eth;&oacute;&iacute;&egrave; &acirc;i&auml;&aacute;&oacute;&acirc;&agrave;&thorn;&ograve;&uuml;&ntilde;&yuml; &acirc;&aelig;&aring; &iacute;&agrave;&acirc;&ecirc;&icirc;&euml;&icirc; &ccedil;&igrave;i&ugrave;&aring;&iacute;&icirc;&atilde;&icirc; &iuml;&icirc;&euml;&icirc;&aelig;&aring;&iacute;&iacute;&yuml; &eth;i&acirc;&iacute;&icirc;&acirc;&agrave;&atilde;&egrave;, &ograve;&icirc;&aacute;&ograve;&icirc;, &auml;&icirc; &ccedil;&igrave;i&ugrave;&aring;&iacute;&uuml;, &iuml;&icirc;&acirc;'&yuml;&ccedil;&agrave;&iacute;&egrave;&otilde; &ccedil; &ecirc;&icirc;&euml;&egrave;&acirc;&agrave;&iacute;&iacute;&yuml;&igrave;&egrave;, &aacute;&oacute;&auml;&aring; &auml;&icirc;&auml;&agrave;&acirc;&agrave;&ograve;&egrave;&ntilde;&uuml;
&iuml;&icirc;&ntilde;&ograve;i&eacute;&iacute;&aring; &acirc; &divide;&agrave;&ntilde;i &ccedil;&igrave;i&ugrave;&aring;&iacute;&iacute;&yuml; AR0 .
l
&Iuml;&eth;&egrave;&iuml;&oacute;&ntilde;&ograve;&egrave;&igrave;&icirc; &ograve;&aring;&iuml;&aring;&eth;, &ugrave;&icirc; 0 ψ(x)dx 6= 00. &Ograve;&icirc;&auml;i &iuml;&eth;&egrave; t = 0 i&igrave;&iuml;&oacute;&euml;&uuml;&ntilde; &ouml;&aring;&iacute;-
Rl
&ograve;&eth;&oacute; &igrave;&agrave;&ntilde; &ntilde;&ograve;&eth;&oacute;&iacute;&egrave; &auml;&icirc;&eth;i&acirc;&iacute;&thorn;&sup1; 0 ρψ(x)dx i, &icirc;&ntilde;&ecirc;i&euml;&uuml;&ecirc;&egrave; &ccedil;&icirc;&acirc;&iacute;i&oslash;&iacute;i &ntilde;&egrave;&euml;&egrave; (&iuml;&icirc;&iuml;&aring;&eth;&aring;&divide;&iacute;i) &acirc;i&auml;&ntilde;&oacute;&ograve;&iacute;i, i&igrave;&iuml;&oacute;&euml;&uuml;&ntilde; &ntilde;&ograve;&eth;&oacute;&iacute;&egrave; &ccedil;&agrave;&euml;&egrave;&oslash;&agrave;&sup1;&ograve;&uuml;&ntilde;&yuml; &iacute;&aring;&ccedil;&igrave;i&iacute;&iacute;&egrave;&igrave;, &ograve;&icirc;&aacute;&ograve;&icirc; &iacute;&agrave; &ecirc;&icirc;&euml;&egrave;&acirc;&agrave;&iacute;&iacute;&yuml;
&ntilde;&ograve;&eth;&oacute;&iacute;&egrave; &iacute;&agrave;&ecirc;&euml;&agrave;&auml;&agrave;&sup1;&ograve;&uuml;&ntilde;&yuml; &eth;i&acirc;&iacute;&icirc;&igrave;i&eth;&iacute;&egrave;&eacute; &eth;&oacute;&otilde; &acirc;&ntilde;i&sup1;&uml; &ntilde;&ograve;&eth;&oacute;&iacute;&egrave; &ccedil;i &oslash;&acirc;&egrave;&auml;&ecirc;i&ntilde;&ograve;&thorn;
Rl
ρψ(x)dx 1
v0 = 0 R l
=
l
ρdx
0
Z
l
ψ(x)dx = B0 .
0
&Icirc;&ograve;&aelig;&aring;, &yuml;&ecirc;&ugrave;&icirc; &ecirc;i&iacute;&ouml;i &ntilde;&ograve;&eth;&oacute;&iacute;&egrave; &acirc;i&euml;&uuml;&iacute;i, &ograve;&icirc; &auml;&euml;&yuml; &ecirc;&icirc;&euml;&egrave;&acirc;&agrave;&iacute;&uuml; &ntilde;&ograve;&eth;&oacute;&iacute;&egrave;, &ccedil;&agrave; &iuml;&aring;&acirc;&iacute;&egrave;&otilde; &oacute;&igrave;&icirc;&acirc;,
&auml;&icirc;&auml;&agrave;&sup1;&ograve;&uuml;&ntilde;&yuml; &ccedil;&igrave;i&ugrave;&aring;&iacute;&iacute;&yuml; &acirc; &iuml;&icirc;&iuml;&aring;&eth;&aring;&divide;&iacute;&icirc;&igrave;&oacute; &iacute;&agrave;&iuml;&eth;&yuml;&igrave;&ecirc;&oacute;, &ugrave;&icirc; &ccedil;&agrave;&euml;&aring;&aelig;&agrave;&ograve;&uuml; &acirc;i&auml; &divide;&agrave;&ntilde;&oacute; &ccedil;&agrave;
&ccedil;&agrave;&ecirc;&icirc;&iacute;&icirc;&igrave; ū = A0 +B0 t, &ugrave;&icirc; i &ccedil;&iacute;&agrave;&eacute;&oslash;&euml;&icirc; &ntilde;&acirc;&icirc;&sup1; &acirc;i&auml;&icirc;&aacute;&eth;&agrave;&aelig;&aring;&iacute;&iacute;&yuml; &oacute; &eth;&icirc;&ccedil;&acirc;'&yuml;&ccedil;&ecirc;&oacute; (3.34)
&Ecirc;&icirc;&euml;&egrave;&acirc;&agrave;&iacute;&iacute;&yuml; &ntilde;&ograve;&eth;&oacute;&iacute;&egrave; &iuml;i&auml; &auml;i&sup1;&thorn; &ccedil;&icirc;&acirc;&iacute;i&oslash;&iacute;i&otilde; &ntilde;&egrave;&euml;.
&Acirc; &iuml;&icirc;&iuml;&aring;&eth;&aring;&auml;&iacute;i&otilde; &eth;&icirc;&ccedil;&auml;i&euml;&agrave;&otilde; &eth;&icirc;&ccedil;&atilde;&euml;&yuml;&iacute;&oacute;&ograve;&icirc; &acirc;i&euml;&uuml;&iacute;i &ecirc;&icirc;&euml;&egrave;&acirc;&agrave;&iacute;&iacute;&yuml; &ntilde;&ograve;&eth;&oacute;&iacute;&egrave;, &ograve;&icirc;&aacute;&ograve;&icirc; &ecirc;&icirc;&euml;&egrave;&acirc;&agrave;&iacute;&iacute;&yuml; &ntilde;&iuml;&eth;&egrave;&divide;&egrave;&iacute;&aring;&iacute;i &euml;&egrave;&oslash;&aring; &acirc;&iacute;&oacute;&ograve;&eth;i&oslash;&iacute;i&igrave;&egrave; &ntilde;&egrave;&euml;&agrave;&igrave;&egrave; &iuml;&eth;&oacute;&aelig;&iacute;&icirc;&ntilde;&ograve;i &ograve;&agrave; &ntilde;&egrave;&euml;&agrave;&igrave;&egrave; &eth;&aring;&agrave;&ecirc;&ouml;i&uml; &iacute;&agrave; &ecirc;i&iacute;&ouml;&yuml;&otilde; &ntilde;&ograve;&eth;&oacute;&iacute;&egrave; &acirc; &ograve;&icirc;&divide;&ecirc;&agrave;&otilde; &uml;&uml; &ccedil;&agrave;&ecirc;&eth;i&iuml;&euml;&aring;&iacute;&iacute;&yuml;. &ETH;&icirc;&ccedil;&atilde;&euml;&yuml;&iacute;&aring;&igrave;&icirc; &ograve;&aring;&iuml;&aring;&eth; &iuml;&eth;&icirc;&ouml;&aring;&ntilde;
&ecirc;&icirc;&euml;&egrave;&acirc;&agrave;&iacute;&iacute;&yuml; &icirc;&aacute;&igrave;&aring;&aelig;&aring;&iacute;&icirc;&uml; &ntilde;&ograve;&eth;&oacute;&iacute;&egrave; &iuml;i&auml; &auml;i&sup1;&thorn; &ccedil;&agrave;&auml;&agrave;&iacute;&icirc;&uml; &ntilde;&egrave;&euml;&egrave;, &ugrave;&icirc; &ccedil;&agrave;&euml;&aring;&aelig;&egrave;&ograve;&uuml; &acirc;i&auml; &divide;&agrave;&ntilde;&oacute; i &eth;&icirc;&ccedil;&iuml;&icirc;&auml;i&euml;&aring;&iacute;&agrave; &iuml;&icirc; &ntilde;&ograve;&eth;&oacute;&iacute;i &ccedil; &euml;i&iacute;i&eacute;&iacute;&icirc;&thorn; &atilde;&oacute;&ntilde;&ograve;&egrave;&iacute;&icirc;&thorn; F (x, t). &Ntilde;&egrave;&euml;&egrave;, &iuml;&eth;&egrave;&ecirc;&euml;&agrave;&auml;&aring;&iacute;i
&oacute; &acirc;&ntilde;i&otilde; &ograve;&icirc;&divide;&ecirc;&agrave;&otilde;, &euml;&aring;&aelig;&agrave;&ograve;&uuml; &acirc; &icirc;&auml;&iacute;i&eacute; &iuml;&euml;&icirc;&ugrave;&egrave;&iacute;i i &igrave;&agrave;&thorn;&ograve;&uuml; &iacute;&agrave;&iuml;&eth;&yuml;&igrave;, &iacute;&icirc;&eth;&igrave;&agrave;&euml;&uuml;&iacute;&egrave;&eacute; &auml;&icirc;
31
&iacute;&aring;&eacute;&ograve;&eth;&agrave;&euml;&uuml;&iacute;&icirc;&atilde;&icirc; &iuml;&icirc;&euml;&icirc;&aelig;&aring;&iacute;&iacute;&yuml; &ntilde;&ograve;&eth;&oacute;&iacute;&egrave;. &Ccedil;i &ccedil;&igrave;i&ntilde;&ograve;&oacute; &euml;i&iacute;i&eacute;&iacute;&icirc;&uml; &atilde;&oacute;&ntilde;&ograve;&egrave;&iacute;&egrave; F (x, t) &acirc;&egrave;&iuml;&euml;&egrave;&acirc;&agrave;&sup1;, &ugrave;&icirc; &iacute;&agrave; &auml;&icirc;&acirc;i&euml;&uuml;&iacute;&egrave;&eacute; &aring;&euml;&aring;&igrave;&aring;&iacute;&ograve; &ntilde;&ograve;&eth;&oacute;&iacute;&egrave; &auml;&icirc;&acirc;&aelig;&egrave;&iacute;&icirc;&thorn; dx &auml;i&sup1; &aring;&euml;&aring;&igrave;&aring;&iacute;&ograve;&agrave;&eth;&iacute;&agrave;
&ntilde;&egrave;&euml;&agrave; F (x, t)dx. &Ccedil;&agrave; &acirc;i&auml;&ntilde;&oacute;&ograve;&iacute;&icirc;&ntilde;&ograve;i i&iacute;&oslash;&egrave;&otilde; &ntilde;&egrave;&euml;, &auml;&agrave;&iacute;&egrave;&eacute; &aring;&euml;&aring;&igrave;&aring;&iacute;&ograve; dx, &igrave;&agrave;&ntilde;&agrave; &yuml;&ecirc;&icirc;&atilde;&icirc;
F (x,t)dx
F (x,t)
&auml;&icirc;&eth;i&acirc;&iacute;&thorn;&sup1; dm = ρdx, &eth;&oacute;&otilde;&agrave;&acirc;&ntilde;&yuml; &aacute; &ccedil; &iuml;&eth;&egrave;&ntilde;&ecirc;&icirc;&eth;&aring;&iacute;&iacute;&yuml;&igrave; f (x, t) = dm = ρ .
&Acirc; &auml;i&eacute;&ntilde;&iacute;&icirc;&ntilde;&ograve;i &acirc; &iuml;&eth;&icirc;&ouml;&aring;&ntilde;i &ecirc;&icirc;&euml;&egrave;&acirc;&agrave;&iacute;&iacute;&yuml; &ntilde;&ograve;&eth;&oacute;&iacute;&egrave; &iacute;&agrave; &acirc;&egrave;&auml;i&euml;&aring;&iacute;&egrave;&eacute; &aring;&euml;&aring;&igrave;&aring;&iacute;&ograve; dx &auml;i&thorn;&ograve;&uuml;
&ograve;&agrave;&ecirc;&icirc;&aelig; &ccedil;&igrave;i&iacute;&iacute;i &ntilde;&egrave;&euml;&egrave; &ccedil; &aacute;&icirc;&ecirc;&oacute; i&iacute;&oslash;&egrave;&otilde; &aring;&euml;&aring;&igrave;&aring;&iacute;&ograve;i&acirc; &ntilde;&ograve;&eth;&oacute;&iacute;&egrave;, &ccedil; &yuml;&ecirc;&egrave;&igrave;&egrave; &auml;&agrave;&iacute;&egrave;&eacute; &aring;&euml;&aring;&igrave;&aring;&iacute;&ograve;
&ccedil;&iacute;&agrave;&otilde;&icirc;&auml;&egrave;&ograve;&uuml;&ntilde;&yuml; &acirc; &ecirc;&icirc;&iacute;&ograve;&agrave;&ecirc;&ograve;i. &Oacute; &ntilde;&acirc;&icirc;&thorn; &divide;&aring;&eth;&atilde;&oacute;, &ccedil;&agrave; &ograve;&eth;&aring;&ograve;i&igrave; &ccedil;&agrave;&ecirc;&icirc;&iacute;&icirc;&igrave; &Iacute;&uuml;&thorn;&ograve;&icirc;&iacute;&agrave;, &aring;&euml;&aring;&igrave;&aring;&iacute;&ograve; &auml;i&sup1; &iacute;&agrave; &ntilde;&oacute;&ntilde;i&auml;&iacute;i &aring;&euml;&aring;&igrave;&aring;&iacute;&ograve;&egrave;, i &acirc; &eth;&aring;&ccedil;&oacute;&euml;&uuml;&ograve;&agrave;&ograve;i &acirc;i&auml;&aacute;&oacute;&acirc;&agrave;&sup1;&ograve;&uuml;&ntilde;&yuml; &ntilde;&agrave;&igrave;&icirc;&oacute;&ccedil;&atilde;&icirc;&auml;&aelig;&aring;&iacute;&egrave;&eacute;
&eth;&oacute;&otilde; &oacute;&ntilde;i&otilde; &aring;&euml;&aring;&igrave;&aring;&iacute;&ograve;i&acirc; &ntilde;&ograve;&eth;&oacute;&iacute;&egrave;.
&Ugrave;&icirc;&aacute; &ccedil;&icirc;&ntilde;&aring;&eth;&aring;&auml;&egrave;&ograve;&egrave;&ntilde;&uuml; &aacute;&aring;&ccedil;&iuml;&icirc;&ntilde;&aring;&eth;&aring;&auml;&iacute;&uuml;&icirc; &iacute;&agrave; &acirc;&egrave;&acirc;&divide;&aring;&iacute;&iacute;i &ntilde;&agrave;&igrave;&aring; &iacute;&agrave;&ntilde;&euml;i&auml;&ecirc;i&acirc; &auml;i&uml; &iacute;&agrave;
&ntilde;&ograve;&eth;&oacute;&iacute;&oacute; &ccedil;&icirc;&acirc;&iacute;i&oslash;&iacute;&uuml;&icirc;&uml; &ntilde;&egrave;&euml;&egrave;, &acirc; &auml;&agrave;&iacute;&icirc;&igrave;&oacute; &eth;&icirc;&ccedil;&auml;i&euml;i &igrave;&agrave;&ecirc;&ntilde;&egrave;&igrave;&agrave;&euml;&uuml;&iacute;&icirc; &ntilde;&iuml;&eth;&icirc;&ntilde;&ograve;&egrave;&igrave;&icirc; &ccedil;&agrave;&auml;&agrave;&divide;&oacute;. &Aacute;&oacute;&auml;&aring;&igrave;&icirc; &acirc;&acirc;&agrave;&aelig;&agrave;&ograve;&egrave;, &ugrave;&icirc; &iuml;&icirc;&divide;&agrave;&ograve;&ecirc;&icirc;&acirc;i &acirc;i&auml;&otilde;&egrave;&euml;&aring;&iacute;&iacute;&yuml; &acirc;i&auml; &iuml;&icirc;&euml;&icirc;&aelig;&aring;&iacute;&iacute;&yuml; &eth;i&acirc;&iacute;&icirc;&acirc;&agrave;&atilde;&egrave;
i &iuml;&icirc;&divide;&agrave;&ograve;&ecirc;&icirc;&acirc;i &oslash;&acirc;&egrave;&auml;&ecirc;&icirc;&ntilde;&ograve;i &ograve;&icirc;&divide;&icirc;&ecirc; &ntilde;&ograve;&eth;&oacute;&iacute;&egrave; &auml;&icirc;&eth;i&acirc;&iacute;&thorn;&thorn;&ograve;&uuml; &iacute;&oacute;&euml;&thorn;, &ograve;&icirc;&aacute;&ograve;&icirc; &acirc; &igrave;&icirc;&igrave;&aring;&iacute;&ograve;
&divide;&agrave;&ntilde;&oacute; t = 0 &ntilde;&ograve;&eth;&oacute;&iacute;&agrave; &ccedil;&iacute;&agrave;&otilde;&icirc;&auml;&egrave;&ograve;&uuml;&ntilde;&yuml; &acirc; &ntilde;&ograve;&agrave;&iacute;i &ntilde;&iuml;&icirc;&ecirc;&icirc;&thorn; &acirc; &ntilde;&acirc;&icirc;&sup1;&igrave;&oacute; &iuml;&icirc;&euml;&icirc;&aelig;&aring;&iacute;&iacute;i &eth;i&acirc;&iacute;&icirc;&acirc;&agrave;&atilde;&egrave;. &Acirc;&acirc;&agrave;&aelig;&agrave;&sup1;&igrave;&icirc; &ograve;&agrave;&ecirc;&icirc;&aelig;, &ugrave;&icirc; &ecirc;i&iacute;&ouml;i &ntilde;&ograve;&eth;&oacute;&iacute;&egrave; &aelig;&icirc;&eth;&ntilde;&ograve;&ecirc;&icirc; &ccedil;&agrave;&ecirc;&eth;i&iuml;&euml;&aring;&iacute;i, &icirc;&ntilde;&ecirc;i&euml;&uuml;&ecirc;&egrave;
&ccedil;&agrave;&auml;&agrave;&divide;&oacute; &iuml;&eth;&icirc; &acirc;i&euml;&uuml;&iacute;i &ecirc;&icirc;&euml;&egrave;&acirc;&agrave;&iacute;&iacute;&yuml; &ntilde;&ograve;&eth;&oacute;&iacute;&egrave; &ntilde;&agrave;&igrave;&aring; &ccedil; &ograve;&agrave;&ecirc;&egrave;&igrave;&egrave; &ecirc;&eth;&agrave;&eacute;&icirc;&acirc;&egrave;&igrave;&egrave; &oacute;&igrave;&icirc;&acirc;&agrave;&igrave;&egrave;
&iacute;&agrave;&eacute;&aacute;i&euml;&uuml;&oslash; &auml;&icirc;&ecirc;&euml;&agrave;&auml;&iacute;&icirc; &eth;&icirc;&ccedil;&atilde;&euml;&yuml;&iacute;&oacute;&ograve;&icirc; &acirc;&egrave;&ugrave;&aring;.
&Igrave;&agrave;&ograve;&aring;&igrave;&agrave;&ograve;&egrave;&divide;&iacute;&agrave; &iuml;&icirc;&ntilde;&ograve;&agrave;&iacute;&icirc;&acirc;&ecirc;&agrave; &ccedil;&agrave;&auml;&agrave;&divide;i &iuml;&eth;&icirc; &ecirc;&icirc;&euml;&egrave;&acirc;&agrave;&iacute;&iacute;&yuml; &icirc;&aacute;&igrave;&aring;&aelig;&aring;&iacute;&icirc;&uml; &ntilde;&ograve;&eth;&oacute;&iacute;&egrave; &auml;&icirc;&acirc;&aelig;&egrave;&iacute;&icirc;&thorn; l &iuml;i&auml; &auml;i&sup1;&thorn; &ccedil;&icirc;&acirc;&iacute;i&oslash;&iacute;i&otilde; &ntilde;&egrave;&euml; &ccedil; &acirc;&eth;&agrave;&otilde;&oacute;&acirc;&agrave;&iacute;&iacute;&yuml;&igrave; &ecirc;&icirc;&iacute;&ecirc;&eth;&aring;&ograve;&iacute;&egrave;&otilde; &oacute;&igrave;&icirc;&acirc;, &ntilde;&ocirc;&icirc;&eth;&igrave;&oacute;&euml;&uuml;&icirc;&acirc;&agrave;&iacute;&egrave;&otilde; &acirc;&egrave;&ugrave;&aring;, &iacute;&agrave;&aacute;&oacute;&acirc;&agrave;&sup1; &acirc;&egrave;&atilde;&euml;&yuml;&auml;&oacute;:
utt = a2 uxx + f (x, t), 0 &lt; x &lt; l, t &gt; 0,
&frac12;
u(0, t) = 0,
u(l, t) = 0;
&frac12;
u(x, 0) = 0,
ut (x, 0) = 0.
(3.35)
(3.36)
(3.37)
&Iacute;&agrave; &acirc;i&auml;&igrave;i&iacute;&oacute; &acirc;i&auml; &eth;i&acirc;&iacute;&yuml;&iacute;&iacute;&yuml; (3.1), &ugrave;&icirc; &icirc;&iuml;&egrave;&ntilde;&oacute;&sup1; &auml;&egrave;&iacute;&agrave;&igrave;i&ecirc;&oacute; &acirc;i&euml;&uuml;&iacute;&egrave;&otilde; &ecirc;&icirc;&euml;&egrave;&acirc;&agrave;&iacute;&uuml;
&ntilde;&ograve;&eth;&oacute;&iacute;&egrave;, &auml;&egrave;&ocirc;&aring;&eth;&aring;&iacute;&ouml;i&agrave;&euml;&uuml;&iacute;&aring; &eth;i&acirc;&iacute;&yuml;&iacute;&iacute;&yuml; (3.35) &sup1; &iacute;&aring;&icirc;&auml;&iacute;&icirc;&eth;i&auml;&iacute;&egrave;&igrave; &divide;&aring;&eth;&aring;&ccedil; &iacute;&agrave;&yuml;&acirc;&iacute;i&ntilde;&ograve;&uuml;
&ocirc;&oacute;&iacute;&ecirc;&ouml;i&uml; f (x, t), &ccedil;&igrave;i&ntilde;&ograve; &yuml;&ecirc;&icirc;&uml; &acirc;&aelig;&aring; &iacute;&agrave;&igrave;&egrave; &icirc;&aacute;&atilde;&icirc;&acirc;&icirc;&eth;&aring;&iacute;&icirc;.
&ETH;&icirc;&ccedil;&acirc;'&yuml;&ccedil;&icirc;&ecirc; &ccedil;&agrave;&auml;&agrave;&divide;i (3.35-3.37) &aacute;&oacute;&auml;&aring;&igrave;&icirc; &oslash;&oacute;&ecirc;&agrave;&ograve;&egrave; &oacute; &acirc;&egrave;&atilde;&euml;&yuml;&auml;i &eth;&yuml;&auml;&oacute; &ccedil;&agrave; &acirc;&euml;&agrave;&ntilde;&iacute;&egrave;&igrave;&egrave;
&ocirc;&oacute;&iacute;&ecirc;&ouml;i&yuml;&igrave;&egrave; &ccedil;&agrave;&auml;&agrave;&divide;i &Oslash;&ograve;&oacute;&eth;&igrave;&agrave;-&Euml;i&oacute;&acirc;i&euml;&euml;&yuml; (3.9),(3.10)
u(x, t) =
∞
X
un (t) sin
n=1
&sup3; πnx &acute;
l
.
(3.38)
&Igrave;&icirc;&aelig;&euml;&egrave;&acirc;i&ntilde;&ograve;&uuml; &ograve;&agrave;&ecirc;&icirc;&atilde;&icirc; &eth;&icirc;&ccedil;&ecirc;&euml;&agrave;&auml;&oacute;
&iuml;&icirc;&acirc;&iacute;&icirc;&ograve;&icirc;&thorn; &icirc;&eth;&ograve;&icirc;&atilde;&icirc;&iacute;&agrave;&euml;&uuml;&iacute;&icirc;&uml; &iuml;&icirc;&ntilde;&euml;i&copy; &icirc;&aacute;&oacute;&igrave;&icirc;&acirc;&euml;&aring;&iacute;&agrave;
&iexcl; πnx &cent;&ordf;
&auml;&icirc;&acirc;&iacute;&icirc;&ntilde;&ograve;i &acirc;&euml;&agrave;&ntilde;&iacute;&egrave;&otilde; &ocirc;&oacute;&iacute;&ecirc;&ouml;i&eacute; sin l
&acirc; &ecirc;&euml;&agrave;&ntilde;i &iacute;&aring;&iuml;&agrave;&eth;&iacute;&egrave;&otilde; &icirc;&aacute;&igrave;&aring;&aelig;&aring;&iacute;&egrave;&otilde; &ocirc;&oacute;&iacute;&ecirc;&ouml;i&eacute;. &Icirc;&ograve;&aelig;&aring;, &auml;&icirc;&acirc;i&euml;&uuml;&iacute;&oacute; &ocirc;&oacute;&iacute;&ecirc;&ouml;i&thorn; u(x, t) (&acirc; &auml;&agrave;&iacute;&icirc;&igrave;&oacute; &acirc;&egrave;&iuml;&agrave;&auml;&ecirc;&oacute; t &ntilde;&euml;i&auml; &eth;&icirc;&ccedil;&atilde;&euml;&yuml;32
&auml;&agrave;&ograve;&egrave; &yuml;&ecirc; &iuml;&agrave;&eth;&agrave;&igrave;&aring;&ograve;&eth;) &ccedil; &ouml;&uuml;&icirc;&atilde;&icirc; &ecirc;&euml;&agrave;&ntilde;&oacute; &igrave;&icirc;&aelig;&iacute;&agrave; &eth;&icirc;&ccedil;&ecirc;&euml;&agrave;&ntilde;&ograve;&egrave; &acirc; &eth;&yuml;&auml; (3.38), i &ouml;&aring;&eacute; &eth;&yuml;&auml;
&ccedil;&aacute;i&atilde;&agrave;&sup1;&ograve;&uuml;&ntilde;&yuml; &eth;i&acirc;&iacute;&icirc;&igrave;i&eth;&iacute;&icirc; &auml;&icirc; &ocirc;&oacute;&iacute;&ecirc;&ouml;i&uml; u(x, t).
&iexcl; &cent;
&Auml;&icirc;&ouml;i&euml;&uuml;&iacute;i&ntilde;&ograve;&uuml; &eth;&icirc;&ccedil;&ecirc;&euml;&agrave;&auml;&oacute; &ntilde;&agrave;&igrave;&aring; &iuml;&icirc; &ocirc;&oacute;&iacute;&ecirc;&ouml;i&yuml;&otilde; sin πnx
&acirc;&egrave;&iuml;&euml;&egrave;&acirc;&agrave;&sup1; &ccedil; &ograve;&icirc;&atilde;&icirc;, &ugrave;&icirc;
l
u(x, t), &iuml;&eth;&aring;&auml;&ntilde;&ograve;&agrave;&acirc;&euml;&aring;&iacute;&agrave; &oacute; &acirc;&egrave;&atilde;&euml;&yuml;&auml;i &eth;&yuml;&auml;&oacute; (3.38), &ccedil;&agrave;&auml;&icirc;&acirc;&icirc;&euml;&uuml;&iacute;&yuml;&sup1; &ouml;i &oacute;&igrave;&icirc;&acirc;&egrave;. &Icirc;&ntilde;&ograve;&agrave;&iacute;&iacute;i&eacute; &ocirc;&agrave;&ecirc;&ograve;&icirc;&eth; &sup1; &acirc;&egrave;&eth;i&oslash;&agrave;&euml;&uuml;&iacute;&egrave;&igrave; &iuml;&eth;&egrave; &acirc;&egrave;&aacute;&icirc;&eth;i &iuml;&icirc;&ntilde;&euml;i&auml;&icirc;&acirc;&iacute;&icirc;&ntilde;&ograve;i &ocirc;&oacute;&iacute;&ecirc;&ouml;i&eacute;, &ccedil;&agrave; &yuml;&ecirc;&egrave;&igrave;&egrave;
&eth;&icirc;&ccedil;&ecirc;&euml;&agrave;&auml;&agrave;&sup1;&ograve;&uuml;&ntilde;&yuml; u(x, t). &Icirc;&ograve;&aelig;&aring;, &yuml;&ecirc;&ugrave;&icirc; &aacute;, &iacute;&agrave;&iuml;&eth;&egrave;&ecirc;&euml;&agrave;&auml;, &acirc; &ccedil;&agrave;&auml;&agrave;&divide;i (3.35-3.37) &ecirc;&eth;&agrave;&eacute;&icirc;&acirc;i &oacute;&igrave;&icirc;&acirc;&egrave; (3.36) &ccedil;&agrave;&igrave;i&iacute;&egrave;&ograve;&egrave; &iacute;&agrave; i&iacute;&oslash;i: u(0, t) = 0, ux (l, t) = 0 (&ograve;&icirc;&aacute;&ograve;&icirc; &ograve;&agrave;&ecirc;i &ugrave;&icirc;
&acirc;i&auml;&iuml;&icirc;&acirc;i&auml;&agrave;&thorn;&ograve;&uuml; &ntilde;&egrave;&ograve;&oacute;&agrave;&ouml;i&uml;, &ecirc;&icirc;&euml;&egrave; &euml;i&acirc;&egrave;&eacute; &ecirc;i&iacute;&aring;&ouml;&uuml; &ntilde;&ograve;&eth;&oacute;&iacute;&egrave; &ccedil;&agrave;&ecirc;&eth;i&iuml;&euml;&aring;&iacute;&egrave;&eacute;,
&acute;o
n &sup3; &agrave; &iuml;&eth;&agrave;&acirc;&egrave;&eacute;
π(2n+1)x
,
&acirc;i&euml;&uuml;&iacute;&egrave;&eacute;), &ograve;&icirc; &ocirc;&oacute;&iacute;&ecirc;&ouml;i&thorn; u(x, t) &ntilde;&euml;i&auml; &aacute;&oacute;&euml;&icirc; &aacute; &eth;&icirc;&ccedil;&ecirc;&euml;&agrave;&auml;&agrave;&ograve;&egrave; &iuml;&icirc; sin
2l
&ograve;&icirc;&aacute;&ograve;&icirc; &iuml;&icirc; &acirc;&euml;&agrave;&ntilde;&iacute;&egrave;&igrave; &ocirc;&oacute;&iacute;&ecirc;&ouml;i&yuml;&igrave; &ccedil;&agrave;&auml;i&divide;i &Oslash;&ograve;&oacute;&eth;&igrave;&agrave;-&Euml;i&oacute;&acirc;i&euml;&euml;&yuml;, &ugrave;&icirc; &acirc;&egrave;&iacute;&egrave;&ecirc;&agrave;&thorn;&ograve;&uuml; &acirc;
&ccedil;&agrave;&auml;&agrave;&divide;i &iuml;&eth;&icirc; &acirc;i&euml;&uuml;&iacute;i &ecirc;&icirc;&euml;&egrave;&acirc;&agrave;&iacute;&iacute;&yuml; &ntilde;&ograve;&eth;&oacute;&iacute;&egrave;, &ccedil; &acirc;i&auml;&iuml;&icirc;&acirc;i&auml;&iacute;&egrave;&igrave;&egrave; &ecirc;&eth;&agrave;&eacute;&icirc;&acirc;&egrave;&igrave;&egrave; &oacute;&igrave;&icirc;&acirc;&agrave;&igrave;&egrave;.
&Ocirc;&oacute;&iacute;&ecirc;&ouml;i&thorn; f (x, t) &ccedil; &eth;i&acirc;&iacute;&yuml;&iacute;&iacute;&yuml; (3.35) &ograve;&agrave;&ecirc;&icirc;&aelig; &iuml;&eth;&aring;&auml;&ntilde;&ograve;&agrave;&acirc;&egrave;&igrave;&icirc; &oacute; &acirc;&egrave;&atilde;&euml;&yuml;&auml;i &eth;&yuml;&auml;&oacute;
∞
X
f (x, t) =
fn (t) sin
&sup3; πnx &acute;
l
n=1
,
(3.39)
&icirc;&aacute; &eth;&oacute;&iacute;&ograve;&icirc;&acirc;&oacute;&thorn;&divide;&egrave; &ograve;&agrave;&ecirc;&oacute; &igrave;&icirc;&aelig;&euml;&egrave;&acirc;i&ntilde;&ograve;&uuml;,
&copy; &iexcl; πnx &cent;&ordf; &yuml;&ecirc; i &auml;&euml;&yuml; &eth;&icirc;&ccedil;&ecirc;&euml;&agrave;&auml;&oacute; u(x, t), &iuml;&icirc;&acirc;&iacute;&icirc;&ograve;&icirc;&thorn;
&iuml;&icirc;&ntilde;&euml;i&auml;&icirc;&acirc;&iacute;&icirc;&ntilde;&ograve;i &ocirc;&oacute;&iacute;&ecirc;&ouml;i&eacute; sin l
. &Ecirc;&icirc;&aring;&ocirc;i&ouml;i&sup1;&iacute;&ograve;&egrave; fn (t) &acirc; (3.39) &ccedil;&iacute;&agrave;&otilde;&icirc;&auml;&egrave;&igrave;&icirc;
&agrave;&aacute;&ntilde;&icirc;&euml;&thorn;&ograve;&iacute;&icirc; &ograve;&agrave;&ecirc; &ntilde;&agrave;&igrave;&icirc;, &yuml;&ecirc; i &ecirc;&icirc;&aring;&ocirc;i&ouml;i&sup1;&iacute;&ograve;&egrave; An &acirc; &eth;&icirc;&ccedil;&ecirc;&euml;&agrave;&auml;i (3.20):
2
fn (t) =
l
Z
l
f (x, t) sin
&sup3; πnx &acute;
l
0
dx,
&agrave;&aacute;&icirc; &ccedil;&acirc;&agrave;&aelig;&agrave;&thorn;&divide;&egrave; &iacute;&agrave; &iuml;&icirc;&auml;&agrave;&euml;&uuml;&oslash;&aring; &ecirc;&icirc;&eth;&egrave;&ntilde;&ograve;&oacute;&acirc;&agrave;&iacute;&iacute;&yuml; &ouml;&egrave;&igrave; &acirc;&egrave;&eth;&agrave;&ccedil;&icirc;&igrave; &iuml;&aring;&eth;&aring;&iuml;&egrave;&oslash;&aring;&igrave;&icirc; &eacute;&icirc;&atilde;&icirc;
&ccedil; i&iacute;&oslash;&egrave;&igrave;&egrave; &iuml;&icirc;&ccedil;&iacute;&agrave;&divide;&aring;&iacute;&iacute;&yuml;&igrave;&egrave; &ccedil;&igrave;i&iacute;&iacute;&icirc;&uml; i&iacute;&ograve;&aring;&atilde;&eth;&oacute;&acirc;&agrave;&iacute;&iacute;&yuml;:
2
fn (t) =
l
Z
l
0
&micro;
πnξ
f (ξ, t) sin
l
&para;
dξ.
(3.40)
&Auml;&egrave;&ocirc;&aring;&eth;&aring;&iacute;&ouml;i&thorn;&thorn;&divide;&egrave; &iuml;&icirc;&divide;&euml;&aring;&iacute;&iacute;&icirc; &eth;&yuml;&auml; (3.38) &iuml;&icirc; x i &iuml;&icirc; t, &ccedil;&iacute;&agrave;&otilde;&icirc;&auml;&egrave;&igrave;&icirc; &auml;&eth;&oacute;&atilde;i &divide;&agrave;&ntilde;&ograve;&egrave;&iacute;&iacute;i
&iuml;&icirc;&otilde;i&auml;&iacute;i &ocirc;&oacute;&iacute;&ecirc;&ouml;i&uml; u(x, t)
utt =
∞
X
ün (t) sin
n=1
uxx = −
∞ &sup3;
X
πn &acute;2
n=1
l
&sup3; πnx &acute;
l
un (t) sin
,
&sup3; πnx &acute;
l
i &iuml;i&auml;&ntilde;&ograve;&agrave;&acirc;&egrave;&igrave;&icirc; &acirc;&egrave;&eth;&agrave;&ccedil;&egrave; &acirc; &eth;i&acirc;&iacute;&yuml;&iacute;&iacute;&yuml; (3.35). &Acirc; &eth;&aring;&ccedil;&oacute;&euml;&uuml;&ograve;&agrave;&ograve;i, &iuml;i&ntilde;&euml;&yuml; &aring;&euml;&aring;&igrave;&aring;&iacute;&ograve;&agrave;&eth;&iacute;&egrave;&otilde;
&iuml;&aring;&eth;&aring;&ograve;&acirc;&icirc;&eth;&aring;&iacute;&uuml;, &igrave;&agrave;&sup1;&igrave;&icirc;:
∞
X
&copy;
n=1
ün (t) +
ωn2 un (t)
&sup3; πnx &acute;
&ordf;
− fn (t) sin
= 0,
l
33
(3.41)
&auml;&aring; ωn = πna
l .
&Euml;i&acirc;&oacute; &divide;&agrave;&ntilde;&ograve;&egrave;&iacute;&oacute; &iexcl;&eth;i&acirc;&iacute;&icirc;&ntilde;&ograve;i
(3.41) &igrave;&icirc;&aelig;&iacute;&agrave; &eth;&icirc;&ccedil;&atilde;&euml;&yuml;&auml;&agrave;&ograve;&egrave; &yuml;&ecirc; &euml;i&iacute;i&eacute;&iacute;&oacute; &ntilde;&oacute;&iuml;&aring;&eth;&iuml;&icirc;&ccedil;&egrave;&cent;
πnx
&ouml;i&thorn; &ocirc;&oacute;&iacute;&ecirc;&ouml;i&eacute; sin l &ccedil; &ecirc;&icirc;&aring;&ocirc;i&ouml;i&sup1;&iacute;&ograve;&agrave;&igrave;&egrave; Cn (t) = ün (t) + ωn2 un (t) − fn (t), &ugrave;&icirc;
&ccedil;&agrave;&euml;&aring;&aelig;&agrave;&ograve;&uuml; &acirc;i&auml; &ccedil;&igrave;i&iacute;&iacute;&icirc;&uml; t, &yuml;&ecirc;
&acirc;i&auml;&cent; &iuml;&agrave;&eth;&agrave;&igrave;&aring;&ograve;&eth;&agrave;.
&iexcl; πnx
&Icirc;&ntilde;&ecirc;i&euml;&uuml;&ecirc;&egrave; &ocirc;&oacute;&iacute;&ecirc;&ouml;i&uml; sin l
&ccedil; &eth;i&ccedil;&iacute;&egrave;&igrave;&egrave; &ccedil;&iacute;&agrave;&divide;&aring;&iacute;&iacute;&yuml;&igrave;&egrave; n &sup1; &euml;i&iacute;i&eacute;&iacute;&icirc; &iacute;&aring;&ccedil;&agrave;&euml;&aring;&aelig;&iacute;&egrave;&igrave;&egrave; (&aacute;&icirc; &icirc;&eth;&ograve;&icirc;&atilde;&icirc;&iacute;&agrave;&euml;&uuml;&iacute;i &ocirc;&oacute;&iacute;&ecirc;&ouml;i&uml; &sup1; &euml;i&iacute;i&eacute;&iacute;&icirc; &iacute;&aring;&ccedil;&agrave;&euml;&aring;&aelig;&iacute;&egrave;&igrave;&egrave; &ccedil;&agrave;&acirc;&aelig;&auml;&egrave;), &ograve;&icirc;
&ograve;&icirc;&ograve;&icirc;&aelig;&iacute;&agrave; &eth;i&acirc;&iacute;i&ntilde;&ograve;&uuml; &iacute;&oacute;&euml;&thorn; &ntilde;&oacute;&igrave;&egrave; (3.41) &igrave;&icirc;&aelig;&euml;&egrave;&acirc;&agrave; &ograve;&icirc;&auml;i i &ograve;i&euml;&uuml;&ecirc;&egrave; &ograve;&icirc;&auml;i, &ecirc;&icirc;&euml;&egrave; &ecirc;&icirc;&aelig;&iacute;&egrave;&eacute; &ccedil; &ecirc;&icirc;&aring;&ocirc;i&ouml;i&sup1;&iacute;&ograve;i&acirc; Cn (t) &auml;&icirc;&eth;i&acirc;&iacute;&thorn;&sup1; &iacute;&oacute;&euml;&thorn;. &Ccedil;&acirc;i&auml;&ntilde;&egrave; &igrave;&agrave;&sup1;&igrave;&icirc;:
ün (t) + ωn2 un (t) − fn (t) = 0, n = 1, 2, 3, ...
&Iuml;i&auml;&iuml;&icirc;&eth;&yuml;&auml;&ecirc;&icirc;&acirc;&oacute;&thorn;&divide;&egrave; &ocirc;&oacute;&iacute;&ecirc;&ouml;i&thorn; u(x, t) &oacute; &acirc;&egrave;&atilde;&euml;&yuml;&auml;i &eth;&icirc;&ccedil;&ecirc;&euml;&agrave;&auml;&oacute; (3.38) &icirc;&auml;&iacute;&icirc;&eth;i&auml;&iacute;&egrave;&igrave; &iuml;&icirc;&divide;&agrave;&ograve;&ecirc;&icirc;&acirc;&egrave;&igrave; &oacute;&igrave;&icirc;&acirc;&agrave;&igrave; (3.37), &icirc;&ograve;&eth;&egrave;&igrave;&agrave;&sup1;&igrave;&icirc;:
u(x, 0) =
∞
X
un (0) sin
&sup3; πnx &acute;
l
n=1
ut (x, 0) =
∞
X
u̇n (0) sin
&sup3; πnx &acute;
n=1
l
=0
= 0.
&iexcl;
&cent;
&Ccedil;&acirc;i&auml;&ntilde;&egrave;, &iacute;&agrave; &iuml;i&auml;&ntilde;&ograve;&agrave;&acirc;i, &ccedil;&iacute;&icirc;&acirc;&oacute; &aelig; &ograve;&agrave;&ecirc;&egrave;, &euml;i&iacute;i&eacute;&iacute;&icirc;&uml; &iacute;&aring;&ccedil;&agrave;&euml;&aring;&aelig;&iacute;&icirc;&ntilde;&ograve;i &ocirc;&oacute;&iacute;&ecirc;&ouml;i&eacute; sin πnx
,
l
&acirc;&egrave;&iuml;&euml;&egrave;&acirc;&agrave;&sup1;: un (0) = 0, u̇n (0) = 0. &Ograve;&agrave;&ecirc;&egrave;&igrave; &divide;&egrave;&iacute;&icirc;&igrave;, &auml;&euml;&yuml; &ccedil;&iacute;&agrave;&otilde;&icirc;&auml;&aelig;&aring;&iacute;&iacute;&yuml; &ocirc;&oacute;&iacute;&ecirc;&ouml;i&eacute;
un (t) &iuml;&eth;&egrave;&otilde;&icirc;&auml;&egrave;&igrave;&icirc; &auml;&icirc; &iacute;&aring;&icirc;&aacute;&otilde;i&auml;&iacute;&icirc;&ntilde;&ograve;i &eth;&icirc;&ccedil;&acirc;'&yuml;&ccedil;&agrave;&iacute;&iacute;&yuml; &ccedil;&agrave;&auml;&agrave;&divide;i &Ecirc;&icirc;&oslash;i:
ün (t) + ωn2 un (t) = fn (t),
&frac12;
un (0) = 0,
u̇n (0) = 0.
t &gt; 0,
(3.42)
(3.43)
&Iacute;&aring;&icirc;&auml;&iacute;&icirc;&eth;i&auml;&iacute;&aring; &auml;&egrave;&ocirc;&aring;&eth;&aring;&iacute;&ouml;i&agrave;&euml;&uuml;&iacute;&aring; &eth;i&acirc;&iacute;&yuml;&iacute;&iacute;&yuml; &ccedil; &iuml;&icirc;&ntilde;&ograve;i&eacute;&iacute;&egrave;&igrave;&egrave; &ecirc;&icirc;&aring;&ocirc;i&ouml;i&sup1;&iacute;&ograve;&agrave;&igrave;&egrave; (3.42),
&ccedil; &iuml;&icirc;&divide;&agrave;&ograve;&ecirc;&icirc;&acirc;&egrave;&igrave;&egrave; &oacute;&igrave;&icirc;&acirc;&agrave;&igrave;&egrave; (3.43) &igrave;&icirc;&aelig;&iacute;&agrave; &eth;&icirc;&ccedil;&acirc;'&yuml;&ccedil;&agrave;&ograve;&egrave;, &iacute;&agrave;&iuml;&eth;&egrave;&ecirc;&euml;&agrave;&auml;, &icirc;&iuml;&aring;&eth;&agrave;&ouml;i&eacute;&iacute;&egrave;&igrave; &igrave;&aring;&ograve;&icirc;&auml;&icirc;&igrave;.
&Ccedil;&agrave;&ntilde;&ograve;&icirc;&ntilde;&oacute;&sup1;&igrave;&icirc; &iuml;&aring;&eth;&aring;&ograve;&acirc;&icirc;&eth;&aring;&iacute;&iacute;&yuml; &Euml;&agrave;&iuml;&euml;&agrave;&ntilde;&agrave; &auml;&icirc; &ocirc;&oacute;&iacute;&ecirc;&ouml;i&eacute; un (x, t) &ograve;&agrave; fn (t):
un (t) : Un (p), fn (t) : Fn (p).
&Ograve;&icirc;&auml;i, &ccedil; &acirc;&eth;&agrave;&otilde;&oacute;&acirc;&agrave;&iacute;&iacute;&yuml;&igrave; &iuml;&icirc;&divide;&agrave;&ograve;&ecirc;&icirc;&acirc;&egrave;&otilde; &oacute;&igrave;&icirc;&acirc; (3.43): ün (t) : p2 Un (p), &iuml;&aring;&eth;&aring;&ograve;&acirc;&icirc;&eth;&aring;&iacute;&aring; &eth;i&acirc;&iacute;&yuml;&iacute;&iacute;&yuml; &iacute;&agrave;&aacute;&oacute;&acirc;&agrave;&sup1; &acirc;&egrave;&atilde;&euml;&yuml;&auml;&oacute;
p2 Un (p) + ωn2 Un (p) = Fn (p).
&Ccedil;&acirc;i&auml;&ntilde;&egrave; &ccedil;&icirc;&aacute;&eth;&agrave;&aelig;&aring;&iacute;&iacute;&yuml; &Euml;&agrave;&iuml;&euml;&agrave;&ntilde;&agrave; &oslash;&oacute;&ecirc;&agrave;&iacute;&icirc;&uml; &ocirc;&oacute;&iacute;&ecirc;&ouml;i&uml; &igrave;&icirc;&aelig;&iacute;&agrave; &iuml;&eth;&aring;&auml;&ntilde;&ograve;&agrave;&acirc;&egrave;&ograve;&egrave; &oacute; &acirc;&egrave;&atilde;&euml;&yuml;&auml;i
Un (p) =
1 ωn
Fn (p).
ωn p2 + ωn2
34
(3.44)
n
&Icirc;&ntilde;&ecirc;i&euml;&uuml;&ecirc;&egrave; Un (p) : un (t); p2ω+ω
2 : sin ωn t; Fn (p) : fn (t), &ograve;&icirc; &ccedil; (3.44)
n
&icirc;&aacute;&aring;&eth;&iacute;&aring;&iacute;&egrave;&igrave; &iuml;&aring;&eth;&aring;&ograve;&acirc;&icirc;&eth;&aring;&iacute;&iacute;&yuml;&igrave; &Euml;&agrave;&iuml;&euml;&agrave;&ntilde;&agrave; &ccedil;&iacute;&agrave;&otilde;&icirc;&auml;&egrave;&igrave;&icirc;:
1
un (t) =
ωn
Z
t
sin ωn (t − τ )fn (τ )dτ.
0
(3.45)
&Ograve;&oacute;&ograve; &ocirc;&oacute;&iacute;&ecirc;&ouml;i&yuml; un (t) &iuml;&eth;&aring;&auml;&ntilde;&ograve;&agrave;&acirc;&euml;&aring;&iacute;&agrave; &oacute; &acirc;&egrave;&atilde;&euml;&yuml;&auml;i &ccedil;&atilde;&icirc;&eth;&ograve;&ecirc;&egrave;, &icirc;&ntilde;&ecirc;i&euml;&uuml;&ecirc;&egrave; &uml;&uml; &ccedil;&icirc;&aacute;&eth;&agrave;&aelig;&aring;&iacute;&iacute;&yuml; &Euml;&agrave;&iuml;&euml;&agrave;&ntilde;&agrave; Un (p) &sup1; &auml;&icirc;&aacute;&oacute;&ograve;&ecirc;&icirc;&igrave; &auml;&acirc;&icirc;&otilde; &ccedil;&icirc;&aacute;&eth;&agrave;&aelig;&aring;&iacute;&uuml;.
&Acirc;&eth;&agrave;&otilde;&icirc;&acirc;&oacute;&thorn;&divide;&egrave; (3.40), &acirc;&egrave;&eth;&agrave;&ccedil; (3.45) &ccedil;&agrave;&iuml;&egrave;&oslash;&aring;&igrave;&icirc; &oacute; &acirc;&egrave;&atilde;&euml;&yuml;&auml;i:
2
un (t) =
ωn l
Z tZ
0
l
0
&micro;
πnξ
sin ωn (t − τ ) sin
l
&para;
f (ξτ )dτ dξ.
(3.46)
&Iuml;i&auml;&ntilde;&ograve;&agrave;&acirc;&egrave;&acirc;&oslash;&egrave; &icirc;&ograve;&eth;&egrave;&igrave;&agrave;&iacute;&egrave;&eacute; &acirc;&egrave;&eth;&agrave;&ccedil; &auml;&euml;&yuml; un (t) &acirc; &eth;&icirc;&ccedil;&ecirc;&euml;&agrave;&auml; (3.38), &ccedil;&igrave;i&iacute;&egrave;&igrave;&icirc;
&iuml;&icirc;&ntilde;&euml;i&auml;&icirc;&acirc;&iacute;i&ntilde;&ograve;&uuml; &icirc;&iuml;&aring;&eth;&agrave;&ouml;i&eacute; &iuml;i&auml;&ntilde;&oacute;&igrave;&icirc;&acirc;&oacute;&acirc;&agrave;&iacute;&iacute;&yuml; i i&iacute;&ograve;&aring;&atilde;&eth;&oacute;&acirc;&agrave;&iacute;&iacute;&yuml; i &ccedil;&agrave;&iuml;&egrave;&oslash;&aring;&igrave;&icirc; &icirc;&ntilde;&ograve;&agrave;&ograve;&icirc;&divide;&iacute;&egrave;&eacute; &acirc;&egrave;&eth;&agrave;&ccedil; &auml;&euml;&yuml; &eth;&icirc;&ccedil;&acirc;'&yuml;&ccedil;&ecirc;&oacute; &ccedil;&agrave;&auml;&agrave;&divide;i (3.35-3.37) &iuml;&eth;&icirc; &ecirc;&icirc;&euml;&egrave;&acirc;&agrave;&iacute;&iacute;&yuml; &icirc;&aacute;&igrave;&aring;&aelig;&aring;&iacute;&icirc;&uml;
&ntilde;&ograve;&eth;&oacute;&iacute;&egrave; &iuml;i&auml; &acirc;&iuml;&euml;&egrave;&acirc;&icirc;&igrave; &ccedil;&icirc;&acirc;&iacute;i&oslash;&iacute;i&otilde; &ntilde;&egrave;&euml;:
Z tZ
u(x, t) =
G(x, ξ; t − τ )f (ξτ )dτ dξ,
0
&auml;&aring;
l
0
(3.47)
&micro;
&para;
∞
πnξ
2X 1
G(x, ξ; t − τ ) =
sin ωn (t − τ ) sin
.
l n=1 ωn
l
&Ocirc;&oacute;&iacute;&ecirc;&ouml;i&yuml; G(x, ξ; t − τ ) &igrave;&agrave;&sup1; &iacute;&agrave;&ccedil;&acirc;&oacute; &ocirc;&oacute;&iacute;&ecirc;&ouml;i&uml; &acirc;&iuml;&euml;&egrave;&acirc;&oacute; &ograve;&icirc;&divide;&ecirc;&icirc;&acirc;&icirc;&atilde;&icirc; &auml;&aelig;&aring;&eth;&aring;&euml;&agrave;, &agrave;&aacute;&icirc;
&ocirc;&oacute;&iacute;&ecirc;&ouml;i&uml; &Atilde;&eth;i&iacute;&agrave;. &Ccedil;'&yuml;&ntilde;&oacute;&sup1;&igrave;&icirc; &ocirc;i&ccedil;&egrave;&divide;&iacute;&egrave;&eacute; &ccedil;&igrave;i&ntilde;&ograve; &ouml;i&sup1;&uml; &ocirc;&oacute;&iacute;&ecirc;&ouml;i&uml;.
&Iuml;&eth;&egrave;&iuml;&oacute;&ntilde;&ograve;&egrave;&igrave;&icirc;, &ugrave;&icirc; &iuml;&eth;&icirc;&ograve;&yuml;&atilde;&icirc;&igrave; &iuml;&aring;&acirc;&iacute;&icirc;&atilde;&icirc; &divide;&agrave;&ntilde;&oacute;, &iuml;&icirc;&divide;&egrave;&iacute;&agrave;&thorn;&divide;&egrave; &ccedil; &igrave;&icirc;&igrave;&aring;&iacute;&ograve;&oacute; t = 0,
&iacute;&agrave; &ntilde;&ograve;&eth;&oacute;&iacute;&oacute; &iacute;&aring; &auml;i&thorn;&ograve;&uuml; &ccedil;&icirc;&acirc;&iacute;i&oslash;&iacute;i &ntilde;&egrave;&euml;&egrave;, &agrave; &iuml;&icirc;&ograve;i&igrave; &acirc; &auml;&aring;&yuml;&ecirc;&egrave;&eacute; &igrave;&icirc;&igrave;&aring;&iacute;&ograve; &divide;&agrave;&ntilde;&oacute; t = t0 &auml;&icirc;
&ntilde;&ograve;&eth;&oacute;&iacute;&egrave; &iuml;&eth;&egrave;&ecirc;&euml;&agrave;&auml;&agrave;&sup1;&ograve;&uuml;&ntilde;&yuml; &ntilde;&egrave;&euml;&agrave; &igrave;&egrave;&ograve;&ograve;&sup1;&acirc;&icirc;&uml; &auml;i&uml; &acirc; &igrave;&agrave;&euml;&icirc;&igrave;&oacute; &icirc;&ecirc;&icirc;&euml;i &ograve;&icirc;&divide;&ecirc;&egrave; &ccedil; &ecirc;&icirc;&icirc;&eth;&auml;&egrave;&iacute;&agrave;&ograve;&icirc;&thorn; x0 . &Ograve;&icirc;&auml;i &euml;i&iacute;i&eacute;&iacute;&oacute; &divide;&agrave;&ntilde;&ograve;&egrave;&iacute;&oacute; &eth;&icirc;&ccedil;&iuml;&icirc;&auml;i&euml;&oacute; &ntilde;&egrave;&euml;&egrave; F (x, t) &igrave;&icirc;&aelig;&iacute;&agrave; &ccedil;&igrave;&icirc;&auml;&aring;&euml;&thorn;&acirc;&agrave;&ograve;&egrave;
&ccedil;&agrave; &auml;&icirc;&iuml;&icirc;&igrave;&icirc;&atilde;&icirc;&thorn; δ -&ocirc;&oacute;&iacute;&ecirc;&ouml;i&uml; &Auml;i&eth;&agrave;&ecirc;&agrave;:
F (x, t) = Cδ(x − x0 )δ(t − t0 ),
(3.48)
&auml;&aring; C &auml;&aring;&yuml;&ecirc;&agrave; &ecirc;&icirc;&iacute;&ntilde;&ograve;&agrave;&iacute;&ograve;&agrave;, &yuml;&ecirc;&agrave; &iuml;&aring;&acirc;&iacute;&egrave;&igrave; &divide;&egrave;&iacute;&icirc;&igrave; &otilde;&agrave;&eth;&agrave;&ecirc;&ograve;&aring;&eth;&egrave;&ccedil;&oacute;&sup1; i&iacute;&ograve;&aring;&iacute;&ntilde;&egrave;&acirc;&iacute;i&ntilde;&ograve;&uuml;
&iuml;&eth;&egrave;&ecirc;&euml;&agrave;&auml;&aring;&iacute;&icirc;&uml; &auml;&icirc; &ntilde;&ograve;&eth;&oacute;&iacute;&egrave; &ntilde;&egrave;&euml;&egrave;.
&Ccedil;&iacute;&agrave;&eacute;&auml;&aring;&igrave;&icirc; &ccedil;&iacute;&agrave;&divide;&aring;&iacute;&iacute;&yuml; i&igrave;&iuml;&oacute;&euml;&uuml;&ntilde;&oacute;, &ugrave;&icirc; &iuml;&aring;&eth;&aring;&auml;&agrave;&iacute;&icirc; &ntilde;&ograve;&eth;&oacute;&iacute;i:
Z lZ
Z lZ
∞
P =
∞
F (x, t)dxdt = C
0
0
0
35
0
δ(x − x0 )δ(t − t0 )dxdt = C.
&Icirc;&ograve;&aelig;&aring;, &ecirc;&icirc;&iacute;&ntilde;&ograve;&agrave;&iacute;&ograve;&agrave; C , &ugrave;&icirc; &acirc;&otilde;&icirc;&auml;&egrave;&uuml; &acirc; (3.48), &igrave;&agrave;&sup1; &ccedil;&igrave;i&ntilde;&ograve; i&igrave;&iuml;&oacute;&euml;&uuml;&ntilde;&oacute;, &ugrave;&icirc; &iuml;&aring;&eth;&aring;&auml;&agrave;&sup1;&ograve;&uuml;&ntilde;&yuml; &ntilde;&ograve;&eth;&oacute;&iacute;i &acirc; &ouml;i&euml;&icirc;&igrave;&oacute; &oslash;&euml;&yuml;&otilde;&icirc;&igrave; &igrave;&egrave;&ograve;&ograve;&sup1;&acirc;&icirc;&uml; &auml;i&uml; &ntilde;&egrave;&euml;&egrave; (3.48). &Ocirc;&oacute;&iacute;&ecirc;&ouml;i&yuml; f (x, t), &ugrave;&icirc;
&acirc;&otilde;&icirc;&auml;&egrave;&ograve;&uuml; &acirc; (3.35) &iuml;&icirc;&acirc;'&yuml;&ccedil;&agrave;&iacute;&agrave; &ccedil; F (x, t) &iuml;&eth;&icirc;&ntilde;&ograve;&egrave;&igrave; &ntilde;&iuml;i&acirc;&acirc;i&auml;&iacute;&icirc;&oslash;&aring;&iacute;&iacute;&yuml;&igrave; &acirc;&egrave;&atilde;&euml;&yuml;&auml;&oacute;:
f (x, t) =
P
δ(x − x0 )δ(t − t0 ).
ρ
(3.49)
&Iuml;i&auml;&ntilde;&ograve;&agrave;&acirc;&egrave;&igrave;&icirc; &ouml;&thorn; &ocirc;&oacute;&iacute;&ecirc;&ouml;i&thorn; &acirc; &iuml;i&auml;i&iacute;&ograve;&aring;&atilde;&eth;&agrave;&euml;&uuml;&iacute;&egrave;&eacute; &acirc;&egrave;&eth;&agrave;&ccedil; &eth;i&acirc;&iacute;&icirc;&ntilde;&ograve;i (3.47):
P
u(x, t) =
ρ
Z tZ
0
l
G(x, ξ, t − τ )δ(ξ − x0 )δ(τ − t0 )dτ dξ.
0
0
&Icirc;&ntilde;&ecirc;i&euml;&uuml;&ecirc;&egrave; x ∈ (0, l), &ograve;&icirc; &iacute;&agrave; &iuml;i&auml;&ntilde;&ograve;&agrave;&acirc;i &acirc;i&auml;&icirc;&igrave;&icirc;&uml; &acirc;&euml;&agrave;&ntilde;&ograve;&egrave;&acirc;&icirc;&ntilde;&ograve;i δ -&ocirc;&oacute;&iacute;&ecirc;&ouml;i&uml;, &acirc; &eth;&aring;&ccedil;&oacute;&euml;&uuml;&ograve;&agrave;&ograve;i i&iacute;&ograve;&aring;&atilde;&eth;&oacute;&acirc;&agrave;&iacute;&iacute;&yuml; &iuml;&icirc; ξ &icirc;&ograve;&eth;&egrave;&igrave;&agrave;&sup1;&igrave;&icirc;
P
u(x, t) =
ρ
Z
t
G(x, x0 , t − τ )δ(ξ − x0 )δ(τ − t0 )dτ dξ.
0
&szlig;&ecirc;&ugrave;&icirc; t &lt; t0 , &ograve;&icirc; t0 ∈/ (0, t) i, &icirc;&ograve;&aelig;&aring;, i&iacute;&ograve;&aring;&atilde;&eth;&agrave;&euml; &auml;&icirc;&eth;i&acirc;&iacute;&thorn;&sup1; &iacute;&oacute;&euml;&thorn;, &icirc;&ntilde;&ecirc;i&euml;&uuml;&ecirc;&egrave;
δ(τ − t0 ) &auml;&icirc;&eth;i&acirc;&iacute;&thorn;&sup1; &iacute;&oacute;&euml;&thorn; &iacute;&agrave; &acirc;&ntilde;&uuml;&icirc;&igrave;&oacute; &iuml;&eth;&icirc;&igrave;i&aelig;&ecirc;&oacute; i&iacute;&ograve;&aring;&atilde;&eth;&oacute;&acirc;&agrave;&iacute;&iacute;&yuml;.
&Iuml;&eth;&egrave; t &gt; t0 , t0 ∈ (0, t) i &ograve;&icirc;&auml;i
Z
t
G(x, x0 ; t − τ )δ(τ − t0 )dτ = G(x, x0 ; t − t0 ).
0
&Ograve;&agrave;&ecirc;&egrave;&igrave; &divide;&egrave;&iacute;&icirc;&igrave;, &ecirc;&icirc;&euml;&egrave;&acirc;&agrave;&iacute;&iacute;&yuml; &ntilde;&ograve;&eth;&oacute;&iacute;&egrave; &iuml;i&auml; &acirc;&iuml;&euml;&egrave;&acirc;&icirc;&igrave; &igrave;&egrave;&ograve;&ograve;&sup1;&acirc;&icirc;&uml; &ograve;&icirc;&divide;&ecirc;&icirc;&acirc;&icirc;&uml; &auml;i&uml; &iacute;&agrave;
&ntilde;&ograve;&eth;&oacute;&iacute;&oacute; &ntilde;&egrave;&euml;&icirc;&thorn; (3.48), &acirc;i&auml;&icirc;&aacute;&eth;&agrave;&aelig;&agrave;&sup1;&ograve;&uuml;&ntilde;&yuml; &ocirc;&oacute;&iacute;&ecirc;&ouml;i&sup1;&thorn;
F (x, t) =
(
0,
t &lt; t0
P
0
ρ G(x, x ; t
− t0 ), t &gt; t0 .
(3.50)
&ETH;i&acirc;&iacute;i&ntilde;&ograve;&uuml; &iacute;&oacute;&euml;&thorn; &ocirc;&oacute;&iacute;&ecirc;&ouml;i&uml; u(x, t) &iuml;&eth;&egrave; t &lt; t0 &igrave;&agrave;&sup1; &iuml;&eth;&icirc;&ntilde;&ograve;&aring; &iuml;&icirc;&yuml;&ntilde;&iacute;&aring;&iacute;&iacute;&yuml;: &icirc;&ntilde;&ecirc;i&euml;&uuml;&ecirc;&egrave;
&acirc;i&auml; &iuml;&icirc;&divide;&agrave;&ograve;&ecirc;&oacute; &acirc;i&auml;&euml;i&ecirc;&oacute; &divide;&agrave;&ntilde;&oacute; t = 0 &auml;&icirc; &igrave;&icirc;&igrave;&aring;&iacute;&ograve;&oacute; t = t0 &iacute;&agrave; &ntilde;&ograve;&eth;&oacute;&iacute;&oacute; &iacute;&aring; &auml;i&yuml;&euml;&egrave; &ccedil;&icirc;&acirc;&iacute;i&oslash;&iacute;i &ntilde;&egrave;&euml;&egrave; (&iacute;&agrave;&atilde;&agrave;&auml;&agrave;&sup1;&igrave;&icirc;, &ugrave;&icirc; &acirc;i&auml;&iuml;&icirc;&acirc;i&auml;&iacute;&icirc; &auml;&icirc; &iuml;&icirc;&divide;&agrave;&ograve;&ecirc;&icirc;&acirc;&egrave;&otilde; &oacute;&igrave;&icirc;&acirc; (3.37), &iuml;&icirc;&divide;&agrave;&ograve;&ecirc;&icirc;&acirc;i
&acirc;i&auml;&otilde;&egrave;&euml;&aring;&iacute;&iacute;&yuml; &ograve;&agrave; &iuml;&icirc;&divide;&agrave;&ograve;&ecirc;&icirc;&acirc;i &oslash;&acirc;&egrave;&auml;&ecirc;&icirc;&ntilde;&ograve;i &ograve;&icirc;&divide;&icirc;&ecirc; &ntilde;&ograve;&eth;&oacute;&iacute;&egrave; &ograve;&agrave;&ecirc;&icirc;&aelig; &auml;&icirc;&eth;i&acirc;&iacute;&thorn;&thorn;&ograve;&uuml; &iacute;&oacute;&euml;&thorn;), &ograve;&icirc; &auml;&icirc; &iuml;&icirc;&divide;&agrave;&ograve;&ecirc;&oacute; &auml;i&uml; &ntilde;&egrave;&euml; &ntilde;&ograve;&eth;&oacute;&iacute;&agrave; &ccedil;&agrave;&euml;&egrave;&oslash;&agrave;&sup1;&ograve;&uuml;&ntilde;&yuml; &acirc; &ntilde;&ograve;&agrave;&iacute;i &ntilde;&iuml;&icirc;&ecirc;&icirc;&thorn;.
&szlig;&ecirc;&ugrave;&icirc; P/ρ = 1 (&acirc; &iuml;&aring;&acirc;&iacute;&egrave;&otilde; &icirc;&auml;&egrave;&iacute;&egrave;&ouml;&yuml;&otilde; &acirc;&egrave;&igrave;i&eth;&oacute;), &ograve;&icirc; &ecirc;&icirc;&aring;&ocirc;i&ouml;i&sup1;&iacute;&ograve; &iuml;&eth;&egrave; δ &ocirc;&oacute;&iacute;&ecirc;&ouml;i&yuml;&otilde; &acirc; (3.49) &auml;&icirc;&eth;i&acirc;&iacute;&thorn;&sup1; &icirc;&auml;&egrave;&iacute;&egrave;&ouml;i, &agrave; &ograve;&icirc;&auml;i &acirc;i&auml;&iuml;&icirc;&acirc;i&auml;&iacute;&egrave;&eacute; &ecirc;&icirc;&aring;&ocirc;i&ouml;i&sup1;&iacute;&ograve; &acirc; (3.50)
&ograve;&agrave;&ecirc;&icirc;&aelig; &auml;&icirc;&eth;i&acirc;&iacute;&thorn;&sup1; &icirc;&auml;&egrave;&iacute;&egrave;&ouml;i.
&Acirc;&egrave;&eth;&agrave;&ccedil; (3.50) &auml;&icirc;&ccedil;&acirc;&icirc;&euml;&yuml;&sup1; &iacute;&agrave;&auml;&agrave;&ograve;&egrave; &ocirc;&oacute;&iacute;&ecirc;&ouml;i&uml; &acirc;&iuml;&euml;&egrave;&acirc;&oacute; &ograve;&icirc;&divide;&ecirc;&icirc;&acirc;&icirc;&atilde;&icirc; &auml;&aelig;&aring;&eth;&aring;&euml;&agrave; &iuml;&aring;&acirc;&iacute;&aring;
&ocirc;i&ccedil;&egrave;&divide;&iacute;&aring; &ograve;&euml;&oacute;&igrave;&agrave;&divide;&aring;&iacute;&iacute;&yuml;, &agrave; &ntilde;&agrave;&igrave;&aring;: &ocirc;&oacute;&iacute;&ecirc;&ouml;i&yuml; G(x, x0 ; t − t0 ) &icirc;&iuml;&egrave;&ntilde;&oacute;&sup1; &iuml;&eth;&egrave; t &gt; t0
&ecirc;&icirc;&euml;&egrave;&acirc;&agrave;&iacute;&iacute;&yuml; &ntilde;&ograve;&eth;&oacute;&iacute;&egrave;, &yuml;&ecirc;&ugrave;&icirc; &acirc; &igrave;&icirc;&igrave;&aring;&iacute;&ograve; &divide;&agrave;&ntilde;&oacute; t = t0 &iuml;&icirc;&auml;i&yuml;&euml;&egrave; &igrave;&egrave;&ograve;&ograve;&sup1;&acirc;&icirc;&thorn; &ograve;&icirc;&divide;&ecirc;&icirc;&acirc;&icirc;&thorn;
&ntilde;&egrave;&euml;&icirc;&thorn;, &yuml;&ecirc;&agrave; &iuml;&aring;&eth;&aring;&auml;&agrave;&sup1; &ntilde;&ograve;&eth;&oacute;&iacute;i i&igrave;&iuml;&oacute;&euml;&uuml;&ntilde; P , &divide;&egrave;&ntilde;&euml;&icirc;&acirc;&aring; &ccedil;&iacute;&agrave;&divide;&aring;&iacute;&iacute;&yuml; &yuml;&ecirc;&icirc;&atilde;&icirc; &auml;&icirc;&eth;i&acirc;&iacute;&thorn;&sup1;
ρ.
36
&ETH;i&acirc;&iacute;&yuml;&iacute;&iacute;&yuml; &atilde;i&iuml;&aring;&eth;&aacute;&icirc;&euml;i&divide;&iacute;&icirc;&atilde;&icirc; &ograve;&egrave;&iuml;&oacute;.
3.1. &Ccedil;&iacute;&agrave;&eacute;&ograve;&egrave; &ccedil;&agrave;&ecirc;&icirc;&iacute; &acirc;i&euml;&uuml;&iacute;&egrave;&otilde; &ecirc;&icirc;&euml;&egrave;&acirc;&agrave;&iacute;&uuml; &ntilde;&ograve;&eth;&oacute;&iacute;&egrave; &iacute;&agrave; &acirc;i&auml;&eth;i&ccedil;&ecirc;&oacute; x ∈ [0, l], &yuml;&ecirc;&ugrave;&icirc;
&ecirc;i&iacute;&ouml;i &ntilde;&ograve;&eth;&oacute;&iacute;&egrave; &aelig;&icirc;&eth;&ntilde;&ograve;&ecirc;&icirc; &ccedil;&agrave;&ecirc;&eth;i&iuml;&euml;&aring;&iacute;i, &iuml;&icirc;&divide;&agrave;&ograve;&ecirc;&icirc;&acirc;&agrave; &oslash;&acirc;&egrave;&auml;&ecirc;i&ntilde;&ograve;&uuml; &auml;&icirc;&eth;i&acirc;&iacute;&thorn;&sup1; &iacute;&oacute;&euml;&thorn;, &iuml;&icirc;&divide;&agrave;&ograve;&ecirc;&icirc;&acirc;&aring; &ccedil;&igrave;i&ugrave;&aring;&iacute;&iacute;&yuml; (&agrave;) &igrave;&agrave;&sup1; &ocirc;&icirc;&eth;&igrave;&oacute;, &ccedil;&icirc;&aacute;&eth;&agrave;&aelig;&aring;&iacute;&oacute; &iacute;&agrave; &eth;&egrave;&ntilde;&oacute;&iacute;&ecirc;&oacute;; (&aacute;)
l
sin(πx/l).
&icirc;&iuml;&egrave;&ntilde;&oacute;&sup1;&ograve;&uuml;&ntilde;&yuml; &ocirc;&icirc;&eth;&igrave;&oacute;&euml;&icirc;&thorn; u(x, 0) = 100
3.2. &Ntilde;&ograve;&eth;&oacute;&iacute;&agrave; &auml;&icirc;&acirc;&aelig;&egrave;&iacute;&icirc;&thorn; l &ccedil; &aelig;&icirc;&eth;&ntilde;&ograve;&ecirc;&icirc; &ccedil;&agrave;&ecirc;&eth;i&iuml;&euml;&aring;&iacute;&egrave;&igrave;&egrave; &ecirc;i&iacute;&ouml;&yuml;&igrave;&egrave; &ccedil;&aacute;&oacute;&auml;&aelig;&oacute;&sup1;&ograve;&uuml;&ntilde;&yuml;
&acirc; &iuml;&icirc;&divide;&agrave;&ograve;&ecirc;&icirc;&acirc;&egrave;&eacute; &igrave;&icirc;&igrave;&aring;&iacute;&ograve; &divide;&agrave;&ntilde;&oacute; &oacute;&auml;&agrave;&eth;&icirc;&igrave; &iuml;&euml;&icirc;&ntilde;&ecirc;&icirc;&atilde;&icirc; &igrave;&icirc;&euml;&icirc;&ograve;&icirc;&divide;&ecirc;&agrave;, &ugrave;&icirc; &iacute;&agrave;&auml;&agrave;&sup1;
&oslash;&acirc;&egrave;&auml;&ecirc;i&ntilde;&ograve;&uuml; v0 &ograve;&icirc;&divide;&ecirc;&agrave;&igrave; &ntilde;&ograve;&eth;&oacute;&iacute;&egrave; &iacute;&agrave; &acirc;i&auml;&eth;i&ccedil;&ecirc;&oacute; 0 &lt; x1 ≤ x ≤ x2 &lt; l.
&Ccedil;&iacute;&agrave;&eacute;&ograve;&egrave; &acirc;i&auml;&otilde;&egrave;&euml;&aring;&iacute;&iacute;&yuml; &ntilde;&ograve;&eth;&oacute;&iacute;&egrave; &acirc;i&auml; &iuml;&icirc;&euml;&icirc;&aelig;&aring;&iacute;&iacute;&yuml; &eth;i&acirc;&iacute;&icirc;&acirc;&agrave;&atilde;&egrave;, &yuml;&ecirc;&ugrave;&icirc; &iuml;&icirc;&divide;&agrave;&ograve;&ecirc;&icirc;&acirc;&aring;
&ccedil;&igrave;i&ugrave;&aring;&iacute;&iacute;&yuml; &auml;&icirc;&eth;i&acirc;&iacute;&thorn;&sup1; &iacute;&oacute;&euml;&thorn;.
3.3. &ETH;&icirc;&ccedil;&acirc;'&yuml;&ccedil;&agrave;&ograve;&egrave; &ccedil;&agrave;&auml;&agrave;&divide;&oacute; &iuml;&eth;&icirc; &ecirc;&icirc;&euml;&egrave;&acirc;&agrave;&iacute;&iacute;&yuml; &ntilde;&ograve;&eth;&oacute;&iacute;&egrave; &auml;&icirc;&acirc;&aelig;&egrave;&iacute;&icirc;&thorn; l i&ccedil; &ccedil;&agrave;&ecirc;&eth;i&iuml;&euml;&aring;&iacute;&egrave;&igrave;&egrave; &ecirc;i&iacute;&ouml;&yuml;&igrave;&egrave;, &ugrave;&icirc; &ccedil;&aacute;&oacute;&auml;&aelig;&oacute;&sup1;&ograve;&uuml;&ntilde;&yuml; &oacute;&auml;&agrave;&eth;&icirc;&igrave; &ograve;&icirc;&iacute;&aring;&iacute;&uuml;&ecirc;&icirc;&atilde;&icirc; &igrave;&icirc;&euml;&icirc;&ograve;&icirc;&divide;&ecirc;&agrave; &acirc; &ograve;&icirc;&divide;&ouml;i
x = x0 , &ugrave;&icirc; &iuml;&aring;&eth;&aring;&auml;&agrave;&sup1; &ntilde;&ograve;&eth;&oacute;&iacute;i i&igrave;&iuml;&oacute;&euml;&uuml;&ntilde; I . &Iuml;&icirc;&divide;&agrave;&ograve;&ecirc;&icirc;&acirc;&agrave; &oslash;&acirc;&egrave;&auml;&ecirc;i&ntilde;&ograve;&uuml; &auml;&icirc;&eth;i&acirc;&iacute;&thorn;&sup1;
&iacute;&oacute;&euml;&thorn;.
3.4. &ETH;&icirc;&ccedil;&acirc;'&yuml;&ccedil;&agrave;&ograve;&egrave; &ccedil;&agrave;&auml;&agrave;&divide;&oacute; &iuml;&eth;&icirc; &iuml;&icirc;&acirc;&ccedil;&auml;&icirc;&acirc;&aelig;&iacute;i &ecirc;&icirc;&euml;&egrave;&acirc;&agrave;&iacute;&iacute;&yuml; &acirc; &iuml;&eth;&oacute;&aelig;&iacute;&uuml;&icirc;&igrave;&oacute; &ntilde;&ograve;&aring;&eth;&aelig;&iacute;i
&auml;&icirc;&acirc;&aelig;&egrave;&iacute;&icirc;&thorn; l &ccedil; &acirc;i&euml;&uuml;&iacute;&egrave;&igrave;&egrave; &ecirc;i&iacute;&ouml;&yuml;&igrave;&egrave;, &yuml;&ecirc;&ugrave;&icirc; &iuml;&icirc;&divide;&agrave;&ograve;&ecirc;&icirc;&acirc;i &ccedil;&igrave;i&ugrave;&aring;&iacute;&iacute;&yuml; i &iuml;&icirc;&divide;&agrave;&ograve;&ecirc;&icirc;&acirc;i &oslash;&acirc;&egrave;&auml;&ecirc;&icirc;&ntilde;&ograve;i &acirc; &iuml;&icirc;&acirc;&ccedil;&auml;&icirc;&acirc;&aelig;&iacute;&uuml;&icirc;&igrave;&oacute; &iacute;&agrave;&iuml;&eth;&yuml;&igrave;&ecirc;&oacute; &auml;&icirc;&acirc;i&euml;&uuml;&iacute;i. &Acirc;&eth;&agrave;&otilde;&oacute;&acirc;&agrave;&ograve;&egrave; &igrave;&icirc;&aelig;&euml;&egrave;&acirc;i&ntilde;&ograve;&uuml; &eth;i&acirc;&iacute;&icirc;&igrave;i&eth;&iacute;&icirc;&atilde;&icirc; &iuml;&icirc;&ntilde;&ograve;&oacute;&iuml;&agrave;&euml;&uuml;&iacute;&icirc;&atilde;&icirc; &eth;&oacute;&otilde;&oacute; &ntilde;&ograve;&aring;&eth;&aelig;&iacute;&yuml;.
3.5. &ETH;&icirc;&ccedil;&acirc;'&yuml;&ccedil;&agrave;&ograve;&egrave; &ccedil;&agrave;&auml;&agrave;&divide;&oacute; &iuml;&eth;&icirc; &iuml;&icirc;&acirc;&ccedil;&auml;&icirc;&acirc;&aelig;&iacute;i &ecirc;&icirc;&euml;&egrave;&acirc;&agrave;&iacute;&iacute;&yuml; &ntilde;&ograve;&aring;&eth;&aelig;&iacute;&yuml; &auml;&icirc;&acirc;&aelig;&egrave;&iacute;&icirc;&thorn; l
&iuml;&eth;&egrave; &iuml;&icirc;&divide;&agrave;&ograve;&ecirc;&icirc;&acirc;&egrave;&otilde; &oacute;&igrave;&icirc;&acirc;&agrave;&otilde; u(x, 0) = kx, ut (x, 0) = 0, &yuml;&ecirc;&ugrave;&icirc; &ecirc;i&iacute;&aring;&ouml;&uuml; x = 0
&ccedil;&agrave;&ecirc;&eth;i&iuml;&euml;&aring;&iacute;&egrave;&eacute;, &agrave; &ecirc;i&iacute;&aring;&ouml;&uuml; x = l &acirc;i&euml;&uuml;&iacute;&egrave;&eacute;.
3.6. &Ccedil;&iacute;&agrave;&eacute;&ograve;&egrave; &iuml;&icirc;&acirc;&ccedil;&auml;&icirc;&acirc;&aelig;&iacute;i &ccedil;&igrave;i&ugrave;&aring;&iacute;&iacute;&yuml; &ograve;&icirc;&divide;&icirc;&ecirc; &ntilde;&ograve;&aring;&eth;&aelig;&iacute;&yuml;, &yuml;&ecirc;&ugrave;&icirc; &ecirc;i&iacute;&aring;&ouml;&uuml; x = 0
&ccedil;&agrave;&ecirc;&eth;i&iuml;&euml;&aring;&iacute;&egrave;&eacute; &iuml;&eth;&oacute;&aelig;&iacute;&uuml;&icirc;, &agrave; &ecirc;i&iacute;&aring;&ouml;&uuml; x = l &acirc;i&euml;&uuml;&iacute;&egrave;&eacute;. &Iuml;&eth;&oacute;&aelig;&iacute;&sup1; &ccedil;&agrave;&ecirc;&eth;i&iuml;&euml;&aring;&iacute;&iacute;&yuml;
&icirc;&ccedil;&iacute;&agrave;&divide;&agrave;&sup1;, &ugrave;&icirc; &iacute;&agrave; &ecirc;i&iacute;&aring;&ouml;&uuml; &auml;i&sup1; &iuml;&icirc;&acirc;&ccedil;&auml;&icirc;&acirc;&aelig;&iacute;&yuml; &ntilde;&egrave;&euml;&agrave;, &iuml;&eth;&icirc;&iuml;&icirc;&eth;&ouml;i&eacute;&iacute;&agrave; &ccedil;&igrave;i&ugrave;&aring;&iacute;&iacute;&thorn;
i &iacute;&agrave;&iuml;&eth;&agrave;&acirc;&euml;&aring;&iacute;&agrave; &acirc; &iuml;&eth;&icirc;&ograve;&egrave;&euml;&aring;&aelig;&iacute;&icirc;&igrave;&oacute; &iacute;&agrave;&iuml;&eth;&yuml;&igrave;&ecirc;&oacute;.
3.7. &Auml;&icirc; &ntilde;&ograve;&aring;&euml;i &euml;i&ocirc;&ograve;&agrave;, &ugrave;&icirc; &eth;i&acirc;&iacute;&icirc;&igrave;i&eth;&iacute;&icirc; &eth;&oacute;&otilde;&agrave;&sup1;&ograve;&uuml;&ntilde;&yuml; i&ccedil; &oslash;&acirc;&egrave;&auml;&ecirc;i&ntilde;&ograve;&thorn; v0 , &aelig;&icirc;&eth;&ntilde;&ograve;&ecirc;&icirc;
&ccedil;&agrave;&ecirc;&eth;i&iuml;&euml;&aring;&iacute;&egrave;&eacute; &ntilde;&ograve;&aring;&eth;&aelig;&aring;&iacute;&uuml; &ecirc;i&iacute;&ouml;&aring;&igrave; x = 0, &iuml;&eth;&egrave; &ouml;&uuml;&icirc;&igrave;&oacute; &ecirc;i&iacute;&aring;&ouml;&uuml; x = l &acirc;i&euml;&uuml;37
&iacute;&egrave;&eacute;. &Ccedil;&iacute;&agrave;&eacute;&ograve;&egrave; &iuml;&icirc;&acirc;&ccedil;&auml;&icirc;&acirc;&aelig;&iacute;i &ecirc;&icirc;&euml;&egrave;&acirc;&agrave;&iacute;&iacute;&yuml; &ntilde;&ograve;&aring;&eth;&aelig;&iacute;&yuml; &iuml;i&ntilde;&euml;&yuml; &igrave;&egrave;&ograve;&ograve;&sup1;&acirc;&icirc;&uml; &ccedil;&oacute;&iuml;&egrave;&iacute;&ecirc;&egrave;
&euml;i&ocirc;&ograve;&agrave;. &Iuml;&icirc;&euml;&icirc;&aelig;&aring;&iacute;&iacute;&yuml; &ntilde;&ograve;&aring;&eth;&aelig;&iacute;&yuml; &acirc;&aring;&eth;&ograve;&egrave;&ecirc;&agrave;&euml;&uuml;&iacute;&aring;
3.8. &Ccedil;&iacute;&agrave;&eacute;&ograve;&egrave; &acirc;i&auml;&otilde;&egrave;&euml;&aring;&iacute;&iacute;&yuml; &acirc;i&auml; &iuml;&icirc;&euml;&icirc;&aelig;&aring;&iacute;&iacute;&yuml; &eth;i&acirc;&iacute;&icirc;&acirc;&agrave;&atilde;&egrave; &iuml;&eth;&yuml;&igrave;&icirc;&ecirc;&oacute;&ograve;&iacute;&icirc;&uml; &igrave;&aring;&igrave;&aacute;&eth;&agrave;&iacute;&egrave;
i&ccedil; &ntilde;&ograve;&icirc;&eth;&icirc;&iacute;&agrave;&igrave;&egrave; x1 , y1 , &yuml;&ecirc;&ugrave;&icirc; &ecirc;&eth;&agrave;&uml; &igrave;&aring;&igrave;&aacute;&eth;&agrave;&iacute;&egrave; &aelig;&icirc;&eth;&ntilde;&ograve;&ecirc;&icirc; &ccedil;&agrave;&ecirc;&eth;i&iuml;&euml;&aring;&iacute;i. &Iuml;&icirc;&divide;&agrave;&ograve;&ecirc;&icirc;&acirc;i &oacute;&igrave;&icirc;&acirc;&egrave; &auml;&icirc;&acirc;i&euml;&uuml;&iacute;i: u(x, y, 0) = φ(x, y) , ut (x, y, 0) = ψ(x, y).
3.9. &ETH;&icirc;&ccedil;&acirc;'&yuml;&ccedil;&agrave;&ograve;&egrave; &ccedil;&agrave;&auml;&agrave;&divide;&oacute; &iuml;&eth;&icirc; &ecirc;&icirc;&euml;&egrave;&acirc;&agrave;&iacute;&iacute;&yuml; &igrave;&aring;&igrave;&aacute;&eth;&agrave;&iacute;&egrave;, &ugrave;&icirc; &igrave;&agrave;&sup1; &ocirc;&icirc;&eth;&igrave;&oacute; &eth;i&acirc;&iacute;&icirc;&aacute;&aring;-
&auml;&eth;&aring;&iacute;&icirc;&atilde;&icirc; &iuml;&eth;&yuml;&igrave;&icirc;&ecirc;&oacute;&ograve;&iacute;&icirc;&atilde;&icirc; &ograve;&eth;&egrave;&ecirc;&oacute;&ograve;&iacute;&egrave;&ecirc;&agrave; &ccedil; &ecirc;&agrave;&ograve;&aring;&ograve;&agrave;&igrave;&egrave; &eth;i&acirc;&iacute;&egrave;&igrave;&egrave; l. &Ecirc;&eth;&agrave;&uml; &igrave;&aring;&igrave;&aacute;&eth;&agrave;&iacute;&egrave; &ccedil;&agrave;&ecirc;&eth;i&iuml;&euml;&aring;&iacute;i. &Iuml;&icirc;&divide;&agrave;&ograve;&ecirc;&icirc;&acirc;i &oacute;&igrave;&icirc;&acirc;&egrave; &auml;&icirc;&acirc;i&euml;&uuml;&iacute;i: u(x, y, 0) = φ(x, y),
ut (x, y, 0) = ψ(x, y).
3.10. &ETH;&icirc;&ccedil;&acirc;'&yuml;&ccedil;&agrave;&ograve;&egrave; &ccedil;&agrave;&auml;&agrave;&divide;&oacute; utt = a2 uxx + Ash(kx) &auml;&euml;&yuml; t &gt; 0, x ∈ [0, l] i&ccedil;
&iacute;&oacute;&euml;&uuml;&icirc;&acirc;&egrave;&igrave;&egrave; &iuml;&icirc;&divide;&agrave;&ograve;&ecirc;&icirc;&acirc;&egrave;&igrave;&egrave; i &ecirc;&eth;&agrave;&eacute;&icirc;&acirc;&egrave;&igrave;&egrave; &oacute;&igrave;&icirc;&acirc;&agrave;&igrave;&egrave;.
3.11. &Ccedil;&iacute;&agrave;&eacute;&ograve;&egrave; &ecirc;&icirc;&euml;&egrave;&acirc;&agrave;&iacute;&iacute;&yuml; &ntilde;&ograve;&eth;&oacute;&iacute;&egrave; &auml;&icirc;&acirc;&aelig;&egrave;&iacute;&icirc;&thorn; l &iuml;i&auml; &auml;i&sup1;&thorn; &ccedil;&icirc;&acirc;&iacute;i&oslash;&iacute;&uuml;&icirc;&uml; &ntilde;&egrave;&euml;&egrave; &ccedil; &euml;i&iacute;i&eacute;&iacute;&icirc;&thorn; &atilde;&oacute;&ntilde;&ograve;&egrave;&iacute;&icirc;&thorn; F (x) = f0 x(l − x)t2 . &Acirc;i&auml;&otilde;&egrave;&euml;&aring;&iacute;&iacute;&yuml; i &oslash;&acirc;&egrave;&auml;&ecirc;&icirc;&ntilde;&ograve;i &ograve;&icirc;&divide;&icirc;&ecirc;
&ntilde;&ograve;&eth;&oacute;&iacute;&egrave; &acirc; &iuml;&icirc;&divide;&agrave;&ograve;&ecirc;&icirc;&acirc;&egrave;&eacute; &igrave;&icirc;&igrave;&aring;&iacute;&ograve; &divide;&agrave;&ntilde;&oacute; &auml;&icirc;&eth;i&acirc;&iacute;&thorn;&thorn;&ograve;&uuml; &iacute;&oacute;&euml;&thorn;.
&ETH;i&acirc;&iacute;&yuml;&iacute;&iacute;&yuml; &iuml;&agrave;&eth;&agrave;&aacute;&icirc;&euml;i&divide;&iacute;&icirc;&atilde;&icirc; &ograve;&egrave;&iuml;&oacute;.
&Iuml;&eth;&egrave;&ecirc;&euml;&agrave;&auml; 3
&ETH;&icirc;&ccedil;&acirc;'&yuml;&ccedil;&agrave;&ograve;&egrave; &ecirc;&eth;&agrave;&eacute;&icirc;&acirc;&oacute; &ccedil;&agrave;&auml;&agrave;&divide;&oacute;:
ut = a2 uxx − βu
u(0, t) = ux (l, t) = 0, u(x, 0) = sin πx
2l , x ∈ (0, l), t &gt; 0.
&Aacute;&oacute;&auml;&aring;&igrave;&icirc; &oslash;&oacute;&ecirc;&agrave;&ograve;&egrave; &eth;&icirc;&ccedil;&acirc;'&yuml;&ccedil;&icirc;&ecirc; &oacute; &acirc;&egrave;&atilde;&euml;&yuml;&auml;i: u(x, t) = e−βt v(x, t). &Iuml;i&auml;&ntilde;&ograve;&agrave;&acirc;&egrave;&acirc;&egrave;&acirc;&oslash;&egrave; &acirc; &eth;i&acirc;&iacute;&yuml;&iacute;&iacute;&yuml;, &iuml;&icirc;&divide;&agrave;&ograve;&ecirc;&icirc;&acirc;i i &ecirc;&eth;&agrave;&eacute;&icirc;&acirc;i &oacute;&igrave;&icirc;&acirc;&egrave; &ouml;&aring;&eacute; &acirc;&egrave;&eth;&agrave;&ccedil; &icirc;&ograve;&eth;&egrave;&igrave;&agrave;&sup1;&igrave;&icirc;
&auml;&euml;&yuml; &ocirc;&oacute;&iacute;&ecirc;&ouml;i&uml; v(x, t) &icirc;&auml;&iacute;&icirc;&eth;i&auml;&iacute;&oacute; &ecirc;&eth;&agrave;&eacute;&icirc;&acirc;&oacute; &ccedil;&agrave;&auml;&agrave;&divide;&oacute;:
vt = a2 vxx
v(0, t) = vx (l, t) = 0, v(x, 0) = sin πx
2l , x ∈ (0, l), t &gt; 0.
&Aacute;&oacute;&auml;&aring;&igrave;&icirc; &eth;&icirc;&ccedil;&acirc;'&yuml;&ccedil;&oacute;&acirc;&agrave;&ograve;&egrave; &ouml;&thorn; &ccedil;&agrave;&auml;&agrave;&divide;&oacute; &igrave;&aring;&ograve;&icirc;&auml;&icirc;&igrave; &eth;&icirc;&ccedil;&auml;i&euml;&aring;&iacute;&iacute;&yuml; &ccedil;&igrave;i&iacute;&iacute;&egrave;&otilde;, &iuml;&eth;&aring;&auml;&ntilde;&ograve;&agrave;&acirc;&euml;&yuml;&thorn;&divide;&egrave; &eth;&icirc;&ccedil;&acirc;'&yuml;&ccedil;&icirc;&ecirc; &oacute; &acirc;&egrave;&atilde;&euml;&yuml;&auml;i v(x, t) = X(x)T (t).
X 00 (x)
Ṫ (t)
= 2
= &micro;,
X(x)
a T (t)
&auml;&aring; &micro; &auml;&aring;&yuml;&ecirc;&agrave; &ecirc;&icirc;&iacute;&ntilde;&ograve;&agrave;&iacute;&ograve;&agrave;.
38
&Auml;&euml;&yuml; &ocirc;&oacute;&iacute;&ecirc;&ouml;i&uml; X(x) &acirc;&egrave;&ecirc;&icirc;&eth;&egrave;&ntilde;&ograve;&icirc;&acirc;&oacute;&thorn;&divide;&egrave; &ecirc;&eth;&agrave;&eacute;&icirc;&acirc;i &oacute;&igrave;&icirc;&acirc;&egrave;, &icirc;&ograve;&eth;&egrave;&igrave;&agrave;&sup1;&igrave;&icirc; &ccedil;&agrave;&auml;&agrave;&divide;&oacute; &Oslash;&ograve;&oacute;&eth;&igrave;&agrave;-&Euml;i&oacute;&acirc;i&euml;&yuml;:
X 00 (x) − &micro;X(x) = 0;
(3.51)
X(0) = 0; X 0 (l) = 0,
(3.52)
&ETH;&icirc;&ccedil;&atilde;&euml;&yuml;&iacute;&aring;&igrave;&icirc; &acirc;&ntilde;i &igrave;&icirc;&aelig;&euml;&egrave;&acirc;i &ccedil;&iacute;&agrave;&divide;&aring;&iacute;&iacute;&yuml; &ecirc;&icirc;&iacute;&ntilde;&ograve;&agrave;&iacute;&ograve;&egrave; &micro;.
• &iuml;&eth;&egrave; &micro; = λ2 &gt; 0 &ccedil;&agrave;&atilde;&agrave;&euml;&uuml;&iacute;&egrave;&eacute; &eth;&icirc;&ccedil;&acirc;'&yuml;&ccedil;&icirc;&ecirc; &eth;i&acirc;&iacute;&yuml;&iacute;&iacute;&yuml; &igrave;&agrave;&sup1; &acirc;&egrave;&atilde;&euml;&yuml;&auml; X(x) =
Aeλx + Be−λx . &Auml;&agrave;&euml;i &iacute;&agrave;&ecirc;&euml;&agrave;&auml;&agrave;&sup1;&igrave;&icirc; &ecirc;&eth;&agrave;&eacute;&icirc;&acirc;i
&oacute;&igrave;&icirc;&acirc;&egrave; (3.52)
X(0) =
&iexcl;
&cent;
A + B = 0 ⇒ A = −B , X 0 (l) = λA eλl − (−λ)e−λl = 0. &Ccedil;&agrave;&auml;&icirc;&acirc;&icirc;&euml;&uuml;&iacute;&egrave;&ograve;&egrave; &ouml;&thorn; &oacute;&igrave;&icirc;&acirc;&oacute; &igrave;&icirc;&aelig;&iacute;&agrave; &euml;&egrave;&oslash;&aring; &iuml;&eth;&egrave; A = 0. &Ograve;&agrave;&ecirc;&egrave;&igrave; &divide;&egrave;&iacute;&icirc;&igrave; &auml;&euml;&yuml;
&acirc;&egrave;&iuml;&agrave;&auml;&ecirc;&oacute; &micro; &gt; 0 &igrave;&agrave;&sup1;&igrave;&icirc; &euml;&egrave;&oslash;&aring; &ograve;&eth;&egrave;&acirc;i&agrave;&euml;&uuml;&iacute;&egrave;&eacute; &eth;&icirc;&ccedil;&acirc;'&yuml;&ccedil;&icirc;&ecirc;.
• &iuml;&eth;&egrave; &micro; = 0 &iuml;i&auml;&ntilde;&ograve;&agrave;&acirc;&euml;&yuml;&sup1;&igrave;&icirc; &ccedil;&agrave;&atilde;&agrave;&euml;&uuml;&iacute;&egrave;&eacute; &eth;&icirc;&ccedil;&acirc;'&yuml;&ccedil;&icirc;&ecirc; X(x) = Ax + B &acirc;
&oacute;&igrave;&icirc;&acirc;&egrave; (3.52), &ccedil;&acirc;i&auml;&ecirc;&egrave; &icirc;&ograve;&eth;&egrave;&igrave;&agrave;&sup1;&igrave;&icirc; X(0) = B = 0, X 0 (l) = A = 0,
&ograve;&icirc;&aacute;&ograve;&icirc; i &auml;&euml;&yuml; &acirc;&egrave;&iuml;&agrave;&auml;&ecirc;&oacute; &micro; = 0 &ograve;&agrave;&ecirc;&icirc;&aelig; &igrave;&icirc;&aelig;&euml;&egrave;&acirc;&egrave;&eacute; &euml;&egrave;&oslash;&aring; &ograve;&eth;&egrave;&acirc;i&agrave;&euml;&uuml;&iacute;&egrave;&eacute;
&eth;&icirc;&ccedil;&acirc;'&yuml;&ccedil;&icirc;&ecirc;.
• &iuml;&eth;&egrave; &micro; = −λ2 &lt; 0 &ccedil;&agrave;&atilde;&agrave;&euml;&uuml;&iacute;&egrave;&eacute; &eth;&icirc;&ccedil;&acirc;'&yuml;&ccedil;&ecirc;&icirc; X(x) = A sin λx+B cos λx,
&ccedil; &ecirc;&eth;&agrave;&eacute;&icirc;&acirc;&egrave;&otilde; &oacute;&igrave;&icirc;&acirc; &ccedil;&iacute;&agrave;&otilde;&icirc;&auml;&egrave;&igrave;&icirc; X(0) = B = 0, X 0 (l) = Aλ cos λl =
0 ⇒ λl = π2 + πn, &auml;&aring; n &ouml;i&euml;&aring; &divide;&egrave;&ntilde;&euml;&icirc;. &Ograve;&agrave;&ecirc;&egrave;&igrave; &divide;&egrave;&iacute;&icirc;&igrave;, &acirc;&euml;&agrave;&ntilde;&iacute;i &ccedil;&iacute;&agrave;&divide;&aring;&sup3;
&acute;2
π(2n+1)
,
&iacute;&iacute;&yuml; &ccedil;&agrave;&auml;&agrave;&divide;i &Oslash;&ograve;&oacute;&eth;&igrave;&agrave; &Euml;i&oacute;&acirc;i&euml;&euml;&yuml; &igrave;&agrave;&thorn;&ograve;&uuml; &acirc;&egrave;&atilde;&euml;&yuml;&auml;: &micro;n = −
2l
&agrave; &acirc;&euml;&agrave;&ntilde;&iacute;i &ocirc;&oacute;&iacute;&ecirc;&ouml;i&uml;, &acirc;i&auml;&iuml;&icirc;&acirc;i&auml;&iacute;&icirc;, Xn (x) = An sin
π(2n+1)x
.
2l
&ETH;&icirc;&ccedil;&acirc;'&yuml;&aelig;&aring;&igrave;&icirc; &ograve;&aring;&iuml;&aring;&eth; &eth;i&acirc;&iacute;&yuml;&iacute;&iacute;&yuml; &auml;&euml;&yuml; &ocirc;&oacute;&iacute;&ecirc;&ouml;i&uml; T (t):
2
Ṫn (t) = a &micro;n Tn (t) ⇒ Tn (t) = Cn e
−γn t
a2 (2n + 1)2 π 2
, γn =
.
4l2
&Ccedil;&agrave;&atilde;&agrave;&euml;&uuml;&iacute;&egrave;&eacute; &eth;&icirc;&ccedil;&acirc;'&yuml;&ccedil;&icirc;&ecirc; &ecirc;&eth;&agrave;&eacute;&icirc;&acirc;&icirc;&uml; &ccedil;&agrave;&auml;&agrave;&divide;i &igrave;&agrave;&sup1; &acirc;&egrave;&atilde;&euml;&yuml;&auml;:
v(x, t) =
∞
X
Bn e−γn t sin
n=1
π(2n + 1)x
2l
(3.53)
&Ecirc;&icirc;&aring;&ocirc;i&ouml;i&sup1;&iacute;&ograve;&egrave; Bn &ccedil;&iacute;&agrave;&otilde;&icirc;&auml;&egrave;&igrave;&icirc;, &iacute;&agrave;&ecirc;&euml;&agrave;&auml;&agrave;&thorn;&divide;&egrave; &iuml;&icirc;&divide;&agrave;&ograve;&ecirc;&icirc;&acirc;&oacute; &oacute;&igrave;&icirc;&acirc;&oacute;:
v(x, 0) =
∞
X
n=1
Bn sin
π(2n + 1)x
πx
= sin
⇒ Bk = 0, k 6= 0, B0 = 1.
2l
2l
&Icirc;&ntilde;&ograve;&agrave;&ograve;&icirc;&divide;&iacute;&icirc; &igrave;&agrave;&sup1;&igrave;&icirc; &acirc;i&auml;&iuml;&icirc;&acirc;i&auml;&uuml;:
u(x, t) = e
−βt
v(x, t) = e
39
2 2
−βt− π4la2 t
sin
πx
.
2l
3.12. &Ccedil;&iacute;&agrave;&eacute;&ograve;&egrave; &eth;&icirc;&ccedil;&iuml;&icirc;&auml;i&euml; &ograve;&aring;&igrave;&iuml;&aring;&eth;&agrave;&ograve;&oacute;&eth;&egrave; &acirc;&ntilde;&aring;&eth;&aring;&auml;&egrave;&iacute;i &ograve;&icirc;&iacute;&ecirc;&icirc;&atilde;&icirc; &ntilde;&ograve;&aring;&eth;&aelig;&iacute;&yuml; x ∈ [0, l]
i&ccedil; &ograve;&aring;&iuml;&euml;&icirc;i&ccedil;&icirc;&euml;&uuml;&icirc;&acirc;&agrave;&iacute;&icirc;&thorn; &aacute;i&divide;&iacute;&icirc;&thorn; &iuml;&icirc;&acirc;&aring;&eth;&otilde;&iacute;&aring;&thorn;, &yuml;&ecirc;&ugrave;&icirc;
(&agrave;) &ograve;&aring;&igrave;&iuml;&aring;&eth;&agrave;&ograve;&oacute;&eth;&agrave; &eacute;&icirc;&atilde;&icirc; &ecirc;i&iacute;&ouml;i&acirc; &iuml;i&auml;&ograve;&eth;&egrave;&igrave;&oacute;&sup1;&ograve;&uuml;&ntilde;&yuml; &eth;i&acirc;&iacute;&icirc;&thorn; &iacute;&oacute;&euml;&thorn;;
(&aacute;) &ograve;&aring;&igrave;&iuml;&aring;&eth;&agrave;&ograve;&oacute;&eth;&agrave; &ecirc;i&iacute;&ouml;&yuml; x = 0 &iuml;i&auml;&ograve;&eth;&egrave;&igrave;&oacute;&sup1;&ograve;&uuml;&ntilde;&yuml; &eth;i&acirc;&iacute;&icirc;&thorn; &iacute;&oacute;&euml;&thorn;, &agrave; &ecirc;i&iacute;&aring;&ouml;&uuml;
x = l &ograve;&aring;&iuml;&euml;&icirc;i&ccedil;&icirc;&euml;&uuml;&icirc;&acirc;&agrave;&iacute;&egrave;&eacute;;
(&acirc;) &icirc;&aacute;&egrave;&auml;&acirc;&agrave; &ecirc;i&iacute;&ouml;i &ntilde;&ograve;&aring;&eth;&aelig;&iacute;&yuml; &ograve;&aring;&iuml;&euml;&icirc;i&ccedil;&icirc;&euml;&uuml;&icirc;&acirc;&agrave;&iacute;i. &Iuml;&icirc;&divide;&agrave;&ograve;&ecirc;&icirc;&acirc;&agrave; &ograve;&aring;&igrave;&iuml;&aring;&eth;&agrave;&ograve;&oacute;&eth;&agrave;
u(x, 0) = f (x). &ETH;&icirc;&ccedil;&atilde;&euml;&yuml;&iacute;&oacute;&ograve;&egrave; &acirc;&egrave;&iuml;&agrave;&auml;&icirc;&ecirc; f (x) = u0 =const.
3.13. &Ccedil;&iacute;&agrave;&eacute;&ograve;&egrave; &eth;&icirc;&ccedil;&iuml;&icirc;&auml;i&euml; &ograve;&aring;&igrave;&iuml;&aring;&eth;&agrave;&ograve;&oacute;&eth;&egrave; &acirc;&ntilde;&aring;&eth;&aring;&auml;&egrave;&iacute;i &ograve;&icirc;&iacute;&ecirc;&icirc;&atilde;&icirc; &ntilde;&ograve;&aring;&eth;&aelig;&iacute;&yuml; x ∈ [0, l] i&ccedil;
&ograve;&aring;&iuml;&euml;&icirc;i&ccedil;&icirc;&euml;&uuml;&icirc;&acirc;&agrave;&iacute;&icirc;&thorn; &aacute;i&divide;&iacute;&icirc;&thorn; &iuml;&icirc;&acirc;&aring;&eth;&otilde;&iacute;&aring;&thorn;, &yuml;&ecirc;&ugrave;&icirc; &ograve;&aring;&igrave;&iuml;&aring;&eth;&agrave;&ograve;&oacute;&eth;&agrave; &ecirc;i&iacute;&ouml;&yuml; x = 0
&iuml;i&auml;&ograve;&eth;&egrave;&igrave;&oacute;&sup1;&ograve;&uuml;&ntilde;&yuml; &eth;i&acirc;&iacute;&icirc;&thorn; &iacute;&oacute;&euml;&thorn;, &agrave; &iacute;&agrave; &ecirc;i&iacute;&ouml;i x = l &ograve;&aring;&igrave;&iuml;&aring;&eth;&agrave;&ograve;&oacute;&eth;&agrave; &ccedil;&igrave;i&iacute;&thorn;&sup1;&ograve;&uuml;&ntilde;&yuml; &ccedil;&agrave; &ccedil;&agrave;&ecirc;&icirc;&iacute;&icirc;&igrave; u(l, t) = Ae−γt . &Iuml;&icirc;&divide;&agrave;&ograve;&ecirc;&icirc;&acirc;&agrave; &ograve;&aring;&igrave;&iuml;&aring;&eth;&agrave;&ograve;&oacute;&eth;&agrave; &acirc;&ntilde;&aring;&eth;&aring;&auml;&egrave;&iacute;i
&ntilde;&ograve;&aring;&eth;&aelig;&iacute;&yuml;: u(x, 0) = Ax/l.
3.14. &Ccedil;&iacute;&agrave;&eacute;&ograve;&egrave; &ograve;&aring;&igrave;&iuml;&aring;&eth;&agrave;&ograve;&oacute;&eth;&oacute; &acirc; &ograve;&icirc;&iacute;&ecirc;&icirc;&igrave;&oacute; &icirc;&auml;&iacute;&icirc;&eth;i&auml;&iacute;&icirc;&igrave;&oacute; &ecirc;i&euml;&uuml;&ouml;i &auml;&icirc;&acirc;&aelig;&egrave;&iacute;&icirc;&thorn; l, &yuml;&ecirc;&ugrave;&icirc;
&aacute;i&divide;&iacute;&agrave; &iuml;&icirc;&acirc;&aring;&eth;&otilde;&iacute;&yuml; &ecirc;i&euml;&uuml;&ouml;&yuml; &ograve;&aring;&iuml;&euml;&icirc;i&ccedil;&icirc;&euml;&uuml;&icirc;&acirc;&agrave;&iacute;&agrave;, &agrave; &iuml;&icirc;&divide;&agrave;&ograve;&ecirc;&icirc;&acirc;&egrave;&eacute; &eth;&icirc;&ccedil;&iuml;&icirc;&auml;i&euml; &ograve;&aring;&igrave;&iuml;&aring;&eth;&agrave;&ograve;&oacute;&eth;&egrave; &acirc; &ecirc;i&euml;&uuml;&ouml;i &auml;&icirc;&acirc;i&euml;&uuml;&iacute;&egrave;&eacute;: u(x, 0) = f (x)
3.15. &Acirc;&egrave;&ccedil;&iacute;&agrave;&divide;&egrave;&ograve;&egrave; &ecirc;&eth;&egrave;&ograve;&egrave;&divide;&iacute;&oacute; &ograve;&icirc;&acirc;&ugrave;&egrave;&iacute;&oacute; &oslash;&agrave;&eth;&oacute;, &acirc; &yuml;&ecirc;&icirc;&igrave;&oacute; &acirc;i&auml;&aacute;&oacute;&acirc;&agrave;&sup1;&ograve;&uuml;&ntilde;&yuml; &auml;&egrave;&ocirc;&oacute;&ccedil;i&yuml;
&divide;&agrave;&ntilde;&ograve;&egrave;&iacute;&icirc;&ecirc; i&ccedil; &eth;&icirc;&ccedil;&igrave;&iacute;&icirc;&aelig;&aring;&iacute;&iacute;&yuml;&igrave;.
(&agrave;) &Ecirc;&icirc;&iacute;&ouml;&aring;&iacute;&ograve;&eth;&agrave;&ouml;i&yuml; &divide;&agrave;&ntilde;&ograve;&egrave;&iacute;&icirc;&ecirc; &iacute;&agrave; &acirc;&aring;&eth;&otilde;&iacute;i&eacute; i &iacute;&egrave;&aelig;&iacute;i&eacute; &iuml;&icirc;&acirc;&aring;&eth;&otilde;&iacute;i &iacute;&oacute;&euml;&uuml;&icirc;&acirc;&agrave;.
(&aacute;) &Iuml;&icirc;&ograve;i&ecirc; &divide;&agrave;&ntilde;&ograve;&egrave;&iacute;&icirc;&ecirc; &divide;&aring;&eth;&aring;&ccedil; &iacute;&egrave;&aelig;&iacute;&thorn; &iuml;&icirc;&acirc;&aring;&eth;&otilde;&iacute;&thorn; &oslash;&agrave;&eth;&oacute; &auml;&icirc;&eth;i&acirc;&iacute;&thorn;&sup1; &iacute;&oacute;&euml;&thorn;,
&agrave; &iacute;&agrave; &acirc;&aring;&eth;&otilde;&iacute;i&eacute; &iuml;&icirc;&acirc;&aring;&eth;&otilde;&iacute;i &ecirc;&icirc;&iacute;&ouml;&aring;&iacute;&ograve;&eth;&agrave;&ouml;i&yuml; &iacute;&oacute;&euml;&uuml;&icirc;&acirc;&agrave;. &Iuml;&icirc;&divide;&agrave;&ograve;&ecirc;&icirc;&acirc;&egrave;&eacute; &eth;&icirc;&ccedil;&iuml;&icirc;&auml;i&euml;
&ecirc;&icirc;&iacute;&ouml;&aring;&iacute;&ograve;&eth;&agrave;&ouml;i&uml; &acirc;&ntilde;&aring;&eth;&aring;&auml;&egrave;&iacute;i &oslash;&agrave;&eth;&oacute; &auml;&icirc;&acirc;i&euml;&uuml;&iacute;&egrave;&eacute;.
3.16. &Acirc;&egrave;&ccedil;&iacute;&agrave;&divide;&egrave;&ograve;&egrave; &ecirc;&eth;&egrave;&ograve;&egrave;&divide;&iacute;i &eth;&icirc;&ccedil;&igrave;i&eth;&egrave; &ecirc;&oacute;&aacute;&oacute;, &yuml;&ecirc;&ugrave;&icirc; &acirc;&ntilde;&aring;&eth;&aring;&auml;&egrave;&iacute;i &acirc;i&auml;&aacute;&oacute;&acirc;&agrave;&sup1;&ograve;&uuml;&ntilde;&yuml; &auml;&egrave;-
&ocirc;&oacute;&ccedil;i&yuml; &divide;&agrave;&ntilde;&ograve;&egrave;&iacute;&icirc;&ecirc; i&ccedil; &eth;&icirc;&ccedil;&igrave;&iacute;&icirc;&aelig;&aring;&iacute;&iacute;&yuml;&igrave;. &Ecirc;&icirc;&iacute;&ouml;&aring;&iacute;&ograve;&eth;&agrave;&ouml;i&yuml; &divide;&agrave;&ntilde;&ograve;&egrave;&iacute;&icirc;&ecirc; &iacute;&agrave; &iuml;&icirc;&acirc;&aring;&eth;&otilde;&iacute;i &iacute;&oacute;&euml;&uuml;&icirc;&acirc;&agrave;. &Iuml;&icirc;&divide;&agrave;&ograve;&ecirc;&icirc;&acirc;&egrave;&eacute; &eth;&icirc;&ccedil;&iuml;&icirc;&auml;i&euml; &ecirc;&icirc;&iacute;&ouml;&aring;&iacute;&ograve;&eth;&agrave;&ouml;i&uml; u(x, y, z, 0) = f (x, y, z).
3.17. &Acirc;&egrave;&ccedil;&iacute;&agrave;&divide;&egrave;&ograve;&egrave; &ecirc;&eth;&egrave;&ograve;&egrave;&divide;&iacute;i &eth;&icirc;&ccedil;&igrave;i&eth;&egrave; &ecirc;&oacute;&euml;i, &yuml;&ecirc;&ugrave;&icirc; &acirc;&ntilde;&aring;&eth;&aring;&auml;&egrave;&iacute;i &acirc;i&auml;&aacute;&oacute;&acirc;&agrave;&sup1;&ograve;&uuml;&ntilde;&yuml; &auml;&egrave;-
&ocirc;&oacute;&ccedil;i&yuml; &divide;&agrave;&ntilde;&ograve;&egrave;&iacute;&icirc;&ecirc; i&ccedil; &eth;&icirc;&ccedil;&igrave;&iacute;&icirc;&aelig;&aring;&iacute;&iacute;&yuml;&igrave;. &Ecirc;&icirc;&iacute;&ouml;&aring;&iacute;&ograve;&eth;&agrave;&ouml;i&yuml; &divide;&agrave;&ntilde;&ograve;&egrave;&iacute;&icirc;&ecirc; &iacute;&agrave; &iuml;&icirc;&acirc;&aring;&eth;&otilde;&iacute;i &iacute;&oacute;&euml;&uuml;&icirc;&acirc;&agrave;. &Iuml;&icirc;&divide;&agrave;&ograve;&ecirc;&icirc;&acirc;&egrave;&eacute; &eth;&icirc;&ccedil;&iuml;&icirc;&auml;i&euml; &divide;&agrave;&ntilde;&ograve;&egrave;&iacute;&icirc;&ecirc; &ccedil;&agrave;&euml;&aring;&aelig;&egrave;&ograve;&uuml; &euml;&egrave;&oslash;&aring; &acirc;i&auml; &acirc;i&auml;&ntilde;&ograve;&agrave;&iacute;i &auml;&icirc; &ouml;&aring;&iacute;&ograve;&eth;&oacute; &ecirc;&oacute;&euml;i: u(r, 0) = f (r).
3.18. &ETH;&icirc;&ccedil;&acirc;'&yuml;&ccedil;&agrave;&ograve;&egrave; &ccedil;&agrave;&auml;&agrave;&divide;&oacute; &iuml;&eth;&icirc; &icirc;&otilde;&icirc;&euml;&icirc;&auml;&aelig;&aring;&iacute;&iacute;&yuml; &ecirc;&oacute;&euml;i &eth;&agrave;&auml;i&oacute;&ntilde;&oacute; r0 , &yuml;&ecirc;&ugrave;&icirc; &acirc; &iuml;&icirc;&divide;&agrave;&ograve;&ecirc;&icirc;&acirc;&egrave;&eacute; &igrave;&icirc;&igrave;&aring;&iacute;&ograve; &divide;&agrave;&ntilde;&oacute; &eth;&icirc;&ccedil;&iuml;&icirc;&auml;i&euml; &ograve;&aring;&igrave;&iuml;&aring;&eth;&agrave;&ograve;&oacute;&eth;&egrave; &ccedil;&agrave;&euml;&aring;&aelig;&agrave;&acirc; &euml;&egrave;&oslash;&aring; &acirc;i&auml; &acirc;i&auml;&ntilde;&ograve;&agrave;&iacute;i &auml;&icirc;
&ouml;&aring;&iacute;&ograve;&eth;&oacute; &ecirc;&oacute;&euml;i: u(r, 0) = f (r).
(&agrave;) &Iacute;&agrave; &iuml;&icirc;&acirc;&aring;&eth;&otilde;&iacute;i &ecirc;&oacute;&euml;i &iuml;i&auml;&ograve;&eth;&egrave;&igrave;&oacute;&sup1;&ograve;&uuml;&ntilde;&yuml; &iacute;&oacute;&euml;&uuml;&icirc;&acirc;&agrave; &ograve;&aring;&igrave;&iuml;&aring;&eth;&agrave;&ograve;&oacute;&eth;&agrave;;
(&aacute;) &Iuml;&icirc;&acirc;&aring;&eth;&otilde;&iacute;&yuml; &ecirc;&oacute;&euml;i &iuml;i&auml;&ograve;&eth;&egrave;&igrave;&oacute;&thorn;&ograve;&uuml;&ntilde;&yuml; &iuml;&eth;&egrave; &ntilde;&ograve;&agrave;&euml;i&eacute; &ograve;&aring;&igrave;&iuml;&aring;&eth;&agrave;&ograve;&oacute;&eth;i u0 =const;
40
(&acirc;) &Iuml;&icirc;&acirc;&aring;&eth;&otilde;&iacute;&yuml; &ecirc;&oacute;&euml;i &acirc;i&euml;&uuml;&iacute;&icirc; &icirc;&otilde;&icirc;&euml;&icirc;&auml;&aelig;&oacute;&sup1;&ograve;&uuml;&ntilde;&yuml; &ccedil;&agrave; &ccedil;&agrave;&ecirc;&icirc;&iacute;&icirc;&igrave; &Iacute;&uuml;&thorn;&ograve;&icirc;&iacute;&agrave; &acirc; &ntilde;&aring;&eth;&aring;&auml;&icirc;&acirc;&egrave;&ugrave;i &ccedil; &iacute;&oacute;&euml;&uuml;&icirc;&acirc;&icirc;&thorn; &ograve;&aring;&igrave;&iuml;&aring;&eth;&agrave;&ograve;&oacute;&eth;&icirc;&thorn;.
3.19. &Aacute;i&divide;&iacute;&agrave; &iuml;&icirc;&acirc;&aring;&eth;&otilde;&iacute;&yuml; &ecirc;&oacute;&euml;i &eth;&agrave;&auml;i&oacute;&ntilde;&oacute; r0 &icirc;&iuml;&eth;&icirc;&igrave;i&iacute;&thorn;&sup1;&ograve;&uuml;&ntilde;&yuml; &icirc;&auml;&iacute;&icirc;&eth;i&auml;&iacute;&egrave;&igrave; &iuml;&icirc;&ograve;&icirc;&ecirc;&icirc;&igrave;
&ograve;&aring;&iuml;&euml;&agrave; &atilde;&oacute;&ntilde;&ograve;&egrave;&iacute;&icirc;&thorn; q . &Ccedil;&iacute;&agrave;&eacute;&ograve;&egrave; &ograve;&aring;&igrave;&iuml;&aring;&eth;&agrave;&ograve;&oacute;&eth;&oacute; &acirc;&ntilde;&aring;&eth;&aring;&auml;&egrave;&iacute;i &ecirc;&oacute;&euml;i &iuml;&eth;&egrave; t &gt; 0,
&yuml;&ecirc;&ugrave;&icirc; &acirc; &iuml;&icirc;&divide;&agrave;&ograve;&ecirc;&icirc;&acirc;&egrave;&eacute; &igrave;&icirc;&igrave;&aring;&iacute;&ograve; &divide;&agrave;&ntilde;&oacute; &ograve;&aring;&igrave;&iuml;&aring;&eth;&agrave;&ograve;&oacute;&eth;&agrave; &ecirc;&oacute;&euml;i &auml;&icirc;&eth;i&acirc;&iacute;&thorn;&acirc;&agrave;&euml;&agrave; &iacute;&oacute;&euml;&thorn;.
&ETH;i&acirc;&iacute;&yuml;&iacute;&iacute;&yuml; &aring;&euml;i&iuml;&ograve;&egrave;&divide;&iacute;&icirc;&atilde;&icirc; &ograve;&egrave;&iuml;&oacute;.
&Iuml;&eth;&egrave;&ecirc;&euml;&agrave;&auml; 4
&Ccedil;&iacute;&agrave;&eacute;&ograve;&egrave; &aring;&euml;&aring;&ecirc;&ograve;&eth;&icirc;&ntilde;&ograve;&agrave;&ograve;&egrave;&divide;&iacute;&egrave;&eacute; &iuml;&icirc;&ograve;&aring;&iacute;&ouml;i&agrave;&euml; &acirc;&ntilde;&aring;&eth;&aring;&auml;&egrave;&iacute;i &icirc;&aacute;&euml;&agrave;&ntilde;&ograve;i &igrave;i&aelig; &iuml;&eth;&icirc;&acirc;i&auml;&iacute;&egrave;&igrave;&egrave; &iuml;&euml;&agrave;&ntilde;&ograve;&egrave;&iacute;&agrave;&igrave;&egrave; x = 0, y = 0, y = y0 , &yuml;&ecirc;&ugrave;&icirc; &iuml;&euml;&agrave;&ntilde;&ograve;&egrave;&iacute;&agrave; x = 0 &igrave;&agrave;&sup1;
&iuml;&icirc;&ograve;&aring;&iacute;&ouml;i&agrave;&euml; u(0, y) = U0 = A(y0 − y)y/y02 &agrave; &iuml;&euml;&agrave;&ntilde;&ograve;&egrave;&iacute;&egrave; y = 0 &ograve;&agrave; y = y0
&ccedil;&agrave;&ccedil;&aring;&igrave;&euml;&aring;&iacute;i (&auml;&egrave;&acirc;. &eth;&egrave;&ntilde;&oacute;&iacute;&icirc;&ecirc;). &Acirc;&ntilde;&aring;&eth;&aring;&auml;&egrave;&iacute;i &icirc;&aacute;&euml;&agrave;&ntilde;&ograve;i &acirc;i&auml;&ntilde;&oacute;&ograve;&iacute;i &acirc;i&euml;&uuml;&iacute;i &ccedil;&agrave;&eth;&yuml;&auml;&egrave;.
&Ecirc;&eth;&agrave;&eacute;&icirc;&acirc;&agrave; &ccedil;&agrave;&auml;&agrave;&divide;&agrave; &auml;&euml;&yuml; &aring;&euml;&aring;&ecirc;&ograve;&eth;&icirc;&ntilde;&ograve;&agrave;&ograve;&egrave;&divide;&iacute;&icirc;&atilde;&icirc; &iuml;&icirc;&ograve;&aring;&iacute;&ouml;i&agrave;&euml;&oacute; u(x, y) &igrave;&agrave;&sup1; &acirc;&egrave;&atilde;&euml;&yuml;&auml;:
uxx + uyy = 0
u(0, y) = A(y0 − y)y/y02 , u(∞, y) = 0, u(x, 0) = u(x, y0 ) = 0, x ∈
(0, ∞), y ∈ (0, y0 ).
&Iuml;i&ntilde;&euml;&yuml; &eth;&icirc;&ccedil;&auml;i&euml;&aring;&iacute;&iacute;&yuml; &ccedil;&igrave;i&iacute;&iacute;&egrave;&otilde; &icirc;&ograve;&eth;&egrave;&igrave;&agrave;&sup1;&igrave;&icirc;:
X 00 (x) Y 00 (y)
−
=
= &micro;.
X(x)
Y (y)
&Auml;&euml;&yuml; &ocirc;&oacute;&iacute;&ecirc;&ouml;i&uml; Y (y) &icirc;&ograve;&eth;&egrave;&igrave;&agrave;&sup1;&igrave;&icirc; &ccedil;&agrave;&auml;&agrave;&divide;&oacute; &Oslash;&ograve;&oacute;&eth;&igrave;&agrave;-&Euml;i&oacute;&acirc;i&euml;&euml;&yuml;, &agrave;&iacute;&agrave;&euml;&icirc;&atilde;i&divide;&iacute;&oacute;
&auml;&icirc; &eth;&icirc;&atilde;&euml;&yuml;&iacute;&oacute;&ograve;&icirc;&uml; &acirc; &Iuml;&eth;&egrave;&ecirc;&euml;&agrave;&auml;i 2. &Ograve;&icirc;&igrave;&oacute; &igrave;&icirc;&aelig;&iacute;&agrave; &iexcl;&icirc;&auml;&eth;&agrave;&ccedil;&oacute;
&iacute;&agrave;&iuml;&egrave;&ntilde;&agrave;&ograve;&egrave; &acirc;&euml;&agrave;&ntilde;&iacute;i
&cent;
πn 2
&ocirc;&oacute;&iacute;&ecirc;&ouml;i&uml; i &acirc;&euml;&agrave;&ntilde;&iacute;i &ccedil;&iacute;&agrave;&divide;&aring;&iacute;&iacute;&yuml; &ouml;i&uml; &ccedil;&agrave;&auml;&agrave;&divide;i: &micro;n = − l , Yn (y) = An sin(πny/y0 ).
&Acirc;i&auml;&iuml;&icirc;&acirc;i&auml;&iacute;&aring; &eth;i&acirc;&iacute;&yuml;&iacute;&iacute;&yuml; &auml;&euml;&yuml; X(x) &igrave;&agrave;&sup1; &acirc;&egrave;&atilde;&euml;&yuml;&auml;:
Xn00 (x)
−
&sup3; πn &acute;2
l
41
Xn (x) = 0,
&Ccedil;&agrave;&atilde;&agrave;&euml;&uuml;&iacute;&egrave;&eacute; &eth;&icirc;&ccedil;&acirc;'&yuml;&ccedil;&icirc;&ecirc; &ouml;&uuml;&icirc;&atilde;&icirc; &eth;i&acirc;&iacute;&yuml;&iacute;&iacute;&yuml;: Xn (x) = Bn e−πnx/y0 +Cn eπnx/y0 .
I&ccedil; &ecirc;&eth;&agrave;&eacute;&icirc;&acirc;&icirc;&uml; &oacute;&igrave;&icirc;&acirc;&egrave; &iuml;&eth;&egrave; x → ∞ &igrave;&icirc;&aelig;&iacute;&agrave; &acirc;&egrave;&ccedil;&iacute;&agrave;&divide;&egrave;&ograve;&egrave; &icirc;&auml;&iacute;&oacute; &ccedil; &ecirc;&icirc;&iacute;&ntilde;&ograve;&agrave;&iacute;&ograve;:
Xn (∞) = 0 ⇒ Cn = 0.
&Ograve;&agrave;&ecirc;&egrave;&igrave; &divide;&egrave;&iacute;&icirc;&igrave;, &igrave;&agrave;&sup1;&igrave;&icirc; &ccedil;&agrave;&atilde;&agrave;&euml;&uuml;&iacute;&egrave;&eacute; &eth;&icirc;&ccedil;&acirc;'&yuml;&ccedil;&icirc;&ecirc;:
u(x, y) =
∞
X
−πnx/y0
An e
n=1
sin
&sup3; πnx &acute;
l
.
(3.54)
&Iacute;&agrave;&ecirc;&euml;&agrave;&auml;&agrave;&thorn;&divide;&egrave; &oacute;&igrave;&icirc;&acirc;&oacute; u(0, y) = U0 i &acirc;&egrave;&ecirc;&icirc;&eth;&egrave;&ntilde;&ograve;&icirc;&acirc;&oacute;&thorn;&divide;&egrave; &eth;&aring;&ccedil;&oacute;&euml;&uuml;&ograve;&agrave;&ograve;&egrave; &ccedil;&agrave;&auml;&agrave;&divide;i
&Iuml;&eth;&egrave;&ecirc;&euml;&agrave;&auml;&oacute; 2, &icirc;&ntilde;&ograve;&agrave;&ograve;&icirc;&divide;&iacute;&icirc; &icirc;&ograve;&eth;&egrave;&igrave;&agrave;&sup1;&igrave;&icirc;:
∞
8A X (−1)m −(2m+1)πx/y0
(2m + 1)πx
u(x, y) = 3
.
e
sin
π m=0 (2m + 1)3
l
3.20. &Ccedil;&iacute;&agrave;&eacute;&ograve;&egrave; &aring;&euml;&aring;&ecirc;&ograve;&eth;&icirc;&ntilde;&ograve;&agrave;&ograve;&egrave;&divide;&iacute;&egrave;&eacute; &iuml;&icirc;&ograve;&aring;&iacute;&ouml;i&agrave;&euml; &acirc;&ntilde;&aring;&eth;&aring;&auml;&egrave;&iacute;i &iuml;&agrave;&eth;&agrave;&euml;&aring;&euml;i&iuml;i&iuml;&aring;&auml;&oacute; i&ccedil; &iuml;&eth;&icirc;&acirc;i&auml;&iacute;&egrave;&igrave;&egrave; &aacute;i&divide;&iacute;&egrave;&igrave;&egrave; &atilde;&eth;&agrave;&iacute;&yuml;&igrave;&egrave;, &yuml;&ecirc;&ugrave;&icirc; &icirc;&auml;&iacute;&agrave; &atilde;&eth;&agrave;&iacute;&uuml; &igrave;&agrave;&sup1; &iuml;&icirc;&ograve;&aring;&iacute;&ouml;i&agrave;&euml; u0 =const,
&agrave; &acirc;&ntilde;i i&iacute;&oslash;i &atilde;&eth;&agrave;&iacute;i &ccedil;&agrave;&ccedil;&aring;&igrave;&euml;&aring;&iacute;i.
3.21. &Ccedil;&iacute;&agrave;&eacute;&ograve;&egrave; &ntilde;&ograve;&agrave;&ouml;i&icirc;&iacute;&agrave;&eth;&iacute;&egrave;&eacute; &eth;&icirc;&ccedil;&iuml;&icirc;&auml;i&euml; &ograve;&aring;&igrave;&iuml;&aring;&eth;&agrave;&ograve;&oacute;&eth;&egrave; &acirc;&ntilde;&aring;&eth;&aring;&auml;&egrave;&iacute;i &iacute;&aring;&ntilde;&ecirc;i&iacute;&divide;&aring;&iacute;&iacute;&icirc;&atilde;&icirc;
&ouml;&egrave;&euml;i&iacute;&auml;&eth;&oacute; &eth;&agrave;&auml;i&oacute;&ntilde;&oacute; r0 , &yuml;&ecirc;&ugrave;&icirc; &iuml;&icirc;&euml;&icirc;&acirc;&egrave;&iacute;&agrave; &ouml;&egrave;&euml;i&iacute;&auml;&eth;&agrave; ϕ ∈ [0, π] &igrave;&agrave;&sup1; &ograve;&aring;&igrave;&iuml;&aring;&eth;&agrave;&ograve;&oacute;&eth;&oacute; &iuml;&icirc;&acirc;&aring;&eth;&otilde;&iacute;i, &eth;i&acirc;&iacute;&oacute; U1 , &agrave; &auml;&eth;&oacute;&atilde;&agrave; &iuml;&icirc;&euml;&icirc;&acirc;&egrave;&iacute;&agrave; &ograve;&aring;&igrave;&iuml;&aring;&eth;&agrave;&ograve;&oacute;&eth;&oacute; U2 .
3.22. &Ccedil;&iacute;&agrave;&eacute;&ograve;&egrave; &ntilde;&ograve;&agrave;&ouml;i&icirc;&iacute;&agrave;&eth;&iacute;&egrave;&eacute; &eth;&icirc;&ccedil;&iuml;&icirc;&auml;i&euml; &ograve;&aring;&igrave;&iuml;&aring;&eth;&agrave;&ograve;&oacute;&eth;&egrave; &acirc;&ntilde;&aring;&eth;&aring;&auml;&egrave;&iacute;i &ecirc;i&euml;&uuml;&ouml;&aring;&acirc;&icirc;&atilde;&icirc; &ntilde;&aring;-
&ecirc;&ograve;&icirc;&eth;&agrave; r1 &lt; r &lt; r2 , 0 &lt; ϕ &lt; α, &yuml;&ecirc;&ugrave;&icirc; &ograve;&aring;&igrave;&iuml;&aring;&eth;&agrave;&ograve;&oacute;&eth;&agrave; &iacute;&agrave; &atilde;&eth;&agrave;&iacute;&egrave;&ouml;&yuml;&otilde; &ccedil;&agrave;&auml;&agrave;&sup1;&ograve;&uuml;&ntilde;&yuml; &eth;i&acirc;&iacute;&icirc;&ntilde;&ograve;&yuml;&igrave;&egrave;: u(r, 0) = u(r, α) = u(r1 , ϕ) = 0, u(r2 , ϕ) = f (ϕ).
3.23. &Ccedil;&iacute;&agrave;&eacute;&ograve;&egrave; &iuml;&icirc;&ograve;&aring;&iacute;&ouml;i&agrave;&euml; &ccedil;&icirc;&acirc;&iacute;i &iacute;&aring;&ntilde;&ecirc;i&iacute;&divide;&aring;&iacute;&iacute;&icirc;&atilde;&icirc; &iuml;&eth;&icirc;&acirc;i&auml;&iacute;&icirc;&atilde;&icirc; &ccedil;&agrave;&ccedil;&aring;&igrave;&euml;&aring;&iacute;&icirc;&atilde;&icirc; &ouml;&egrave;-
&euml;i&iacute;&auml;&eth;&oacute; &eth;&agrave;&auml;i&ntilde;&oacute; r0 , &ugrave;&icirc; &ccedil;&iacute;&agrave;&otilde;&icirc;&auml;&egrave;&ograve;&ntilde;&yuml; &acirc; &icirc;&auml;&iacute;&icirc;&eth;i&auml;&iacute;&icirc;&igrave;&oacute; &aring;&euml;&aring;&ecirc;&ograve;&eth;&egrave;&divide;&iacute;&icirc;&igrave;&oacute; &iuml;&icirc;&euml;i
~ 0 , &iacute;&agrave;&iuml;&eth;&agrave;&acirc;&euml;&aring;&iacute;&icirc;&igrave;&oacute; &iuml;&aring;&eth;&iuml;&aring;&iacute;&auml;&egrave;&ecirc;&oacute;&euml;&yuml;&eth;&iacute;&icirc; &auml;&icirc; &icirc;&ntilde;i &ouml;&egrave;&euml;i&iacute;&auml;&eth;&oacute;. &Acirc;&egrave;&ccedil;&iacute;&agrave;&divide;&egrave;&ograve;&egrave; &iuml;&icirc;E
&acirc;&aring;&eth;&otilde;&iacute;&aring;&acirc;&oacute; &atilde;&oacute;&ntilde;&ograve;&egrave;&iacute;&oacute; &ccedil;&agrave;&eth;&yuml;&auml;i&acirc; &iacute;&agrave; &ouml;&egrave;&euml;i&iacute;&auml;&eth;i.
42
3.24. &Iacute;&aring;&ntilde;&ecirc;i&iacute;&divide;&aring;&iacute;&egrave;&eacute; &iuml;&eth;&icirc;&acirc;i&auml;&iacute;&egrave;&eacute; &ouml;&egrave;&euml;i&iacute;&auml;&eth; &ccedil;&iacute;&agrave;&otilde;&icirc;&auml;&egrave;&ograve;&uuml;&ntilde;&yuml; &acirc; &ccedil;&icirc;&acirc;&iacute;i&oslash;&iacute;&uuml;&icirc;&igrave;&oacute; &icirc;&auml;&iacute;&icirc;&eth;i-
~ 0 , &iacute;&agrave;&iuml;&eth;&agrave;&acirc;&euml;&aring;&iacute;&icirc;&igrave;&oacute; &acirc;&ccedil;&auml;&icirc;&acirc;&aelig; &icirc;&ntilde;i x. &Ograve;&acirc;i&eth;&iacute;&agrave;
&auml;&iacute;&icirc;&igrave;&oacute; &aring;&euml;&aring;&ecirc;&ograve;&eth;&egrave;&divide;&iacute;&icirc;&igrave;&oacute; &iuml;&icirc;&euml;i E
&ouml;&egrave;&euml;i&iacute;&auml;&eth;&oacute; &iuml;&agrave;&eth;&agrave;&euml;&aring;&euml;&uuml;&iacute;&agrave; &icirc;&ntilde;i z . &Ccedil;&iacute;&agrave;&eacute;&ograve;&egrave; &atilde;&oacute;&ntilde;&ograve;&egrave;&iacute;&oacute; &iuml;&icirc;&acirc;&aring;&eth;&otilde;&iacute;&aring;&acirc;&icirc;&atilde;&icirc; &ccedil;&agrave;&eth;&yuml;&auml;&oacute; &iacute;&agrave;
&ouml;&egrave;&euml;i&iacute;&auml;&eth;i.
3.25. &Ccedil;&iacute;&agrave;&eacute;&ograve;&egrave; &ntilde;&ograve;&agrave;&ouml;i&icirc;&iacute;&agrave;&eth;&iacute;&egrave;&eacute; &eth;&icirc;&ccedil;&iuml;&icirc;&auml;i&euml; &ograve;&aring;&igrave;&iuml;&aring;&eth;&agrave;&ograve;&oacute;&eth;&egrave; &acirc; &ograve;&acirc;&aring;&eth;&auml;&icirc;&igrave;&oacute; &ograve;i&euml;i, &ugrave;&icirc; &icirc;&aacute;&igrave;&aring;&aelig;&aring;&iacute;&aring; &iacute;&aring;&ntilde;&ecirc;i&iacute;&divide;&aring;&iacute;&iacute;&egrave;&igrave;&egrave; &ecirc;&icirc;&agrave;&ecirc;&ntilde;i&agrave;&euml;&uuml;&iacute;&egrave;&igrave;&egrave; &ouml;&egrave;&euml;i&iacute;&auml;&eth;&egrave;&divide;&iacute;&egrave;&igrave;&egrave; &iuml;&icirc;&acirc;&aring;&eth;&otilde;&iacute;&yuml;&igrave;&egrave; i&ccedil;
&eth;&agrave;&auml;i&oacute;&ntilde;&agrave;&igrave;&egrave; r1 i r2 (r1 &lt; r2 ), &yuml;&ecirc;&ugrave;&icirc; &iacute;&agrave; &iuml;&icirc;&acirc;&aring;&eth;&otilde;&iacute;i &acirc;&iacute;&oacute;&ograve;&eth;i&oslash;&iacute;&uuml;&icirc;&atilde;&icirc; &ouml;&egrave;&euml;i&iacute;&auml;&eth;&oacute;
&iuml;i&auml;&ograve;&eth;&egrave;&igrave;&oacute;&sup1;&ograve;&uuml;&ntilde;&yuml; &iuml;&icirc;&ntilde;&ograve;i&eacute;&iacute;&agrave; &ograve;&aring;&igrave;&iuml;&aring;&eth;&agrave;&ograve;&oacute;&eth;&agrave; u0 . &Iuml;&icirc;&euml;&icirc;&acirc;&egrave;&iacute;&agrave; 0 ≤ ϕ ≤ π &ccedil;&icirc;&acirc;&iacute;i&oslash;&iacute;&uuml;&icirc;&atilde;&icirc; &ouml;&egrave;&euml;i&iacute;&auml;&eth;&oacute; &iuml;i&auml;&ograve;&eth;&egrave;&igrave;&oacute;&sup1;&ograve;&uuml;&ntilde;&yuml; &iuml;&eth;&egrave; &iacute;&oacute;&euml;&uuml;&icirc;&acirc;i&eacute; &ograve;&aring;&igrave;&iuml;&eth;&aring;&eth;&agrave;&ograve;&oacute;&eth;i, &agrave; i&iacute;&oslash;&agrave;
&iuml;&icirc;&euml;&icirc;&acirc;&egrave;&iacute;&agrave; π &lt; ϕ ≤ 2π &iuml;&eth;&egrave; &ograve;&aring;&igrave;&iuml;&aring;&eth;&agrave;&ograve;&oacute;&eth;i u0 .
4. &Ccedil;&agrave;&auml;&agrave;&divide;i &ccedil; &acirc;&egrave;&ecirc;&icirc;&eth;&egrave;&ntilde;&ograve;&agrave;&iacute;&iacute;&yuml;&igrave; δ -&ocirc;&oacute;&iacute;&ecirc;&ouml;i&eacute;.
4.1. &Ccedil;&agrave;&iuml;&egrave;&ntilde;&agrave;&ograve;&egrave; &icirc;&aacute;'&sup1;&igrave;&iacute;&oacute; &atilde;&oacute;&ntilde;&ograve;&egrave;&iacute;&oacute; &ccedil;&agrave;&eth;&yuml;&auml;&oacute; &acirc; &auml;&aring;&ecirc;&agrave;&eth;&ograve;&icirc;&acirc;i&eacute;, &ouml;&egrave;&euml;i&iacute;&auml;&eth;&egrave;&divide;&iacute;i&eacute; &agrave;&aacute;&icirc;
&ntilde;&ocirc;&aring;&eth;&egrave;&divide;&iacute;i&eacute; &ntilde;&egrave;&ntilde;&ograve;&aring;&igrave;i &ecirc;&icirc;&icirc;&eth;&auml;&egrave;&iacute;&agrave;&ograve; &auml;&euml;&yuml; &acirc;&egrave;&iuml;&agrave;&auml;&ecirc;&oacute;, &ecirc;&icirc;&euml;&egrave; &ccedil;&agrave;&eth;&yuml;&auml; &eth;i&acirc;&iacute;&icirc;&igrave;i&eth;&iacute;&icirc;
&eth;&icirc;&ccedil;&iuml;&icirc;&auml;i&euml;&aring;&iacute;&egrave;&eacute; &iuml;&icirc; &iuml;&icirc;&acirc;&aring;&eth;&otilde;&iacute;&yuml;&igrave; &agrave;&aacute;&icirc; &acirc;&ccedil;&auml;&icirc;&acirc;&aelig; &euml;i&iacute;i&eacute;:
(a) &Auml;&euml;&yuml; &ecirc;&oacute;&euml;i &eth;&agrave;&auml;i&oacute;&ntilde;&oacute; R i&ccedil; &iuml;&icirc;&acirc;&aring;&eth;&otilde;&iacute;&aring;&acirc;&icirc;&thorn; &atilde;&oacute;&ntilde;&ograve;&egrave;&iacute;&icirc;&thorn; σ .
(&aacute;) &Auml;&euml;&yuml; &iuml;i&acirc;&ecirc;&oacute;&euml;i &eth;&agrave;&auml;i&oacute;&ntilde;&oacute; R i&ccedil; &iuml;&icirc;&acirc;&aring;&eth;&otilde;&iacute;&aring;&acirc;&icirc;&thorn; &atilde;&oacute;&ntilde;&ograve;&egrave;&iacute;&icirc;&thorn; σ .
(&acirc;) &Ograve;&icirc;&iacute;&ecirc;&aring; &ecirc;i&euml;&uuml;&ouml;&aring; &eth;&agrave;&auml;i&oacute;&ntilde;&oacute; R, &ugrave;&icirc; &ccedil;&iacute;&agrave;&otilde;&icirc;&auml;&egrave;&ograve;&uuml;&ntilde;&yuml; &acirc; &iuml;&euml;&icirc;&ugrave;&egrave;&iacute;i (x, y). &Euml;i&iacute;i&eacute;&iacute;&agrave;
&atilde;&oacute;&ntilde;&ograve;&egrave;&iacute;&agrave; &ccedil;&agrave;&eth;&yuml;&auml;&oacute; γ .
(&atilde;) &Iacute;&agrave;&iuml;i&acirc;&ecirc;i&euml;&uuml;&ouml;&aring; &eth;&agrave;&auml;i&oacute;&ntilde;&oacute; R, &euml;i&iacute;i&eacute;&iacute;&agrave; &atilde;&oacute;&ntilde;&ograve;&egrave;&iacute;&agrave; &ccedil;&agrave;&eth;&yuml;&auml;&oacute; γ .
(&auml;) &Ograve;&icirc;&iacute;&ecirc;&egrave;&eacute; &ntilde;&ograve;&aring;&eth;&aelig;&aring;&iacute;&uuml; &auml;&icirc;&acirc;&aelig;&egrave;&iacute;&icirc;&thorn; l, &ugrave;&icirc; &eth;&icirc;&ccedil;&ograve;&agrave;&oslash;&icirc;&acirc;&agrave;&iacute;&egrave;&eacute; &acirc;&ccedil;&auml;&icirc;&acirc;&aelig; &auml;&icirc;&auml;&agrave;&ograve;&iacute;&uuml;&icirc;&uml; &divide;&agrave;&ntilde;&ograve;&egrave;&iacute;&egrave; &icirc;&ntilde;i z . &Euml;i&iacute;i&eacute;&iacute;&agrave; &atilde;&oacute;&ntilde;&ograve;&egrave;&iacute;&agrave; &ccedil;&agrave;&eth;&yuml;&auml;&oacute; γ .
(&aring;) &Ograve;&icirc;&iacute;&ecirc;&egrave;&eacute; &auml;&egrave;&ntilde;&ecirc; &eth;&agrave;&auml;i&oacute;&ntilde;&oacute; R, &ugrave;&icirc; &euml;&aring;&aelig;&egrave;&ograve;&uuml; &acirc; &iuml;&euml;&icirc;&ugrave;&egrave;&iacute;i (x, y), &ccedil;&agrave;&eth;&yuml;&auml;&aelig;&aring;&iacute;&egrave;&eacute;
&ccedil; &iuml;&icirc;&acirc;&aring;&eth;&otilde;&iacute;&aring;&acirc;&icirc;&thorn; &atilde;&oacute;&ntilde;&ograve;&egrave;&iacute;&icirc;&thorn; σ .
(&aelig;) &Ouml;&egrave;&euml;i&iacute;&auml;&eth;&egrave;&divide;&iacute;&agrave; &iuml;&icirc;&acirc;&aring;&eth;&otilde;&iacute;&yuml; &eth;&agrave;&auml;i&oacute;&ntilde;&oacute; R i &acirc;&egrave;&ntilde;&icirc;&ograve;&icirc;&thorn; h, &ccedil;&agrave;&eth;&yuml;&auml;&aelig;&aring;&iacute;&agrave; &ccedil; &iuml;&icirc;&acirc;&aring;&eth;&otilde;&iacute;&aring;&acirc;&icirc;&thorn; &atilde;&oacute;&ntilde;&ograve;&egrave;&iacute;&icirc;&thorn; σ . &Icirc;&ntilde;&iacute;&icirc;&acirc;&agrave; &ouml;&egrave;&euml;i&iacute;&auml;&eth;&oacute; &eth;&icirc;&ccedil;&ograve;&agrave;&oslash;&icirc;&acirc;&agrave;&iacute;&agrave; &acirc; &iuml;&euml;&icirc;&ugrave;&egrave;&iacute;i
(x, y).
4.2. &Auml;&icirc;&acirc;&aring;&ntilde;&ograve;&egrave; &eth;i&acirc;&iacute;&icirc;&ntilde;&ograve;i:
1 ε
= δ(x);
ε→+0 π x2 + ε2
(a) lim
2
dδ(x)
xε
=
−
;
ε→+0 π (x2 + ε2 )2
dx
(&acirc;) lim
43
2
x2 ε
= δ(x)
ε→+0 π (x2 + ε2 )2
(&aacute;) lim
(&auml;) x
dδ(x)
= −δ(x);
dx
1 − cos(nx)
(&aring;) lim
= δ(x);
n→∞
πnx2
1
(&aelig;)
2π
Z
+∞
eikx dk = δ(x)
−∞
4.3. &Auml;&icirc;&acirc;&aring;&ntilde;&ograve;&egrave;, &ugrave;&icirc; &auml;&euml;&yuml; &aacute;&oacute;&auml;&uuml;-&yuml;&ecirc;&icirc;&uml; &atilde;&euml;&agrave;&auml;&ecirc;&icirc;&uml; &ocirc;&oacute;&iacute;&ecirc;&ouml;i&uml; f (x) &igrave;&agrave;&sup1; &igrave;i&ntilde;&ouml;&aring; &eth;i&acirc;&iacute;i&ntilde;&ograve;&uuml;:
f (x)
dδ(x − a)
dδ(x − a)
df (x)
= f (a)
− δ(x − a)
dx
dx
dx
4.4. &Auml;&icirc;&acirc;&aring;&ntilde;&ograve;&egrave;, &ugrave;&icirc; &yuml;&ecirc;&ugrave;&icirc; f 0 (an ) 6= 0, &auml;&aring; {an } &igrave;&iacute;&icirc;&aelig;&egrave;&iacute;&agrave; &iacute;&oacute;&euml;i&acirc; &ocirc;&oacute;&iacute;&ecirc;&ouml;i&uml; f (x):
f (an ) = 0, &ograve;&icirc;
δ(f (x)) =
X δ(x − an )
n
|f 0 (an )|
5. &Igrave;&aring;&ograve;&icirc;&auml; &eth;&icirc;&ccedil;&auml;i&euml;&aring;&iacute;&iacute;&yuml; &ccedil;&igrave;i&iacute;&iacute;&egrave;&otilde; &ccedil; &acirc;&egrave;&ecirc;&icirc;&eth;&egrave;&ntilde;&ograve;&agrave;&iacute;&iacute;&yuml;&igrave;
&ntilde;&iuml;&aring;&ouml;i&agrave;&euml;&uuml;&iacute;&egrave;&otilde; &ocirc;&oacute;&iacute;&ecirc;&ouml;i&eacute;.
&Ccedil;&agrave;&auml;&agrave;&divide;i &ccedil; &acirc;&egrave;&ecirc;&icirc;&eth;&egrave;&ntilde;&ograve;&agrave;&iacute;&iacute;&yuml;&igrave; &ouml;&egrave;&euml;i&iacute;&auml;&eth;&egrave;&divide;&iacute;&egrave;&otilde; &ocirc;&oacute;&iacute;&ecirc;&ouml;i&eacute;.
5.1. &Ccedil;&iacute;&agrave;&eacute;&ograve;&egrave; &acirc;&egrave;&eth;&agrave;&ccedil; &auml;&euml;&yuml; &aring;&euml;&aring;&ecirc;&ograve;&eth;&icirc;&ntilde;&ograve;&agrave;&ograve;&egrave;&divide;&iacute;&icirc;&atilde;&icirc; &iuml;&icirc;&ograve;&aring;&iacute;&ouml;i&agrave;&euml;&oacute; &acirc;&ntilde;&aring;&eth;&aring;&auml;&egrave;&iacute;i &ouml;&egrave;&euml;i&iacute;-
&auml;&eth;&oacute; &eth;&agrave;&auml;i&oacute;&ntilde;&oacute; r0 &ograve;&agrave; &acirc;&egrave;&ntilde;&icirc;&ograve;&icirc;&thorn; h, &yuml;&ecirc;&ugrave;&icirc; &aacute;i&divide;&iacute;&agrave; &iuml;&icirc;&acirc;&aring;&eth;&otilde;&iacute;&yuml; i &acirc;&aring;&eth;&otilde;&iacute;&yuml; &icirc;&ntilde;&iacute;&icirc;&acirc;&agrave;
&ccedil;&agrave;&ccedil;&aring;&igrave;&euml;&aring;&iacute;i, &agrave; &iacute;&egrave;&aelig;&iacute;&yuml; &icirc;&ntilde;&iacute;&icirc;&acirc;&agrave; &igrave;&agrave;&sup1; &iuml;&icirc;&ntilde;&ograve;i&eacute;&iacute;&egrave;&eacute; &iuml;&icirc;&ograve;&aring;&iacute;&ouml;i&agrave;&euml; u0 .
5.2. &Ograve;&aring;&igrave;&iuml;&aring;&eth;&agrave;&ograve;&oacute;&eth;&agrave; &iacute;&egrave;&aelig;&iacute;&uuml;&icirc;&uml; &icirc;&ntilde;&iacute;&icirc;&acirc;&egrave; i &aacute;i&divide;&iacute;&icirc;&uml; &iuml;&icirc;&acirc;&aring;&eth;&otilde;&iacute;i &ouml;&egrave;&euml;i&iacute;&auml;&eth;&oacute; &auml;&icirc;&eth;i&acirc;&iacute;&thorn;&sup1;
&iacute;&oacute;&euml;&thorn;. &ETH;&agrave;&auml;i&oacute;&ntilde; &ouml;&egrave;&euml;i&iacute;&auml;&eth;&oacute; r0 , &acirc;&egrave;&ntilde;&icirc;&ograve;&agrave; &auml;&icirc;&eth;i&acirc;&iacute;&thorn;&sup1; h. &Ccedil;&iacute;&agrave;&eacute;&ograve;&egrave; &ntilde;&ograve;&agrave;&ouml;i&icirc;&iacute;&agrave;&eth;&iacute;&egrave;&eacute;
&eth;&icirc;&ccedil;&iuml;&icirc;&auml;i&euml; &ograve;&aring;&igrave;&iuml;&aring;&eth;&agrave;&ograve;&oacute;&eth;&egrave; &acirc;&ntilde;&aring;&eth;&aring;&auml;&egrave;&iacute;i &ouml;&egrave;&euml;i&iacute;&auml;&eth;&oacute;. &szlig;&ecirc;&ugrave;&icirc; &ograve;&aring;&igrave;&iuml;&aring;&eth;&agrave;&ograve;&oacute;&eth;&agrave; &acirc;&aring;&eth;&otilde;&iacute;&uuml;&icirc;&uml; &icirc;&ntilde;&iacute;&icirc;&acirc;&egrave; (&agrave;) &igrave;&agrave;&sup1; &agrave;&ecirc;&ntilde;i&agrave;&euml;&uuml;&iacute;&icirc;-&ntilde;&egrave;&igrave;&aring;&ograve;&eth;&egrave;&divide;&iacute;&egrave;&eacute; &eth;&icirc;&ccedil;&iuml;&icirc;&auml;i&euml;: u(r, h) = f (r);
(&aacute;) &igrave;&agrave;&sup1; &eth;&icirc;&ccedil;&iuml;&icirc;&auml;i&euml;, &ugrave;&icirc; &ccedil;&agrave;&auml;&agrave;&sup1;&ograve;&uuml;&ntilde;&yuml; &ocirc;&icirc;&eth;&igrave;&oacute;&euml;&icirc;&thorn;: u(r, ϕ, h) = f (r, ϕ).
5.3. &Ccedil;&iacute;&agrave;&eacute;&ograve;&egrave; &ecirc;&eth;&egrave;&ograve;&egrave;&divide;&iacute;&egrave;&eacute; &eth;&agrave;&auml;i&oacute;&ntilde; &iacute;&aring;&ntilde;&ecirc;i&iacute;&divide;&aring;&iacute;&iacute;&icirc; &auml;&icirc;&acirc;&atilde;&icirc;&atilde;&icirc; &ouml;&egrave;&euml;i&iacute;&auml;&eth;&oacute;, &acirc; &yuml;&ecirc;&icirc;&igrave;&oacute; &acirc;i&auml;&aacute;&oacute;&acirc;&agrave;&sup1;&ograve;&uuml;&ntilde;&yuml; &auml;&egrave;&ocirc;&oacute;&ccedil;i&yuml; &divide;&agrave;&ntilde;&ograve;&egrave;&iacute;&icirc;&ecirc; &ccedil; &eth;&icirc;&ccedil;&igrave;&iacute;&icirc;&aelig;&aring;&iacute;&iacute;&yuml;&igrave;. &Ecirc;&icirc;&iacute;&ouml;&aring;&iacute;&ograve;&eth;&agrave;&ouml;i&yuml; &divide;&agrave;&ntilde;&ograve;&egrave;&iacute;&icirc;&ecirc; &iacute;&agrave; &iuml;&icirc;&acirc;&aring;&eth;&otilde;&iacute;i &ouml;&egrave;&euml;i&iacute;&auml;&eth;&oacute; &auml;&icirc;&eth;i&acirc;&iacute;&thorn;&sup1; &iacute;&oacute;&euml;&thorn;.
5.4. &ETH;&icirc;&ccedil;&acirc;'&yuml;&ccedil;&agrave;&ograve;&egrave; &ccedil;&agrave;&auml;&agrave;&divide;&oacute; &iuml;&eth;&icirc; &iuml;&icirc;&iuml;&aring;&eth;&aring;&divide;&iacute;i &ecirc;&icirc;&euml;&egrave;&acirc;&agrave;&iacute;&iacute;&yuml; &igrave;&aring;&igrave;&aacute;&eth;&agrave;&iacute;&egrave;, &ugrave;&icirc; &igrave;&agrave;&sup1; &ocirc;&icirc;&eth;&igrave;&oacute; &ecirc;&icirc;&euml;&agrave;. &Ecirc;&eth;&agrave;&eacute; &igrave;&aring;&igrave;&aacute;&eth;&agrave;&iacute;&egrave; &ccedil;&agrave;&ecirc;&eth;i&iuml;&euml;&aring;&iacute;&egrave;&eacute;. &Iuml;&icirc;&divide;&agrave;&ograve;&ecirc;&icirc;&acirc;&egrave;&eacute; &eth;&icirc;&ccedil;&iuml;&icirc;&auml;i&euml; &oslash;&acirc;&egrave;&auml;&ecirc;&icirc;&ntilde;&ograve;&aring;&eacute; i &ccedil;&igrave;i&ugrave;&aring;&iacute;&uuml; &sup1; &eth;&agrave;&auml;i&agrave;&euml;&uuml;&iacute;&icirc;-&ntilde;&egrave;&igrave;&aring;&ograve;&eth;&egrave;&divide;&iacute;&egrave;&igrave;.
5.5. (&agrave;) &Auml;&icirc;&acirc;&aring;&ntilde;&ograve;&egrave;
g(x, t) = e
x
2 (t−1/t)
=
+∞
X
n
Jn (x)t , &auml;&aring; Jn (x) =
n=−∞
+∞
X
s=0
44
(−1)s &sup3; x &acute;n+2s
.
s!(n + s)! 2
(&aacute;) &ETH;&icirc;&ccedil;&atilde;&euml;&yuml;&iacute;&oacute;&acirc;&oslash;&egrave; &auml;&icirc;&aacute;&oacute;&ograve;&icirc;&ecirc; &ograve;&acirc;i&eth;&iacute;&egrave;&otilde; &ocirc;&oacute;&iacute;&ecirc;&ouml;i&eacute; g(x, t)g(−x, t), &auml;&icirc;&acirc;&aring;&ntilde;&ograve;&egrave;
&ugrave;&icirc;
1=
J02 (x)
+2
+∞
X
1
Jn2 (x), |J0 (x)| ≤ 1; |Jn (x)| ≤ √ .
2
n=1
(&acirc;) &Acirc;&egrave;&ecirc;&icirc;&eth;&egrave;&ntilde;&ograve;&icirc;&acirc;&oacute;&thorn;&divide;&egrave; &acirc;&euml;&agrave;&ntilde;&ograve;&egrave;&acirc;&icirc;&ntilde;&ograve;i &ograve;&acirc;i&eth;&iacute;&egrave;&otilde; &ocirc;&oacute;&iacute;&ecirc;&ouml;i&eacute; g(x+y, t) = g(x, t)g(y, t),
&auml;&icirc;&acirc;&aring;&ntilde;&ograve;&egrave;
Jn (x + y) =
+∞
X
Js (x)Jn−s (y).
s=−∞
5.6. &Ccedil;&iacute;&agrave;&eacute;&ograve;&egrave; &aring;&euml;&aring;&ecirc;&ograve;&eth;&icirc;&ntilde;&ograve;&agrave;&ograve;&egrave;&divide;&iacute;&egrave;&eacute; &iuml;&icirc;&ograve;&aring;&iacute;&ouml;i&agrave;&euml; &acirc;&ntilde;&aring;&eth;&aring;&auml;&egrave;&iacute;i &ouml;&egrave;&euml;i&iacute;&auml;&eth;&egrave;&divide;&iacute;&icirc;&uml; &ecirc;&icirc;&eth;&icirc;&aacute;-
&ecirc;&egrave; &eth;&agrave;&auml;i&oacute;&ntilde;&oacute; r0 &acirc;&egrave;&ntilde;&icirc;&ograve;&icirc;&thorn; h, &yuml;&ecirc;&ugrave;&icirc; &acirc;&aring;&eth;&otilde;&iacute;&yuml; i &iacute;&egrave;&aelig;&iacute;&yuml; &icirc;&ntilde;&iacute;&icirc;&acirc;&agrave; &ccedil;&agrave;&ccedil;&aring;&igrave;&euml;&aring;&iacute;i, &agrave;
&aacute;i&divide;&iacute;&agrave; &iuml;&icirc;&acirc;&aring;&eth;&otilde;&iacute;&yuml; &ccedil;&agrave;&eth;&yuml;&auml;&aelig;&aring;&iacute;&agrave; &auml;&icirc; &iuml;&icirc;&ograve;&aring;&iacute;&ouml;i&agrave;&euml;&oacute; u0 . &Acirc;&egrave;&ccedil;&iacute;&agrave;&divide;&egrave;&ograve;&egrave; &iacute;&agrave;&iuml;&eth;&oacute;&aelig;&aring;&iacute;i&ntilde;&ograve;&uuml; &iuml;&icirc;&euml;&yuml; &iacute;&agrave; &icirc;&ntilde;i &ouml;&egrave;&euml;i&iacute;&auml;&eth;&oacute;.
&Ccedil;&agrave;&auml;&agrave;&divide;i &ccedil; &acirc;&egrave;&ecirc;&icirc;&eth;&egrave;&ntilde;&ograve;&agrave;&iacute;&iacute;&yuml;&igrave; &iuml;&icirc;&euml;i&iacute;&icirc;&igrave;i&acirc; &Euml;&aring;&aelig;&agrave;&iacute;&auml;&eth;&agrave; i &ntilde;&ocirc;&aring;&eth;&egrave;&divide;&iacute;&egrave;&otilde;
&ocirc;&oacute;&iacute;&ecirc;&ouml;i&eacute;.
5.7. &Ccedil;&iacute;&agrave;&eacute;&ograve;&egrave; &aring;&euml;&aring;&ecirc;&ograve;&eth;&icirc;&ntilde;&ograve;&agrave;&ograve;&egrave;&divide;&iacute;&aring; &iuml;&icirc;&euml;&aring; &ograve;&icirc;&divide;&ecirc;&icirc;&acirc;&icirc;&atilde;&icirc; &ccedil;&agrave;&eth;&yuml;&auml;&oacute; q &acirc; &iuml;&eth;&egrave;&ntilde;&oacute;&ograve;&iacute;&icirc;&ntilde;&ograve;i &iuml;&eth;&icirc;&acirc;i&auml;&iacute;&icirc;&uml; &ccedil;&agrave;&ccedil;&aring;&igrave;&euml;&aring;&iacute;&icirc;&uml; &ecirc;&oacute;&euml;i &eth;&agrave;&auml;i&oacute;&ntilde;&oacute; r0
(&agrave;) &ccedil;&agrave;&eth;&yuml;&auml; &ccedil;&iacute;&agrave;&otilde;&icirc;&auml;&egrave;&ograve;&uuml;&ntilde;&yuml; &iacute;&agrave; &acirc;i&auml;&ntilde;&ograve;&agrave;&iacute;i a &gt; r0 &acirc;i&auml; &ouml;&aring;&iacute;&ograve;&eth;&oacute; &ecirc;&oacute;&euml;i, &ccedil;&iacute;&agrave;&eacute;&ograve;&egrave;
&iuml;&icirc;&ograve;&aring;&iacute;&ouml;i&agrave;&euml; &ccedil;&icirc;&acirc;&iacute;&sup1; &ecirc;&oacute;&euml;i;
(&aacute;) &ccedil;&agrave;&eth;&yuml;&auml; &ccedil;&iacute;&agrave;&otilde;&icirc;&auml;&egrave;&ograve;&uuml;&ntilde;&yuml; &iacute;&agrave; &acirc;i&auml;&ntilde;&ograve;&agrave;&iacute;i a &lt; r0 &acirc;i&auml; &ouml;&aring;&iacute;&ograve;&eth;&oacute; &ecirc;&oacute;&euml;i, &ccedil;&iacute;&agrave;&eacute;&ograve;&egrave;
&iuml;&icirc;&ograve;&aring;&iacute;&ouml;i&agrave;&euml; &acirc;&ntilde;&aring;&eth;&aring;&auml;&egrave;&iacute;i &ecirc;&oacute;&euml;i.
5.8. &Ccedil;&iacute;&agrave;&eacute;&ograve;&egrave; &iuml;&icirc;&ograve;&aring;&iacute;&ouml;i&agrave;&euml; &acirc; &icirc;&aacute;&euml;&agrave;&ntilde;&ograve;i &igrave;i&aelig; &auml;&acirc;&icirc;&igrave;&agrave; &ecirc;&icirc;&iacute;&ouml;&aring;&iacute;&ograve;&eth;&egrave;&divide;&iacute;&egrave;&igrave;&egrave; &iuml;&eth;&icirc;&acirc;i&auml;&iacute;&egrave;&igrave;&egrave;
&ccedil;&agrave;&ccedil;&aring;&igrave;&euml;&aring;&iacute;&egrave;&igrave;&egrave; &ecirc;&oacute;&euml;&yuml;&igrave;&egrave; (r1 &lt; r &lt; r2 , &auml;&aring; r1 , r2 &eth;&agrave;&auml;i&oacute;&ntilde;&egrave; &ecirc;&oacute;&euml;&uuml;), &yuml;&ecirc;&ugrave;&icirc; &iacute;&agrave;
&acirc;i&auml;&ntilde;&ograve;&agrave;&iacute;i a (r1 &lt; a &lt; r2 ) &eth;&icirc;&ccedil;&ograve;&agrave;&oslash;&icirc;&acirc;&agrave;&iacute;&egrave;&eacute; &ograve;&icirc;&divide;&ecirc;&icirc;&acirc;&egrave;&eacute; &ccedil;&agrave;&eth;&yuml;&auml; q .
5.9. &Ntilde;&ocirc;&aring;&eth;&egrave;&divide;&iacute;&agrave; &iuml;&icirc;&ntilde;&oacute;&auml;&egrave;&iacute;&agrave; &ccedil; &ograve;&acirc;&aring;&eth;&auml;&egrave;&igrave;&egrave; &ntilde;&ograve;i&iacute;&ecirc;&agrave;&igrave;&egrave;, &iacute;&agrave;&iuml;&icirc;&acirc;&iacute;&aring;&iacute;&agrave; &atilde;&agrave;&ccedil;&icirc;&igrave;, &ograve;&eth;&egrave;&acirc;&agrave;-
&euml;&egrave;&eacute; &divide;&agrave;&ntilde; &eth;&oacute;&otilde;&agrave;&euml;&agrave;&ntilde;&uuml; &eth;i&acirc;&iacute;&icirc;&igrave;i&eth;&iacute;&icirc; i&ccedil; &oslash;&acirc;&egrave;&auml;&ecirc;i&ntilde;&ograve;&thorn; v0 . &Acirc; &igrave;&icirc;&igrave;&aring;&iacute;&ograve; &divide;&agrave;&ntilde;&oacute; t = 0
&acirc;&icirc;&iacute;&agrave; &igrave;&egrave;&ograve;&ograve;&sup1;&acirc;&icirc; &ccedil;&oacute;&iuml;&egrave;&iacute;&egrave;&euml;&agrave;&ntilde;&uuml; i &ccedil;&agrave;&euml;&egrave;&oslash;&agrave;&euml;&agrave;&ntilde;&uuml; &iacute;&aring;&eth;&oacute;&otilde;&icirc;&igrave;&icirc;&thorn;. &Ccedil;&iacute;&agrave;&eacute;&ograve;&egrave; &ecirc;&icirc;&euml;&egrave;&acirc;&agrave;&iacute;&iacute;&yuml;, &ugrave;&icirc; &acirc;&ntilde;&ograve;&agrave;&iacute;&icirc;&acirc;&egrave;&euml;&egrave;&ntilde;&uuml; &acirc;&ntilde;&aring;&eth;&aring;&auml;&egrave;&iacute;i &iuml;&icirc;&ntilde;&oacute;&auml;&egrave;&iacute;&egrave;.
5.10. &Ograve;&acirc;&aring;&eth;&auml;&agrave; &ecirc;&oacute;&euml;&yuml; &eth;&oacute;&otilde;&agrave;&sup1;&ograve;&uuml;&ntilde;&yuml; &ccedil; &iuml;&icirc;&ntilde;&ograve;i&eacute;&iacute;&icirc;&thorn; &oslash;&acirc;&egrave;&auml;&ecirc;i&ntilde;&ograve;&thorn; v0 &acirc; &iacute;&aring;&ntilde;&ograve;&egrave;&ntilde;&euml;&egrave;&acirc;i&eacute; &eth;i&auml;&egrave;&iacute;i, &ugrave;&icirc; &ccedil;&iacute;&agrave;&otilde;&icirc;&auml;&egrave;&ograve;&ntilde;&yuml; &oacute; &ntilde;&iuml;&icirc;&ecirc;&icirc;&uml; &iacute;&agrave; &auml;&agrave;&euml;&aring;&ecirc;&egrave;&otilde; &acirc;i&auml;&ntilde;&ograve;&agrave;&iacute;&yuml;&otilde; &acirc;i&auml; &ecirc;&oacute;&euml;i. &ETH;&agrave;&auml;i&oacute;&ntilde;
&ecirc;&oacute;&euml;i r0 . &Ccedil;&iacute;&agrave;&eacute;&ograve;&egrave; &iuml;&icirc;&ograve;&aring;&iacute;&ouml;i&agrave;&euml; &oslash;&acirc;&egrave;&auml;&ecirc;&icirc;&ntilde;&ograve;&aring;&eacute; &eth;i&auml;&egrave;&iacute;&egrave;.
5.11. &Ccedil;&iacute;&agrave;&eacute;&ograve;&egrave; &aring;&euml;&aring;&ecirc;&ograve;&eth;&icirc;&ntilde;&ograve;&agrave;&ograve;&egrave;&divide;&iacute;&egrave;&eacute; &iuml;&icirc;&ograve;&aring;&iacute;&ouml;i&agrave;&euml; &ccedil;&icirc;&acirc;&iacute;i &iuml;&eth;&icirc;&acirc;i&auml;&iacute;&icirc;&uml; &ccedil;&agrave;&ccedil;&aring;&igrave;&euml;&aring;&iacute;&icirc;&uml; &ecirc;&oacute;&euml;i
&eth;&agrave;&auml;i&oacute;&ntilde;&oacute; r0 , &ugrave;&icirc; &ccedil;&iacute;&agrave;&otilde;&icirc;&auml;&egrave;&ograve;&ntilde;&yuml; &acirc; &ccedil;&icirc;&acirc;&iacute;i&oslash;&iacute;&uuml;&icirc;&igrave;&oacute; &icirc;&auml;&iacute;&icirc;&eth;i&auml;&iacute;&icirc;&igrave;&oacute; &aring;&euml;&aring;&ecirc;&ograve;&eth;&egrave;&divide;&iacute;&icirc;&igrave;&oacute;
~ 0.
&iuml;&icirc;&euml;i E
45
5.12. &Icirc;&aacute;&divide;&egrave;&ntilde;&euml;&egrave;&ograve;&egrave; Pn (0), Pn (1).
5.13. &Icirc;&aacute;&divide;&egrave;&ntilde;&euml;&egrave;&ograve;&egrave;
Z
(&agrave;)
1
xPn (x)dx;
0
Z
(&aacute;)
1
Pn (x)dx.
0
5.14. &Ccedil;&iacute;&agrave;&eacute;&ograve;&egrave; &ntilde;&ograve;&agrave;&ouml;i&icirc;&iacute;&agrave;&eth;&iacute;&egrave;&eacute; &eth;&icirc;&ccedil;&iuml;&icirc;&auml;i&euml; &ograve;&aring;&igrave;&iuml;&aring;&eth;&agrave;&ograve;&oacute;&eth;&egrave; &acirc;&ntilde;&aring;&eth;&aring;&auml;&egrave;&iacute;i &iuml;i&acirc;&ecirc;&oacute;&euml;i &eth;&agrave;&auml;i&oacute;&ntilde;&oacute; r0 , &yuml;&ecirc;&ugrave;&icirc; &ntilde;&ocirc;&aring;&eth;&egrave;&divide;&iacute;&agrave; &divide;&agrave;&ntilde;&ograve;&egrave;&iacute;&agrave; &iuml;&icirc;&acirc;&aring;&eth;&otilde;&iacute;i &iuml;i&acirc;&ecirc;&oacute;&euml;i &iuml;i&auml;&ograve;&eth;&egrave;&igrave;&oacute;&sup1;&ograve;&uuml;&ntilde;&yuml; &iuml;&eth;&egrave;
&ntilde;&ograve;&agrave;&euml;i&eacute; &ograve;&aring;&igrave;&iuml;&aring;&eth;&agrave;&ograve;&oacute;&eth;i u0 , &agrave; &icirc;&ntilde;&iacute;&icirc;&acirc;&agrave; &igrave;&agrave;&sup1; &iacute;&oacute;&euml;&uuml;&icirc;&acirc;&oacute; &ograve;&aring;&igrave;&iuml;&aring;&eth;&agrave;&ograve;&oacute;&eth;&oacute;.
5.15. &Ccedil;&iacute;&agrave;&eacute;&ograve;&egrave; &atilde;&agrave;&eth;&igrave;&icirc;&iacute;i&eacute;&iacute;&oacute; &ocirc;&oacute;&iacute;&ecirc;&ouml;i&thorn; &acirc;&ntilde;&aring;&eth;&aring;&auml;&egrave;&iacute;i &ecirc;&oacute;&euml;i &eth;&agrave;&auml;i&oacute;&ntilde;&oacute; r0 , &ugrave;&icirc; &iacute;&agrave;&aacute;&oacute;&acirc;&agrave;&sup1;
&iacute;&agrave; &iuml;&icirc;&acirc;&aring;&eth;&otilde;&iacute;i &ccedil;&iacute;&agrave;&divide;&aring;&iacute;&iacute;&yuml;: u(r0 , θ) = cos2 θ.
5.16. &ETH;&icirc;&ccedil;&acirc;'&yuml;&ccedil;&agrave;&ograve;&egrave; &ccedil;&agrave;&auml;&agrave;&divide;&oacute; &iuml;&eth;&icirc; &icirc;&otilde;&icirc;&euml;&icirc;&auml;&aelig;&aring;&iacute;&iacute;&yuml; &ecirc;&oacute;&euml;i &eth;&agrave;&auml;i&oacute;&ntilde;&oacute; r0 , &yuml;&ecirc;&ugrave;&icirc; &iuml;&icirc;&divide;&agrave;&ograve;&ecirc;&icirc;&acirc;&egrave;&eacute; &eth;&icirc;&ccedil;&iuml;&icirc;&auml;i&euml; &ograve;&aring;&igrave;&iuml;&aring;&eth;&agrave;&ograve;&oacute;&eth;&egrave; &ccedil;&agrave;&auml;&agrave;&iacute;&egrave;&eacute;: u(r, θ, ϕ, 0) = f (r, θ, ϕ).
(a) &Iacute;&agrave; &iuml;&icirc;&acirc;&aring;&eth;&otilde;&iacute;i &ecirc;&oacute;&euml;i &iuml;i&auml;&ograve;&eth;&egrave;&igrave;&oacute;&sup1;&ograve;&uuml;&ntilde;&yuml; &iacute;&oacute;&euml;&uuml;&icirc;&acirc;&agrave; &ograve;&aring;&igrave;&iuml;&aring;&eth;&agrave;&ograve;&oacute;&eth;&agrave;.
(&aacute;) &Iacute;&agrave; &iuml;&icirc;&acirc;&aring;&eth;&otilde;&iacute;i &ecirc;&oacute;&euml;i &iuml;i&auml;&ograve;&eth;&egrave;&igrave;&oacute;&sup1;&ograve;&uuml;&ntilde;&yuml; &ntilde;&ograve;&agrave;&euml;&agrave; &ograve;&aring;&igrave;&iuml;&aring;&eth;&agrave;&ograve;&oacute;&eth;&agrave; u0 .
(&acirc;) &Iuml;&icirc;&acirc;&aring;&eth;&otilde;&iacute;&yuml; &ecirc;&oacute;&euml;i &acirc;i&euml;&uuml;&iacute;&icirc; &icirc;&otilde;&icirc;&euml;&icirc;&auml;&aelig;&oacute;&sup1;&ograve;&uuml;&ntilde;&yuml; &acirc; &ntilde;&aring;&eth;&aring;&auml;&icirc;&acirc;&egrave;&ugrave;i i&ccedil; &iacute;&oacute;&euml;&uuml;&icirc;&acirc;&icirc;&thorn;
&ograve;&aring;&igrave;&iuml;&aring;&eth;&agrave;&ograve;&oacute;&eth;&icirc;&thorn;.
5.17. &Ccedil;&iacute;&agrave;&eacute;&ograve;&egrave; &ograve;&aring;&igrave;&iuml;&aring;&eth;&agrave;&ograve;&oacute;&eth;&oacute; &acirc;&ntilde;&aring;&eth;&aring;&auml;&egrave;&iacute;i &ntilde;&ocirc;&aring;&eth;&egrave;&divide;&iacute;&icirc;&uml; &icirc;&aacute;&icirc;&euml;&icirc;&iacute;&ecirc;&egrave; &igrave;i&aelig; &auml;&acirc;&icirc;&igrave;&agrave; &ecirc;&icirc;&iacute;-
&ouml;&aring;&iacute;&ograve;&eth;&egrave;&divide;&iacute;&egrave;&igrave;&egrave; &ntilde;&ocirc;&aring;&eth;&egrave;&divide;&iacute;&egrave;&igrave;&egrave; &iuml;&icirc;&acirc;&aring;&eth;&otilde;&iacute;&yuml;&igrave;&egrave; (r1 &lt; r &lt; r2 ). &Iuml;&icirc;&divide;&agrave;&ograve;&ecirc;&icirc;&acirc;&egrave;&eacute;
&eth;&icirc;&ccedil;&iuml;&icirc;&auml;i&euml; &ograve;&aring;&igrave;&iuml;&aring;&eth;&agrave;&ograve;&oacute;&eth;&egrave; &ccedil;&agrave;&auml;&agrave;&iacute;&egrave;&eacute;: u(r, θ, ϕ, 0) = f (r, θ, ϕ).
(&agrave;) &Ograve;&aring;&igrave;&iuml;&aring;&eth;&agrave;&ograve;&oacute;&eth;&agrave; &iacute;&agrave; &acirc;&iacute;&oacute;&ograve;&eth;i&oslash;&iacute;i&eacute; i &ccedil;&icirc;&acirc;&iacute;i&oslash;&iacute;i&eacute; &iuml;&icirc;&acirc;&aring;&eth;&otilde;&iacute;&yuml;&otilde; &icirc;&aacute;&icirc;&euml;&icirc;&iacute;&ecirc;&egrave; &iuml;i&auml;&ograve;&eth;&egrave;&igrave;&oacute;&thorn;&sup1;&ograve;&uuml;&ntilde;&yuml; &eth;i&acirc;&iacute;&icirc;&thorn; &iacute;&oacute;&euml;&thorn;.
(&aacute;) &Ccedil;&icirc;&acirc;&iacute;i&oslash;&iacute;&yuml; &iuml;&icirc;&acirc;&aring;&eth;&otilde;&iacute;&yuml; &icirc;&otilde;&icirc;&euml;&icirc;&auml;&aelig;&oacute;&sup1;&ograve;&uuml;&ntilde;&yuml; &acirc; &ntilde;&aring;&eth;&aring;&auml;&icirc;&acirc;&egrave;&ugrave;i &ccedil; &iacute;&oacute;&euml;&uuml;&icirc;&acirc;&icirc;&thorn; &ograve;&aring;&igrave;&iuml;&aring;&eth;&agrave;&ograve;&oacute;&eth;&icirc;&thorn;, &agrave; &acirc;&iacute;&oacute;&ograve;&eth;i&oslash;&iacute;&yuml; &iuml;&icirc;&acirc;&aring;&eth;&otilde;&iacute;&yuml; &icirc;&iuml;&eth;&icirc;&igrave;i&iacute;&thorn;&sup1;&ograve;&uuml;&ntilde;&yuml; &eth;i&acirc;&iacute;&icirc;&igrave;i&eth;&iacute;&egrave;&igrave; &iuml;&icirc;&ograve;&icirc;&ecirc;&icirc;&igrave; &ograve;&aring;&iuml;&euml;&agrave; &ccedil; &atilde;&oacute;&ntilde;&ograve;&egrave;&iacute;&icirc;&thorn; Q.
5.18. &ETH;&icirc;&ccedil;&ecirc;&euml;&agrave;&ntilde;&ograve;&egrave; &iuml;&euml;&icirc;&ntilde;&ecirc;&oacute; &otilde;&acirc;&egrave;&euml;&thorn; ei~k~r &iuml;&icirc; &iuml;&icirc;&euml;i&iacute;&icirc;&igrave;&agrave;&igrave; &Euml;&aring;&aelig;&agrave;&iacute;&auml;&eth;&agrave; i &ocirc;&oacute;&iacute;&ecirc;&ouml;i&yuml;&igrave;
&Aacute;&aring;&ntilde;&aring;&euml;&yuml;.
5.19. &ETH;&icirc;&ccedil;&acirc;'&yuml;&ccedil;&agrave;&ograve;&egrave; &ccedil;&agrave;&auml;&agrave;&divide;&oacute; &iuml;&eth;&icirc; &eth;&icirc;&ccedil;&ntilde;i&yuml;&iacute;&iacute;&yuml; &iuml;&euml;&icirc;&ntilde;&ecirc;&icirc;&uml; &agrave;&ecirc;&oacute;&ntilde;&ograve;&egrave;&divide;&iacute;&icirc;&uml; &otilde;&acirc;&egrave;&euml;i &iacute;&agrave; &ograve;&acirc;&aring;&eth;-
&auml;&icirc;&igrave;&oacute; &ouml;&egrave;&euml;i&iacute;&auml;&eth;i &eth;&agrave;&auml;i&oacute;&ntilde;&oacute; r0 . &Ograve;&acirc;i&eth;&iacute;&agrave; &ouml;&egrave;&euml;i&iacute;&auml;&eth;&oacute; &iuml;&agrave;&eth;&agrave;&euml;&aring;&euml;&uuml;&iacute;&agrave; &otilde;&acirc;&egrave;&euml;&uuml;&icirc;&acirc;i&eacute; &iuml;&icirc;&acirc;&aring;&eth;&otilde;&iacute;i &iuml;&euml;&icirc;&ntilde;&ecirc;&icirc;&uml; &otilde;&acirc;&egrave;&euml;i.
5.20. &ETH;&icirc;&ccedil;&acirc;'&yuml;&ccedil;&agrave;&ograve;&egrave; &ccedil;&agrave;&auml;&agrave;&divide;&oacute; &iuml;&eth;&icirc; &eth;&icirc;&ccedil;&ntilde;i&yuml;&iacute;&iacute;&yuml; &iuml;&euml;&icirc;&ntilde;&ecirc;&icirc;&uml; &agrave;&ecirc;&oacute;&ntilde;&ograve;&egrave;&divide;&iacute;&icirc;&uml; &otilde;&acirc;&egrave;&euml;i &iacute;&agrave; &ograve;&acirc;&aring;&eth;&auml;i&eacute;
&ecirc;&oacute;&euml;i &eth;&agrave;&auml;i&oacute;&ntilde;&oacute; r0 .
46
6. I&iacute;&ograve;&aring;&atilde;&eth;&agrave;&euml;&uuml;&iacute;i &eth;i&acirc;&iacute;&iacute;&yuml;&iacute;&iacute;&yuml;.
6.1. &ETH;&icirc;&ccedil;&acirc;'&yuml;&ccedil;&agrave;&ograve;&egrave; i&iacute;&ograve;&aring;&atilde;&eth;&agrave;&euml;&uuml;&iacute;&aring; &eth;i&acirc;&iacute;&yuml;&iacute;&iacute;&yuml; &igrave;&aring;&ograve;&icirc;&auml;&icirc;&igrave; &iuml;&icirc;&ntilde;&euml;i&auml;&icirc;&acirc;&iacute;&egrave;&otilde; &iacute;&agrave;&aacute;&euml;&egrave;&aelig;&aring;&iacute;&uuml;:
Z
(a) ϕ(x) = 1 − 2
x
tϕ(t)dt;
Z0
1
(&aacute;) ϕ(x) = x −
2
1
(t − x)ϕ(t)dt.
−1
6.2. &ETH;&icirc;&ccedil;&acirc;'&yuml;&ccedil;&agrave;&ograve;&egrave; i&iacute;&ograve;&aring;&atilde;&eth;&agrave;&euml;&uuml;&iacute;&aring; &eth;i&acirc;&iacute;&yuml;&iacute;&iacute;&yuml; &ccedil; &acirc;&egrave;&eth;&icirc;&auml;&aelig;&aring;&iacute;&egrave;&igrave; &yuml;&auml;&eth;&icirc;&igrave;:
Z
(a) ϕ(x) = 1 + λ
1
(x − t)ϕ(t)dt;
0
Z
+∞
2
(&aacute;) ϕ(x) = −x +
2
e−x
−t2
ϕ(t)dt;
−∞
1
Z
x2 t2 ϕ(t)dt;
(&acirc;) ϕ(x) = −x2 +
Z
−1
1
(1 + x2 t2 )ϕ(t)dt;
(&atilde;) ϕ(x) = x3 + λ
Z0
x
(&auml;) ϕ(x) = e + 2λ
1
ex+t ϕ(t)dt;
0
1
Z
(e) ϕ(x) = ex + λ
ex−t (1 + xt)ϕ(t)dt;
0
Z 1
2
4
(xt + x2 t2 )ϕ(t)dt.
(&aelig;) ϕ(x) = x + x + λ
−1
6.3. &ETH;&icirc;&ccedil;&acirc;'&yuml;&ccedil;&agrave;&ograve;&egrave; i&iacute;&ograve;&aring;&atilde;&eth;&agrave;&euml;&uuml;&iacute;i &eth;i&acirc;&iacute;&yuml;&iacute;&iacute;&yuml;:
Z
(a) ϕ(x) = π − 2x + λ
Z
(&aacute;) ϕ(x) = cos 3x +
sin(2x + t)ϕ(t)dt;
0
2π
(cos x cos t + cos 2x cos 2t)ϕ(t)dt;
Z
(&acirc;) ϕ(x) = cos x +
π
0
2π
(cos x cos t + 2 sin 2x sin 2t)ϕ(t)dt.
0
6.4. &ETH;&icirc;&ccedil;&acirc;'&yuml;&ccedil;&agrave;&ograve;&egrave; i&iacute;&ograve;&aring;&atilde;&eth;&agrave;&euml;&uuml;&iacute;&aring; &eth;i&acirc;&iacute;&yuml;&iacute;&iacute;&yuml;:
Z
π
ϕ(x) = λ
cos(x + t)ϕ(t)dt + a sin x + b
0
&auml;&euml;&yuml; &acirc;&ntilde;i&otilde; λ, &ograve;&agrave; &acirc;&ntilde;i&otilde; &ccedil;&iacute;&agrave;&divide;&aring;&iacute;&uuml; &iuml;&agrave;&eth;&agrave;&igrave;&aring;&ograve;&eth;i&acirc; a,b.
47
6.5. &Ccedil;&iacute;&agrave;&eacute;&ograve;&egrave; &acirc;&ntilde;i &ccedil;&iacute;&agrave;&divide;&aring;&iacute;&iacute;&yuml; &iuml;&agrave;&eth;&agrave;&igrave;&aring;&ograve;&eth;i&acirc; a, b, c, &iuml;&eth;&egrave; &yuml;&ecirc;&egrave;&otilde; i&iacute;&ograve;&aring;&atilde;&eth;&agrave;&euml;&uuml;&iacute;&aring; &eth;i&acirc;&iacute;&yuml;&iacute;&iacute;&yuml;
Z
2
1
ϕ(x) = ax + bx + c + λ
(xt + x2 t2 )ϕ(t)dt
−1
&igrave;&agrave;&sup1; &eth;&icirc;&ccedil;&acirc;'&yuml;&ccedil;&ecirc;&egrave; &iuml;&eth;&egrave; &aacute;&oacute;&auml;&uuml;-&yuml;&ecirc;&egrave;&otilde; λ.
6.6. &Ccedil;&iacute;&agrave;&eacute;&ograve;&egrave; &acirc;&euml;&agrave;&ntilde;&iacute;i &ccedil;&iacute;&agrave;&divide;&aring;&iacute;&iacute;&yuml; i &acirc;&euml;&agrave;&ntilde;&iacute;i &ocirc;&oacute;&iacute;&ecirc;&ouml;i&uml; &acirc;&egrave;&eth;&icirc;&auml;&aelig;&aring;&iacute;&icirc;&atilde;&icirc; &yuml;&auml;&eth;&agrave; K(x, t) =
t + x &iacute;&agrave; &iuml;&eth;&icirc;&igrave;i&aelig;&ecirc;&oacute; [−1, 1].
6.7. &Ccedil;&iacute;&agrave;&eacute;&ograve;&egrave; &acirc;&euml;&agrave;&ntilde;&iacute;i &ccedil;&iacute;&agrave;&divide;&aring;&iacute;&iacute;&yuml; i &acirc;&euml;&agrave;&ntilde;&iacute;i &ocirc;&oacute;&iacute;&ecirc;&ouml;i&uml; &yuml;&auml;&eth;&agrave; K(x, t) &iacute;&agrave; &iuml;&eth;&icirc;&igrave;i&aelig;&ecirc;&oacute;
[0, 2π]:
(&agrave;) K(x, t) = cos(t − x);
(&aacute;) K(x, t) =
1
2
+ sin(x + t).
6.8. &Ccedil;&iacute;&agrave;&eacute;&ograve;&egrave; &acirc;&euml;&agrave;&ntilde;&iacute;i &ccedil;&iacute;&agrave;&divide;&aring;&iacute;&iacute;&yuml; i &acirc;&euml;&agrave;&ntilde;&iacute;i &ocirc;&oacute;&iacute;&ecirc;&ouml;i&uml; &yuml;&auml;&eth;&agrave; K(x, t) &iacute;&agrave; &iuml;&eth;&icirc;&igrave;i&aelig;&ecirc;&oacute;
[0, 1]:
2
45 ;
2/5
(&agrave;) K(x, t) = x2 t2 −
(&aacute;) K(x, t) = (x/y)
+ (y/x)2/5 .
6.9. &Iuml;&icirc;&aacute;&oacute;&auml;&oacute;&acirc;&agrave;&ograve;&egrave; &eth;&aring;&ccedil;&icirc;&euml;&uuml;&acirc;&aring;&iacute;&ograve;&oacute; &yuml;&auml;&eth;&agrave; K(x, t) i&iacute;&ograve;&aring;&atilde;&eth;&agrave;&euml;&uuml;&iacute;&icirc;&atilde;&icirc; &eth;i&acirc;&iacute;&yuml;&iacute;&iacute;&yuml; &Ocirc;&eth;&aring;&auml;&atilde;&icirc;&euml;&uuml;&igrave;&agrave;
Z
1
ϕ(x) = f (x) +
K(x, t)ϕ(t)dt,
0
&yuml;&ecirc;&ugrave;&icirc;
(a) K(x, t) = ex−t ;
(&aacute;) K(x, t) = xt;
(&acirc;) K(x, t) = 1.
6.10. &Ccedil;&iacute;&agrave;&eacute;&ograve;&egrave; &eth;&aring;&ccedil;&icirc;&euml;&uuml;&acirc;&aring;&iacute;&ograve;&oacute; i &eth;&icirc;&ccedil;&acirc;'&yuml;&ccedil;&agrave;&ograve;&egrave; i&iacute;&ograve;&aring;&atilde;&eth;&agrave;&euml;&uuml;&iacute;&aring; &eth;i&acirc;&iacute;&yuml;&iacute;&iacute;&yuml;:
Z
(a) ϕ(x) = f (x) + λ
Z
(&aacute;) ϕ(x) = f (x) +
π
sin(x + t)ϕ(t)dt;
0
+π
(x sin t + cos t)ϕ(t)dt.
−π
6.11. &Acirc;&egrave;&ecirc;&icirc;&eth;&egrave;&ntilde;&ograve;&icirc;&acirc;&oacute;&thorn;&divide;&egrave; &iuml;&aring;&eth;&aring;&ograve;&acirc;&icirc;&eth;&aring;&iacute;&iacute;&yuml; &Ocirc;&oacute;&eth;'&sup1; &eth;&icirc;&ccedil;&acirc;'&yuml;&ccedil;&agrave;&ograve;&egrave; i&iacute;&ograve;&aring;&atilde;&eth;&agrave;&euml;&uuml;&iacute;i &eth;i&acirc;&iacute;&yuml;&iacute;&iacute;&yuml;:
Z
(a)
+∞
1 2
ϕ(t) cos(xt)dt = e− 2 x ;
0
48
Z
(&aacute;)
+∞
ϕ(t) cos(xt)dt =
−∞
1
.
a2 + x2
6.12. &ETH;&icirc;&ccedil;&acirc;'&yuml;&ccedil;&agrave;&ograve;&egrave; i&iacute;&ograve;&aring;&atilde;&eth;&agrave;&euml;&uuml;&iacute;&aring; &eth;i&acirc;&iacute;&yuml;&iacute;&iacute;&yuml; &Oacute;&eth;&egrave;&ntilde;&icirc;&iacute;&agrave;:
Z
+∞
2
ϕ(t)ϕ(x − t)dt = e−x
−∞
49
50
&Aacute;i&aacute;&euml;i&icirc; &eth;&agrave;&ocirc;i&yuml;
[1] &Aacute;.&Igrave;. &Aacute;&oacute;&auml;&agrave;&ecirc;, &Agrave;.&Agrave;. &Ntilde;&agrave;&igrave;&agrave;&eth;&ntilde;&ecirc;&egrave;&eacute;, &Agrave;.&Iacute;. &Ograve;&egrave;&otilde;&icirc;&iacute;&icirc;&acirc; &Ntilde;&aacute;&icirc;&eth;&iacute;&egrave;&ecirc; &ccedil;&agrave;&auml;&agrave;&divide; &iuml;&icirc; &igrave;&agrave;&ograve;&aring;&igrave;&agrave;&ograve;&egrave;&divide;&aring;&ntilde;&ecirc;&icirc;&eacute; &ocirc;&egrave;&ccedil;&egrave;&ecirc;&aring;, -&Igrave;.: &Iacute;&agrave;&oacute;&ecirc;&agrave;, 1956.
[2] &Agrave;.&Iacute;. &Ograve;&egrave;&otilde;&icirc;&iacute;&icirc;&acirc;, &Agrave;.&Agrave;. &Ntilde;&agrave;&igrave;&agrave;&eth;&ntilde;&ecirc;&egrave;&eacute;. &Oacute;&eth;&agrave;&acirc;&iacute;&aring;&iacute;&egrave;&yuml; &igrave;&agrave;&ograve;&aring;&igrave;&agrave;&ograve;&egrave;&divide;&aring;&ntilde;&ecirc;&icirc;&eacute; &ocirc;&egrave;&ccedil;&egrave;&ecirc;&egrave;,
-&Igrave;.: &Iacute;&agrave;&oacute;&ecirc;&agrave;, 1972.
[3] &Acirc;.&szlig;. &Agrave;&eth;&ntilde;&aring;&iacute;&egrave;&iacute;. &Igrave;&aring;&ograve;&icirc;&auml;&ucirc; &igrave;&agrave;&ograve;&aring;&igrave;&agrave;&ograve;&egrave;&divide;&aring;&ecirc;&ntilde;&ecirc;&icirc;&eacute; &ocirc;&egrave;&ccedil;&egrave;&ecirc;&egrave; &egrave; &ntilde;&iuml;&aring;&ouml;&egrave;&agrave;&euml;&uuml;&iacute;&ucirc;&aring;
&ocirc;&oacute;&iacute;&ecirc;&ouml;&egrave;&egrave;, -&Igrave;.: &Iacute;&agrave;&oacute;&ecirc;&agrave;, 1974.
[4] &Egrave;.&Acirc;. &Ecirc;&icirc;&euml;&icirc;&ecirc;&icirc;&euml;&icirc;&acirc;, &Aring;.&Agrave;. &Ecirc;&oacute;&ccedil;&iacute;&aring;&ouml;&icirc;&acirc;, &Agrave;.&Egrave;. &Igrave;&egrave;&euml;&uuml;&oslash;&ograve;&aring;&eacute;&iacute;, &Aring;.&Acirc;. &Iuml;&icirc;&auml;&egrave;&acirc;&egrave;&euml;&icirc;&acirc;,
&Agrave;.&Egrave;. &times;&aring;&eth;&iacute;&ucirc;&otilde;, &Auml;.&Agrave;. &Oslash;&agrave;&iuml;&egrave;&eth;&icirc;, &Aring;.&Atilde;. &Oslash;&agrave;&iuml;&egrave;&eth;&icirc; &Ccedil;&agrave;&auml;&agrave;&divide;&egrave; &iuml;&icirc; &igrave;&agrave;&ograve;&aring;&igrave;&agrave;&ograve;&egrave;&divide;&aring;&ntilde;&ecirc;&egrave;&igrave; &igrave;&aring;&ograve;&icirc;&auml;&agrave;&igrave; &ocirc;&egrave;&ccedil;&egrave;&ecirc;&egrave;, -&Igrave;.: &Yacute;&auml;&egrave;&ograve;&icirc;&eth;&egrave;&agrave;&euml; &Oacute;&ETH;&Ntilde;&Ntilde;, 2000. -288 &ntilde;.
[5] &Igrave;.&Agrave;. &Euml;&agrave;&acirc;&eth;&aring;&iacute;&ograve;&uuml;&aring;&acirc;, &Aacute;.&Acirc;. &Oslash;&agrave;&aacute;&agrave;&ograve; &Igrave;&aring;&ograve;&icirc;&auml;&ucirc; &ocirc;&oacute;&iacute;&ecirc;&ouml;&egrave;&eacute; &ecirc;&icirc;&igrave;&iuml;&euml;&aring;&ecirc;&ntilde;&iacute;&icirc;&atilde;&icirc; &iuml;&aring;&eth;&aring;&igrave;&aring;&iacute;&iacute;&icirc;&atilde;&icirc;, -&Igrave;.: &Iacute;&agrave;&oacute;&ecirc;&agrave;, 1957.
51
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