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RST UVWXYUZ [\WU]
^^T _`VVaW
bUc
bdefghij kX l Xm nedogp _X l Xm qdejrstfua vX bX w xuyuzr r g{eu|}
bUc sdsra {i ~ud~urzdtfi€ r‚rfdƒ RzdpX {itiprdX „ pXƒ
†‚y}ji „ p‡ˆnR ‰lˆn†“ m c\VŠX ‹ tX
ŒŽ ‹
utt~uerju‘ta feudj’d ‚uyuzr yha ip’fsijdss’“ yrdeds”ruh•}
s’“ geujsdsr€ r geujsdsr€ j zuts’“ {eir‚jiys’“X erjiyrta ed–dsrd
‚uyuz ~diyi~ —ge•dX „iyde|r pih•–id fihrzdtji {er~deij r g{eu|}
sdsr€ yha yi~u–sd€ r tu~itiadh•si€ eupi’ tgydsijX „iijdtjgd
eupizr~ {eioeu~~u~ yrt”r{hrs’ ‰bdiy’ ~ud~urzdtfi€ r‚rfr“ zd}
jdeioi td~dteu ufgh•du ˆhdfeisrfr r {aioi td~dteu ˜fe’ioi
ufgh•duX
edysu‚suzdsi yha tgydsij jtd“ su{eujhdsr€ ufgh•du ˆhdfei}
srfr r ˜fe’ioi ufgh•duX
RST UVWXYUZ [\WU]
^^T _`VVaW
d”ds‚ds’ƒ fudyeu j’t–d€ ~ud~urfr „ p‡ R™
y}e r‚X}~uX sugfm {eiX šX†X ^dhi{ih•tfua [„ p‡k„R]X
Rjde|ydsi
edyuf”rissi}r‚yudh•tfr~ tijdi~ gsrjdetrdu
j fuzdtjd gzdpsioi {itipra
ŒŽ ‹
› „ p‡ˆnR œlˆn†m c\VŠ
10*ž*87*
Sussid r‚yusrd ajhadta tpiesrfi~ ‚uyuz {i sdfiie’~ eu‚ydhu~
yrt”r{hrs’ ‰bdiy’ ~ud~urzdtfi€ r‚rfr“ m r‚gzud~i€ tgydsu~r u}
fgh•du Ÿhdfeisrfr r ife’ioi ufgh•duX xuyuzsrf pu‚regdta r yi}
{ihsad gzdpsid {itiprd V¡X
_ {deji~ eu‚ydhd eutt~uerju‘ta feudj’d ‚uyuzr yha ip’fsijds}
s’“ yrdeds”ruh•s’“ geujsdsr€X vu {er~deu“ {ifu‚us’ itipdssitrm
fiie’d ji‚srfu‘ {er ed–dsrr ufr“ ‚uyuzX
_iei€ eu‚ydh {itja¢ds ed–dsr‘ ‚uyuzr £ge~u¤lrgjrhhaX u‚i}
peusi pih•–id fihrzdtji ‚uyuz su“i|ydsra tiptjdss’“ zrtdh r tip}
tjdss’“ gsf”r€ eu‚hrzs’“ yrdeds”ruh•s’“ i{deuieij jieioi {i}
eayfuX
u‚ydh’ `m Šm ¥ tiyde|u eu‚siipeu‚s’d feudj’d ‚uyuzr yha yrd}
eds”ruh•s’“ geujsdsr€ j zuts’“ {eir‚jiys’“ eu‚s’“ r{ij yha ed–d}
sra fiie’“ rt{ih•‚ijuhta iyrs r‚ surpihdd zuti {er~dsad~’“ ~diyij
~ud~urzdtfi€ r‚rfr ¤ ~diy —ge•dX ˆiy ~diy {i‚jihad su“iyr•
ed–dsrd ‚uyuzr j jryd eayum j fiiei~ eu‚hi|dsrd j’{ihsadta {i tip}
tjdss’~ gsf”ra~ yrdeds”ruh•sioi i{deuieum tja‚ussioi t eutt~u}
erjud~i€ ‚uyuzd€X
_ eu‚ydhd U {erjiyata {er~de’ {er~dsdsra ~diyu Suhu~pdeuX
_ {itiprr {iyeipsi eu‚ipeus’ ~sioizrthdss’d ‚uyuzrm fiie’dm fuf
{ifu‚’jud i{’m j’‚’ju‘ g tgydsij egysitrX ˆi m {ed|yd jtdoim ‚u}
yuzrm {er ed–dsrr fiie’“ edpgdta j’{ihsr• {ded“iy f ferjihrsd€si€
trtd~d fiieyrsuX
_ fis”d fu|yioi eu‚ydhu tiyde|uta g{eu|sdsra yha tu~itiadh•}
sioi ed–dsraX _td g{eu|sdsra r~d‘ ijd’X
`
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utt~ier~ hrsd€sid yrdeds”ruh•sid geujsdsrd jieioi {ieayfum
‚u{rtussid j tr~~derzsi€ ie~dƒ
[VXV]
1
(p (x)y ) + q (x)y = f (x),
−
ρ(x)
oyd ρ(x), p (x), p (x), q (x), f (x) ¤ ‚uyuss’dm sd{ede’js’d su [ a, b ] gsf}
”rrm {er Ÿi~ j’{ihsa‘ta gthijra ρ(x) ≥ ρ > 0, p (x) ≥ p > 0 X
˜p¢dd ed–dsrd ufioi geujsdsra r~dd jry c¡ƒ
0 0
0
0
0
y(x) = C1 y1 (x) + C2 y2 (x) + ỹ(x).
xydt• y (x) r y (x) ¤ yju hrsd€si sd‚ujrtr~’“ ed–dsra iysieiysioi geuj}
sdsram tiijdtjg‘¢doi geujsdsr‘ [VXV]m u ỹ(x) ¤ zutsid ed–dsrd geuj}
sdsra [VXV]m C m C ¤ {eir‚jih•s’d {itiass’dX
gt• su fis”u“ {ei~d|gfu [a, b] gsf”ra y(x) gyijhdjiead gthi}
jra~ƒ
(
[VXc]
R y (a) − S y(a) = t ,
R y (b) + S y(b) = t ,
oyd R m R m S m S m t m t ¤ ‚uyuss’d {itiass’dm |R | + |S | =6 0 m
m R , R , S , S ≥ 0X
|R | +
|S
|
=
6
0
nufrd gthijra su‚’ju‘ta feudj’~r rhr oeusrzs’~rX
xuyuzu su“i|ydsra su rsdejuhd (a, b) ed–dsra yrdeds”ruh•sioi
geujsdsra [VXV]m gyijhdjiea‘¢doi j izfu“ x = a r x = b feudj’~ gthi}
jra~ [VXc]m su‚’judta feudji€ ‚uyuzd€X
utt~ier~ sdfiie’d zuts’d thgzur feudj’“ gthijr€X
Rthijraƒ R y (a) − S y(a) = 0, R y (b) + S y(b) = 0
su‚’ju‘ta iysieiys’~rX
(t =Rthijraƒ
t = 0)
y(a)
= t , y(b)
=t
su‚’ju‘ta
feudj’~r
gthijra~r V}oi eiyu rhr gthijra~r
(R
=
R
=
0)
Srer“hdX
Rthijraƒ y (a) = t , y (b) = t
su‚’ju‘ta feudj’~r gthijra~r c}oi eiyu rhr gthijra~r
(S
=
S
=
0)
vd€~usuX
Rthijraƒ R y (a) − S y(a) = t , R y (b) + S y(b) = t
su‚’ju‘ta feudj’~r gthijra~r `}oi eiyuX
˜ysieiys’~r feudj’~r gthijra~r ajha‘ta uf|d gthijra {deri}
yrzsitrƒ y(a) = y(b), y (a) = y (b).
¬thr gsf”rr ρ(x) m p(x) m p (x) m q(x) sd{ede’js’ su ife’i~ rsde}
juhd (a, b) rhr dthr gsf”rr ρ(x) rhr p(x) ipeu¢u‘ta j sgh• j oeusrzsi€
Š
1
2
1
1
2
2
2
1
2
1
1
1
2
1
2
0
1
2
0
2
1
2
2
1
0
1
1
2
1
2
0
2
2
1
1
2
0
1
2
0
1
2
2
1
0
1
0
1
0
0
2
0
2
2
1
izfdm i j ufr“ thgzua“ j fuzdtjd feudjioi gthijra rt{ih•‚gdta gthi}
jrd ioeusrzdssitr y(x) {er x → a + 0 rhr x → b − 0 X nufrd feudj’d
gthijra ajha‘ta iysieiys’~rX
vu fis”u“ {ei~d|gfu [a, b] ~iog p’• ‚uyus’ feudj’d gthijra eu‚}
s’“ r{ijX
gt• edpgdta ed–r• feudjg‘ ‚uyuzg yha iysieiysioi hrsd€sioi
yrdeds”ruh•sioi geujsdsraƒ
(
{er a < x < b,
−y + qy = 0
00
R1 y 0 (a) − S1 y(a) = t1 ,
˜p¢dd ed–dsrd geujsdsra −y
R2 y 0 (b) + S2 y(b) = t2 .
r~dd jryƒ
00
+ qy = 0
y(x) = C1 y1 (x) + C2 y2 (x),
oyd gsf”rr y (x) m y (x) ¤ Ÿi yju hrsd€si sd‚ujrtr~’“ ed–dsra geuj}
sdsraX vdegysi ‚u~dr•m zi gsf”rr y (x − a) m y (x − a) uf|d fuf r
gsf”rr y (b − x) m y (b − x) pgyg yegor~r {ueu~r hrsd€si sd‚ujrtr~’“
ed–dsr€X iŸi~g ip¢dd ed–dsrd geujsdsra ~i|si {edytujr• j jrydƒ
1
2
1
1
2
2
y(x) = C y (x − a) + C y (x − a),
rhr j jrydƒ
y(x) = C y (b − x) + C y (b − x).
_’pie jryu ip¢doi ed–dsra
geujsdsra ‚ujrtr i ‚uyuss’“ feudj’“ gthi}
jr€X
Teudjua ‚uyuzu ~i|d r~d• dyrstjdssid ed–dsrdm ~i|d r~d• pdt}
fisdzsi ~sioi ed–dsr€ rhr ~i|d sd r~d• ed–dsr€X
vu {er~deu“ eu‚pded~ sdfiie’d itipdssitrm fiie’d ji‚srfu‘ {er
ed–dsrr feudj’“ ‚uyuzX
­ P®P® vu€r ed–dsrd feudji€ ‚uyuzr
(
{er a < x < b,
−y − y = 0
1 1
2 2
1 1
2 2
00
y(a) = 0,
y(b) = 2.
„suzuhu eutt~ier~ iysieiysid yrdeds”ruh•sid geujsdsrdX ¬oi
“ueufdertrzdtfid geujsdsrd −p − 1 = 0 X qrthu p = i m p = −i ¤ fiesr
geujsdsraX er Ÿi~ gsf”rr y (x) = cos x r y (x) = sin x ipeu‚g‘ gs}
yu~dsuh•sg‘ trtd~g ed–dsr€X xsuzr ip¢dd ed–dsrd geujsdsra r~dd
jry
cos x + C sin x.
˜{edydhr~ ‚suzdsra C y(x)
r C ={erC fiie’“
su€ydssua gsf”ra gyijhdji}
ead ‚uyuss’~ feudj’~ gthijra~X
2
1
1
2
1
1
2
2
(
C1 cos a + C2 sin a = 0
C1 cos b + C2 sin b = 2.
U
2
Sha su“i|ydsra C r C thdygd ed–r• trtd~g hrsd€s’“ geujsdsr€X
bi|si g{eitr• su“i|ydsrd ed–dsra feudji€ ‚uyuzrm dthr ip¢dd
ed–dsrd geujsdsra rtfu• j jrydƒ
1
2
y(x) = C1 cos(x − a) + C2 sin(x − a).
ˆi gyipsim {itfih•fg feudjid gthijrd j izfd x = a ¤ iysieiysidX †‚
{dejioi feudjioi gthijra {ihgzr~ eujdstji C cos 0 + C sin 0 = 0. xsuzr
r y(x) = C sin(x − a) X Sha Ÿi€ gsf”rr yih|si j’{ihsa•ta
C
=
0
jieid feudjid gthijrdƒ
1
1
2
2
C sin(b − a) = 2.
d–dsrd Ÿioi geujsdsra ‚ujrtr
i ‚suzdsra b − a X
¬thr b − a = πk, k ∈ Z m i
2
C2 sin(πk) = 2
⇔
C2 0 = 2.
Reujsdsrd sd r~dd ed–dsr€X xsuzr r feudjua ‚uyuzu sd pgyd r~d• ed}
–dsr€X
¬thr b−a 6= πk (k ∈ Z) m ioyu sin(πk) 6= 0 r C = 2 X Teudjua
sin(b − a)
‚uyuzu r~dd dyrstjdssid ed–dsrd y(x) = 2 sin(x − a) X
sin(b − a) r X
utt~ier feudj’d ‚uyuzr {er eu‚s’“ ‚suzdsra“
a b
gt• a = π m b = 7π X er Ÿi~ b − a = π r C = 4 X Teudjua ‚uyuzu
r~dd dyrstjdssid ed–dsrd
6
6
X
y(x) =gt•
4 sin(x − π)m
X nioyu b − a = π m sin π = 0 X Teudjua ‚uyuzu
a
=
π
b
=
2
π
ed–dsr€ sd r~ddX
˜jdƒ y(x) = 2 sin(x − a) {er b − a 6= πk (k ∈ Z) ™
− a)
ed–dsr€ sd {er b −sin(b
X
a
=
π
k
(k
∈
Z)
­ P®N® vu€r ed–dsrd feudji€ ‚uyuzr
(
{er a < x < b,
−y − y = 0
2
2
00
y(a) = 1,
y 0 (b) = 0.
Reujsdsrd j eutt~uerjud~i€ ‚uyuzd ufid |dm zi r j {edy’yg¢d€X
itfih•fg iysieiysid feudjid gthijrd ‚ydt• ‚uyusi j izfd x = b m i ip¢dd
ed–dsrd geujsdsra ”dhdtiipeu‚si rtfu• j jrydƒ
y(x) = C1 cos(b − x) + C2 sin(b − x).
vu€yd~ y (x) = C sin(b − x) − C cos(b − x) X †t{ih•‚ga jieid feudjid
gthijrdm {ihgzr~ C = 0 X xsuzr y(x) = C cos(b − x) X iytujr~ Ÿg
¥
0
1
2
2
1
gsf”r‘ j {dejid feudjid gthijrdƒ
C1 cos(b − a) = 1.
_ thgzua“ b − a = π + πk (k ∈ Z) Ÿi geujsdsrd sd r~dd ed–dsr€X
xsuzr r feudjua ‚uyuzu2ed–dsr€ sd r~ddX
¬thr b − a 6= π + πk(k ∈ Z) m i cos(b − a) 6= 0 r C = 1 X
2
cos(b − a)
Teudjua ‚uyuzu r~dd dyrstjdssid ed–dsrd y(x) = cos(b − x) X
cos(b − a)
˜jdƒ y(x) = cos(b − x) {er b − a 6= π + πk(k ∈ Z) ™
− a)
2
ed–dsr€ sd {er b −cos(b
X
π
a = + πk(k ∈ Z)
2ed–dsrd feudji€ ‚uyuzrƒ
­ P®¯® vu€r
(
{er 0 < x < π,
−y − y = 0
1
00
y(0) = 0,
y(π) = 0.
Reujsdsrd j Ÿi~ {er~ded ufid |dm fuf r j g|d eutt~iedss’“X i}
tfih•fg feudjid gthijrd j izfd x = 0 ¤ iysieiysidm i ip¢dd ed–dsrd
geujsdsra gyipsi rtfu• j jrydƒ
y(x) = C1 cos x + C2 sin x.
_it{ih•‚gd~ta feudj’~r gthijra~rƒ
(
C1 + C2 0 = 0
−C1 + C2 0 = 0.
d–ua Ÿg trtd~gm {ihgzr~ C = 0 m C ¤ h‘pid zrthiX xsuzr fu|yua
gsf”ra jryu y(x) = C sin x (C ∈ R) pgyd ed–dsrd~ ‚uyussi€ feudji€
‚uyuzrX Teudjua ‚uyuzu r~dd pdtfisdzsi ~sioi ed–dsr€X
˜jdƒ y(x) = C sin x (C ∈ R) X
­ P®Q® vu€r ed–dsrd feudji€ ‚uyuzr j ‚ujrtr~itr i l
(
{er 0 < x < l,
−y − y = 0
1
2
2
2
2
2
00
y 0 (0) − 2y(0) = 1,
y(l) = 0.
Reujsdsrd j ‚uyuzd ufid |dm fuf r j {edy’yg¢r“X itfih•fg iy}
sieiysid feudjid gthijrd ‚ydt• ‚uyusi j izfd x = l m i ip¢dd ed–dsrd
geujsdsra pgyd~ rtfu• j jrydƒ
y(x) = C1 cos(l − x) + C2 sin(l − x).
W
†‚ jieioi feudjioi gthijra thdygdm zi C
1
vu€yd~ y (x)
gthijrdƒ
0
=0
X iŸi~g
y(x) = C2 sin(l − x).
= −C2 cos(l − x)
r {iytujr~ gsf”r‘ j {dejid feudjid
−C2 cos(l) − 2C2 sin(l) = 1
⇔
¬thr j’{ihsadta eujdstji
cos(l) + 2 sin(l) = 0
−C2 (cos(l) + 2 sin(l)) = 1.
1
l = arctg(− ) + πk (k ∈ Z),
2
⇔
i feudjua ‚uyuzu ed–dsr€ sd r~ddX
_ thgzud cos(l) + 2 sin(l) 6= 0 m XdX {er l 6= arctg(− 1 ) + πk (k ∈ Z)
2
feudjua ‚uyuzu r~dd dyrstjdssid ed–dsrd y(x) = − sin(l
− x) X
cos(l) + 2 sin(l)
˜jdƒ y(x) = − sin(l − x) {er l 6= arctg(− 1 ) + πk (k ∈ Z) ™
cos(l) + 2 sin(l)
2
ed–dsr€ sd {er l = arctg(− 1 ) + πk (k ∈ Z) X
2
­ P®°® vu€r ed–dsrd
feudji€ ‚uyuzr

− 1 d ρy (ρ) + 1 n y (ρ) = 0 {er 0 < ρ < R (n > 0),
ρ dρ
ρ
y (ρ) ioeusrzdsu {er ρ → 0 + 0, y (R) = A.
xuyussid yrdeds”ruh•sid geujsdsrd ¤ Ÿi geujsdsrd t {ded~dss’}
~r fiŸr”rdsu~rX edipeu‚gd~ doim j’zrthrj {eir‚jiys’dƒ
0
2
n
n
2
n
n
1 0
1
yn (ρ) + 2 n2 yn (ρ) = 0
ρ
ρ
00
−yn (ρ) −
⇔
0
00
−ρ2 yn (ρ) − ρ yn (ρ) + n2 yn (ρ) = 0.
ihgzdssid geujsdsrd ajhadta geujsdsrd~ ˆ€hdeuX „ {i~i¢•‘ ‚u~ds’
isi {edipeu‚gdta f yrdeds”ruh•si~g geujsdsr‘ t {itiass’~r
ρ
=
e
fiŸr”rdsu~rX er ufi€ ‚u~dsd t{eujdyhrj’ eujdstju
t
yn (ρ) = yn (et ) = vn (t) = vn (ln(ρ)).
er Ÿi~ v = y ρ m v = y ρ + y ρ X ˜sitrdh•si gsf”rr v {ihgzrta
geujsdsrd t {itiass’~r fiŸr”rdsu~r jryuƒ
0
n
0
n
00
n
00 2
n
0
n
n
00
−vn (t) + n2 vn (t) = 0.
Z
xu{r–d~ yha sdoi “ueufdertrzdtfid geujsdsrdƒ −p + n = 0 X ¬oi fiesrƒ
r p = −n X nioyu ip¢r~ ed–dsrd~ yrdeds”ruh•sioi geujsdsra
ppgyd
= ngsf”ra
2
1
2
2
vn (t) = Cn ent + Dn e−nt .
Rzr’juam zi
t = ln(ρ)
Dn
yn (ρ) = Cn ρn + n
ρ
m su€yd~ ip¢dd ed–dsrd rt“iysioi geujsdsra
X
†‚ gthijra ioeusrzdssitr gsf”rr {er ρ → 0+0 thdygdm zi D = 0
r y (ρ) = C ρ X †t{ih•‚ga jieid feudjid gthijrdm {ihgzr~ C R = A X
xsuzr C = A X d–dsrd~ feudji€ ‚uyuzr pgyd gsf”ra y (ρ) = Aρ X
R
˜jdƒ y R(ρ) = Aρ X
R
­ P®±® vu€r
ed–dsrd feudji€ ‚uyuzr

 1 d (ρy (ρ)) = 0 {er 0 < ρ < R,
ρ dρ
y(ρ) ioeusrzdsu {er ρ → 0 + 0, y(R) = A.
„suzuhum fuf r j {edy’yg¢d~ {er~ded {edipeu‚gd~ geujsdsrdƒ
n
n
n
n
n
n
n
n
n
n
n
n
n
n
0
00
y (ρ) +
1 0
y (ρ) = 0
ρ
00
0
ρ2 y (ρ) + ρ y (ρ) = 0.
⇔
ihgzrhit• geujsdsrd ˆ€hdeuX _’{ihsr~ ‚u~dsg ρ = e X nioyu y(ρ) =
{er Ÿi~ v = y ρ m v = y ρ + y ρ r ‚suzr gsf}
=
y(e
)
=
v(t)
=
v(ln(
ρ
))
”ra v(t) gyijhdjiead geujsdsr‘ƒ v = 0 X ˜p¢r~ ed–dsrd~ Ÿioi geuj}
sdsra ajhadta gsf”ra v(t) = C + D t X Rzr’jua eujdstji t = ln(ρ) m
{ihgzr~ ed–dsrd ‚uyussioi geujsdsra y(ρ) = C + D ln(ρ) X †‚ gthijra
ioeusrzdssitr ed–dsra {er ρ → 0 + 0 thdygdm zi D = 0 X †t{ih•‚ga
jieid feudjid gthijrdm su€yd~ C = A X nioyu ed–dsrd~ ‚uyuzr ajhadta
gsf”ra y(x) = A X
˜jdƒ y(x) = A X
­ P®²® Teudjg‘ ‚uyuzg
(
{er 0 < x < l,
−y + 2y = 3x + 1
t
0
t
0
00
0
00
00 2
0
0
0
0
0
0
00
y 0 (2) − 3y(2) = 4,
y 0 (4) = 1
tjdtr f ‚uyuzd t iysieiys’~r feudj’~r gthijra~rX
er {er~dsdsrr ~diyu —ge•d feudjua ‚uyuzu {edipeu‚gdta f ‚uyuzd
t iysieiys’~r feudj’~r gthijra~rX —gsf”r‘m fiieua ajhadta ed–dsr}
d~ ‚uyussi€ ‚uyuzr {edytujha‘ j jryd tg~~’ yjg“ y(x) = v(x) + w(x) X
—gsf”r‘ w(x) j’preu‘ ufmzip’ isu gyijhdjieahu ufr~ |d feudj’~
gthijra~m zi r gsf”ra y(x) X nioyu gsf”ra v(x) pgyd gyijhdjiea•
Y
iysieiys’~ feudj’~ gthijra~X _i ~sior“ thgzua“ w(x) ‚uyu‘ fuf hr}
sd€sg‘ gsf”r‘ w(x) = αx + β X TiŸr”rds’ α r β su“iya {ithd
{iytusijfr w(x) j feudj’d gthijraX
„jdyd~ ‚uyussg‘ feudjg‘ ‚uyuzg f ‚uyuzd t iysieiys’~r feudj’~r
gthijra~rX gt• y(x) = v(x) + w(x) m oyd w(x) = αx + β X iypded~ α r β
ufm zip’ yha gsf”rr w(x, t) j’{ihsahrt• feudj’d gthijra
w0 (2) − 3w(2) = 4,
w0 (4) = 1.
vu€yd~ w (x) = α rm rt{ih•‚ga feudj’d gthijram {ihgzr~ trtd~g
0
(
α − 3(2α + β) = 4
α=1
(
α=1
⇔
1 − 3(2 + β) = 4
(
α=1
⇔
β = −3.
xsuzr w(x) = x − 3 X er Ÿi~ gsf”ra v(x) gyijhdjiead iysieiy}
s’~ feudj’~ gthijra~X xu{r–d~ feudjg‘ ‚uyuzg yha v(x) m gzr’juam zi
m
m
ƒ
y(x) = v(x) + x − 3 y (x) = v (x) + 1 y (x) = v (x)
(
{er 0 < x < l,
−v + 2(v + x − 3) = 3x + 1
0
0
00
00
00
v 0 (2) − 3v(2) = 0,
v 0 (4) = 0.
edipeu‚gam {ihgzr~ rtfi~g‘ feudjg‘ ‚uyuzg
(
{er 0 < x < l,
−v + 2v = x + 7
00
v 0 (2) − 3v(2) = 0,
v 0 (4) = 0.
˜jdƒ y(x) = v(x) + x − 3 m
{er
(
−v 00 + 2v = x + 7
v 0 (2) − 3v(2) = 0,
0 < x < l,
v (4) = 0.
0
³´µ¶· ¸¹¸º»
_ g{eu|sdsra“ VXV}VXU ed–r• ‚uyuss’d feudj’d ‚uyuzrX
(
{er 0 < x < ,
−y − 4y = 0
1.1
00
π
4
y(0) = 0, y 0 ( π4 ) + 2y( π4 ) = 1.
(
π
−y 00 − y = 0
< x < π3 ,
6
1.2
y 0 ( π6 ) − 2y( π6 ) = 0, y( π3 ) = 21 .
(
−y 00 − 16y = 0
0 < x < π2 ,
1.3
y 0 (0) = 2, y( π2 ) = 0.
{er
{er
V\
{er
(
−y 00 − 9y = 0
0 < x < π6 ,
1.4
y(0) = 0, y 0 ( π6 ) = 0.
(
0
0 < ρ < R,
ρ2 y (ρ) 0 − 6y = 0
1.5
y(ρ)
ρ → 0 + 0, y(R) = 1.
{er
ioeusrzdsu {er
_ g{eu|sdsra“ VX¥}VXZ ed–r• ‚uyuss’d feudj’d ‚uyuzr j ‚ujrtr~itr
i l X
(
{er 0 < x < l,
−y − 4y = 0
00
1.6
y(0) = 12 , y(l) = 0.
(
0 < x < l,
−y 00 − 9y = 0
1.7 0
0
y (0) = 0, y (l) + 3y(l) = 1.
(
0 < x < l,
−y 00 − 4y = 0
1.8
y 0 (0) = 0, y(l) = 0.
{er
{er
_ g{eu|sdsra“ VXY}VXV\ tjdtr ‚uyussg‘ feudjg‘ ‚uyuzg f ‚uyuzd t
iysieiys’~r feudj’~r gthijra~rX
(
{er 2 < x < 3,
−y + 3y = −2x + 3
1.9
00
y(2) = 1, y 0 (3) + 4y(3) = 9.
(
−y 00 − 5y = 4x
0 < x < 4,
1.10
y 0 (0) − 3y(0) = 2, y(4) = 9.
{er
¼½¾¹½¿
VXV y(x) = 1 sin(2x) ™ VXc y(x) = √ 1 (cos(x − π ) + 2 sin(x − π )) ™
2
6
6
3+2
VX` ed–dsr€ sd™ VXŠ y(x) = C sin(3x) m C ∈ R ™ VXU y(ρ) = ρ ™
VX¥ y(x) = sin(l − x) {er l 6= πk m k ∈ Z ™ ed–dsr€ sd {er lR= πk m k ∈ Z ™
2 sin(l)
VXW y(x) =
{er l 6= π + πk m k ∈ Z ™ ed–dsr€ sd {er
cos(3x)
3(cos(3l) − sin(3l))
12
3
™ VXZ y(x) = C cos(2x), C ∈ R {er l = πk m k ∈ Z ™
π
πk m
l =
+
k ∈ Z
12
3
2
{er
™
VXY
m
πk m
y(x) = 0
l 6=
k∈Z
y(x) = v(x) + x − 1
2
{er
™
−v
+
3v
=
−5x
+
6
2
<
x
<
3,
v(2)
=
0,
v
(3)
+
4v(3)
=
0
VXV\ y(x) = v(x) + 5x + 1 m −v − 5v = 20x + 11 {er 0 < x < 4,
X
v (0) − 3v(0) = 0, v(4) = 0
2
2
2
1
00
0
2
1
0
00
VV
À ' ACÁ C5>C <&= &6> Á 2?A) &Â(>? 1>(( D
gt• D(L) ¤ Ÿi ~si|dtji yju|y’ yrdeds”regd~’“ su {ei~d}
|gfd (a, b) gsf”r€m gyijhdjiea‘¢r“ su fis”u“ {ei~d|gfu iysieiy}
s’~ feudj’~ gthijra~m su{er~de gthijra~ jryuƒ
R1 y 0 (a) − S1 y(a) = 0,
R2 y 0 (b) + S2 y(b) = 0.
bsi|dtji D(L) {edytujhad tipi€ hrsd€sid {eiteustjiX Sha Ÿhd}
~dsij f (x) r g(x) Ÿioi ~si|dtju i{edydhr~ tfuhaesid {eir‚jdydsrd {i
ie~ghd
(f, g) =
Zb
f (x)g(x)ρ(x)dx,
oyd ρ(x) ¤ sd{ede’jsua su [a, b] gsf”ram ρ(x) ≥ ρ > 0 X ˜su su‚’judta
jdtiji€ gsf”rd€m rhr jdti~X
—gsf”rr f (x) r g(x) su‚’ju‘ta ieioisuh•s’~rm dthr (f, g) = 0 X
gt• y ∈ D(L) X vie~u gsf”rr y(x) m {iei|ydssua tfuhaes’~ {ei}
r‚jdydsrd~m su“iyrta {i {eujrhgƒ
a
0
v
u b
uZ
u
kyk = t y 2 (x)ρ(x)dx.
nufr~ ipeu‚i~m h‘pua gsf”ra y ∈ D(L) ajhadta Ÿhd~dsi~ {eiteus}
tju L [a, b; ρ(x)] [t~X `¡]X
utt~ier~ hrsd€s’€ yrdeds”ruh•s’€ i{deuiem yd€tjg‘¢r€ r‚
{eiteustju D(L) j hrsd€sid {eiteustji sd{ede’js’“ su [a, b] gsf}
”r€m jryu
a
2
L(y) = −
1
(p(x)y 0 )0 + q(x)y,
ρ(x)
[a, b]
p(x) ≥ p0 > 0 ρ(x)
oyd ρ(x) m p(x) m q(x) ¤ sd{ede’js’d su
gsf”rrm
m
¤ jdtijua gsf”raX nufi€ i{deuie su‚’judta i{deuiei~ £ge~u¤lrg}
jrhhaX
Sha i{deuieu £ge~u¤lrgjrhha L(y) t{eujdyhrj’ thdyg‘¢rd tji€}
tjuƒ
V] i{deuie tr~~derzdsm XdX ∀y, z, ∈ D(L) t{eujdyhrji eujdstji
m
(L(y),
z)
=
(y,
L(z))
c] i{deuie {ihi|rdh•si i{edydhdsm XdX yha ∀y ∈ D(L) t{eujdyhrji
sdeujdstji (L(y), y) ≥ q kyk oyd q = min q(x) X
Sifu‚udh•tji Ÿr“ tji€tj {erjiyrta j {itiprr V¡X
Vc
0
2
0
x∈[a,b]
J N®P ÃÄÅÆÇ λ ÈÉÊËÌÉÍÎ ÅÏ ÅÇÐÅ ÎÌÍÈÈËÑ Ò ÄÅÆÇÑ ÇÓ ÍÔ ÉÕ
Å×Ø Í Å ÎÌ × ÍÎ ÈÍÈ ×Æ ÍÌÉÏ Ù× ÈÚÛ ÄÏ y(x) ∈ D(L) Ö Ü ÆÏ Ú Ç Î ÇÔÇÝ
Î ÇÔ É L Ö Í ÅÆÄ ÔÄ
à Î ÇÑ Ù× ÈÚÛ ÄÏ y(x) ÈÉÊËÌÉÍÎ ÅÏ ÅÇÐÅ ÎÌÍÈÈ ÇÝ Ù× ÈÚÛ Ä Í Ý
L(y) = λy Þ ß
ÇÓ ÍÔ ÉÎ ÇÔ ÉÞ
xuyuzu su“i|ydsra tiptjdss’“ zrtdh r tiptjdss’“ gsf”r€ i{deu}
ieu su‚’judta ‚uyuzd€ £ge~u¤lrgjrhhaƒ
[cXV]
(
L(y) = λy, a < x < b
R1 y 0 (a) − S1 y(a) = 0, R2 y 0 (b) + S2 y(b) = 0.
bsi|dtji jtd“ tiptjdss’“ zrtdh su‚’judta t{dfei~ i{deuieu
rhr t{dfei~ ‚uyuzr [cXV]X
L(y) dedzrthr~ itsijs’d tji€tju tiptjdss’“ zrtdh ‚uyuzr £ge~u¤
lrgjrhhaƒ
V] „iptjdss’d zrthu i{deuieu L(y) jd¢dtjdss’dX
c] „iptjdss’d zrthu i{deuieu L(y) yrtfeds’dm XdX {edytujhad
tipi€ {ithdyijudh•sit• {λ } X
`] ithdyijudh•sit• {λ } ioeusrzdsu tsr‚gƒ λ ≥ min q(x) r
X
lim λ = +∞
Š] er sdfiie’“ {ihi|rdh•s’“ A r B yha jtd“ yituizsi pih•–r“
t{eujdyhrj’ sdeujdstju An ≤ λ ≤ Bn X
n Sha tiptjdss’“ gsf”r€
i{deuieu £ge~u¤lrgjrhha t{eujdyhr}
j’ thdyg‘¢rd gjde|ydsraƒ
V] Tu|yi~g tiptjdssi~g zrthg tiijdtjgd ih•fi iysu [t izsi}
t•‘ yi {itiassioi ~si|rdha] tiptjdssua gsf”raX
c] nuf fuf tiptjdss’d gsf”rrm tiijdtjg‘¢rd eu‚hrzs’~ tip}
tjdss’~ zrthu~m ieioisuh•s’m ‚suzr trtd~u tiptjdss’“ gsf”r€
i{deuieu {y (x)} ajhadta ieioisuh•si€ trtd~i€X
`] ˜eioisuh•sua trtd~u {y (x)} ajhadta {ihsi€ j {eiteustjd
X ˆi i‚suzudm zi h‘pua gsf”ra r‚ Ÿioi {eiteustju ~i}
L
[a,
b;
ρ
(x)]
|d p’• eu‚hi|dsu j eay —ge•d {i trtd~d {y (x)} r Ÿi eay pgyd
t“iyr•ta f gsf”rr j sie~d {eiteustju L [a, b; ρ(x)] X
Tei~d ioim yha gsf”rr r‚ iphutr i{edydhdsra i{deuieu L t{eu}
jdyhrju died~u „dfhijuƒ
%­ N®P ® ß ×Å Îá y ∈ D(L) Þ â ÅÆÄ Ù× ÈÚÛ Äã y(x) Ô ÉÊ ÆÇä Ä Îá Ì
ÔÏ Ü å ×Ô áÍ ÓÇ ÇÔ Î ÇæÇ ÈÉÆ áÈ ÇÝ Ö ÓÇÆ È ÇÝ Ì ÓÔÇÅ Î Ô ÉÈ Å ÎÌÍ L [a, b; ρ(x)] ÅÄÅ ÎÍÕ
Ç à Î Ç Î ÔÏ Ü Ð× ÜÍÎ É ÐÅÇÆã ÎÈ Ç ÅçÇ Ü Ä Î ÅÏ Ú y(x)
ÑÍ Ù× ÈÚÛ ÄÝà {yÇ (x)}
Ö
Î
∀x ∈ [a, b] è Î Î ÔÏ Ü Ñ Çä È Ç ÓÇ Ò Æ ÍÈÈ Ç Ü ÄÙÙ ÍÔ ÍÈÛ ÄÔÇ ÌÉÎá 2 Ô ÉÊÉÞ é Ï Ü Ä Ê
Ó ÍÔ ÌË ç ÓÔÇÄ ÊÌ Ç ÜÈË ç Ð× ÜÍÎ ÓÇ Î Ç ÒÍÒÈ Ç ÅçÇ Ü Ä Îá ÅÏ Ú y (x) Ö É ÔÏ Ü Ä Ê ÌÎ ÇÔ Ë ç
ÓÔÇÄ ÊÌ Ç ÜÈË ç ÅçÇ Ü Ä Î ÅÏ Ú y (x) Ì È ÇÔ ÑÍ ÓÔÇÅ Î Ô ÉÈ Å ÎÌÉ L [a, b; ρ(x)] Þ
V`
+∞
n n=1
n
n→+∞
n
n
2
+∞
k=1
k
2
n
k
+∞
k=1
2
+∞
k=1
k
2
k
2
+∞
k=1
00
0
2
x∈[a,b]
Sha eu‚hrzs’“ i{deuieij £ge~u¤lrgjrhha L(y) su€yd~ tiptjdss’d
zrthu r tiptjdss’d jdfie’X
­ N®P® Sha i{deuieu L (y) = − d y m a < x < b m i{edydhdssioi
dx
su ~si|dtjd gsf”r€m gyijhdjiea‘¢r“ iysieiys’~
gthijra~ Srer“hd
j izfu“ a r b m su€r tiptjdss’d zrthu r tiptjdss’d gsf”rrm gfu‚u•
{eiteustji j fiiei~ tiptjdss’d gsf”rr ipeu‚g‘ {ihsg‘ ieioi}
suh•sg‘ trtd~g r su€r fjuyeu’ sie~ tiptjdss’“ gsf”r€X
Sha Ÿioi i{deuieu ed–r~ ‚uyuzg £ge~u¤lrgjrhhaƒ
2
x
(
2
−y 00 = λy, a < x < b,
y(a) = 0, y(b) = 0.
itfih•fg q(x) ≡ 0 m i λ ≥ 0 X d–r~ tsuzuhu yrdeds”ruh•sid geuj}
sdsrdX xu{r–d~ yha sdoi “ueufdertrzdtfid geujsdsrdƒ −p = λ. ¬thr
m i geujsdsrd r~dd yju iyrsufij’“ fiesa p = p = 0 r gsf”ra
λ=0
pgyd ip¢r~ ed–dsrd~ yrdeds”ruh•sioi geujsdsraX
y(x)
=
C
+
C
x
iytujr~ y(x) j feudj’d gthijraƒ
2
1
1
2
2
(
C1 a + C2 = 0,
C1 b + C2 = 0,
ifgyu C = C = 0 m XdX y ≡ 0 rm thdyijudh•sim λ = 0 sd ajhadta
tiptjdss’~ zrthi~X
gt• λ > 0 m ipi‚suzr~ λ = µ X êueufdertrzdtfid geujsdsrd {er}
~d jry −p = µ . Tiesr Ÿioi geujsdsra fi~{hdfts’d p = µi m p = −µi X
˜p¢dd ed–dsrd yrdeds”ruh•sioi geujsdsra gyipsi ‚u{rtu• j jryd
X Sha i{edydhdsra ‚suzdsr€ C r
y(x)jit{ih•‚gd~ta
= C cos(µ(x −
a))
+
C
sin(
µ
(x
−
a))
feudj’~r gthijra~rƒ
C
1
2
2
2
2
1
1
2
2
1
2
(
C1 = 0,
C1 cos(µ(b − a)) + C2 sin(µ(b − a)) = 0.
ˆu trtd~u r~dd sdsghdjid ed–dsrd C = 0 m C 6= 0 m ih•fi fioyu sin(µ(b−
X ˜pi‚suzr~ yhrsg {ei~d|gfu zded‚ l = b − a m ioyu sin(µl) = 0 X
−a)) = 0
ihi|rdh•s’d ed–dsra Ÿioi geujsdsra µ = πk m k = 1, 2, . . . X nufr~
ipeu‚i~m tiptjdss’~r zrthu~r i{deuieu j thgzudl feudj’“ gthijr€ Sr}
er“hd pgyg λ = µ = πk m k = 1, 2, . . . X ihi|rj {eir‚jih•sg‘
{itiassg‘ C = 1 m {ihgzud~l trtd~g tiptjdss’“ gsf”r€
1
2
k
2
k
2
k
2
yk (x) = sin(µk (x − a)) = sin
VŠ
πk(x − a)
,
l
k = 1, 2, . . . .
Sha i{deuieu L (y) = −y jdtijua gsf”ra ρ(x) = 1 m ‚suzr tiptjdss’d
gsf”rr y (x) ipeu‚g‘ {ihsg‘ ieioisuh•sg‘ trtd~g j {eiteustjd
X
L [a,vu€yd~
b; 1]
fjuyeu’ sie~ tiptjdss’“ gsf”r€ƒ
00
x
k
2
kyk (x)k2 =
Zb
yk2 (x)dx =
a
=
Zb
a
Zb
sin2
πk(x − a)
dx =
l
a
2πk(x − a)
1
l
1
1 − cos
dx = (b − a) = .
2
l
2
2
˜jdƒ V] tX zX λ = πk m k = 1, 2, . . .
l
™ c] tX X y (x) = sin πk(x − a), k = 1, 2, . . . ™
(l = b − a)
l
`] L [a, b; 1] ™ Š] ky (x)k = l X
2
­ N®N® Sha i{deuieu
m i{edydhdssioi
d ym
L (y) = −
a<x<b
dx
su ~si|dtjd gsf”r€m gyijhdjiea‘¢r“ iysieiys’~
gthijra~ vd€~usu
j izfu“ a r b m su€r tiptjdss’d zrthu r tiptjdss’d gsf”rrm gfu‚u•
{eiteustji j fiiei~ tiptjdss’d gsf”rr ipeu‚g‘ {ihsg‘ ieioi}
suh•sg‘ trtd~g r su€r fjuyeu’ sie~ tiptjdss’“ gsf”r€X
d–r~ ‚uyuzg £ge~u¤lrgjrhhaƒ
2
k
k
2
2
k
2
x
(
2
−y 00 = λy, a < x < b,
y 0 (a) = 0, y 0 (b) = 0.
ˆu ‚uyuzu ihrzudta i {edy’yg¢d€ ih•fi feudj’~r gthijra~rX
gt• λ = 0 m ioyu “ueufdertrzdtfid geujsdsrd −p = λ r~dd
iyrsufij’d fiesr p = p = 0 r ip¢r~ ed–dsrd~ yrdeds”ruh•sioi
geujsdsra pgyd gsf”ra y(x) = C + C x X iytujhaa y(x) j feudj’d
gthijram {ihgzr~ƒ
2
1
2
1
(
2
C2 = 0,
C2 = 0.
ihi|r~ C = 1 m ioyu gsf”ra y (x) = 1 ajhadta tiptjdssi€ gsf”rd€
i{deuieu L (y) m tiijdtjg‘¢d€ tiptjdssi~g zrthg λ = 0 X
er λ = µ > 0 m fiesa~r “ueufdertrzdtfioi geujsdsra pgyg zrthu
m
r ip¢dd ed–dsrd yrdeds”ruh•sioi geujsdsra ~i|si
p = µi p = −µi
1
x
1
0
0
2
2
VU
‚u{rtu• j jryd y(x) = C
j feudj’d gthijraƒ
1 cos(µ(x − a)) + C2 sin(µ(x − a))
X iytujr~ y(x)
(
µC2 = 0,
−µC1 sin(µ(b − a)) + µC2 cos(µ(b − a)) = 0.
„rtd~u r~dd sdsghdjid ed–dsrdm dthr sin(µ(b − a)) = 0 m XdX {er µ = πk m
l
X xsuzr yha i{deuieu L (y) = −y j thgzud feudj’“
kgthijr€
= 1, 2, .vd€~usu
. . (l = b−a)
tiptjdss’~r zrthu~r uf|d pgyg zrthu λ = µ =
X „zruam zi C = 1 m {ihgzr~ tiptjdss’d gsf”rrm
πk m
k = 1, 2, . . .
=
l
tiijdtjg‘¢rd
Ÿr~ zrthu~ƒ
k
x
00
k
2
2
k
1
yk (x) = cos(µk (x − a)) = cos
πk(x − a)
, k = 1, 2, . . . .
l
ρ(x) = 1
Sha i{deuieu L (y) = −y jdtijua gsf”ra
m ‚suzr tiptjdss’d
gsf”rr y (x) ipeu‚g‘ {ihsg‘ ieioisuh•sg‘ trtd~g j {eiteustjd
X
L [a,vu€yd~
b; 1]
fjuyeu sie~’ gsf”rr y (x) = 1 ƒ
00
x
k
2
0
ky0 (x)k2 =
Zb
y02 (x)dx =
Zb
1dx = b − a = l.
˜{edydhr~ fjuyeu’ sie~ tiptjdss’“ gsf”r€ y (x) m k = 1, 2, . . . ƒ
a
a
k
kyk (x)k2 =
Zb
yk2 (x)dx =
a
Zb
cos2
πk(x − a)
dx =
l
a
Zb
1
2πk(x − a)
1
l
=
1 + cos
dx = (b − a) = .
2
l
2
2
a
2
πk
λ0 = 0 λk =
k = 1, 2, . . . (l = b − a)
l
πk(x − a)
, k = 1, 2, . . .
y0 (x) = 1 yk (x) = cos
l
l
L2 [a, b; 1]
ky0 (x)k2 = l kyk (x)k2 =
2
d2 y
Lx (y) = − 2 a < x < b
dx
˜jdƒ V] tX zX
m
m
™
c] tX X
m
™
`]
™ Š]
m
X
­ N®¯® Sha i{deuieu
m
m i{edydhdssioi
su ~si|dtjd gsf”r€m gyijhdjiea‘¢r“ iysieiys’~ feudj’~ gthijra~
Ry 0 (a) − Sy(a) = 0,
V¥
y 0 (b) = 0.
m
m su€r tiptjdss’d zrthu r tiptjdss’d gsf”rrm gfu‚u•
{eiteustji j fiiei~ tiptjdss’d gsf”rr ipeu‚g‘ {ihsg‘ ieioi}
suh•sg‘ trtd~g r su€r fjuyeu’ sie~ tiptjdss’“ gsf”r€X
utt~ier~ ‚uyuzg £ge~u¤lrgjrhhaƒ
(R 6= 0 S 6= 0)
(
−y 00 = λy, a < x < b,
Ry 0 (a) − Sy(a) = 0, y 0 (b) = 0.
_ Ÿi€ ‚uyuzd yrdeds”ruh•s’€ i{deuie i |dm zi r j {edy’yg}
¢r“ {er~deu“X xsuzr λ ≥ 0 X
gt• λ = 0 m ioyu ip¢dd ed–dsrd yrdeds”ruh•sioi geujsdsra r~d}
d jry y(x) = C + C x X †t{ih•‚ga feudj’d gthijram {ihgzr~ƒ
1
2
(
RC2 − S(C1 + C2 a) = 0,
C2 = 0.
d–dsrd Ÿi€ trtd~’ƒ C = 0 m C = 0 X xsuzr y(x) = 0 r zrthi λ = 0 sd
ajhadta tiptjdss’~ zrthi~ L (y) X
¬thr λ = µ > 0 m i zrthu p = µi m p = −µi ajha‘ta fiesa~r
“ueufdertrzdtfioi geujsdsra −p = µ X ˜p¢dd ed–dsrd yrdeds”r}
uh•sioi geujsdsra yha eutt~uerjud~i€ ‚uyuzr gyipsi ‚u{rtu• j jryd
X iytujr~ gsf”r‘ y(x) j feu}
y(x)
=
C
cos(
µ
(b
−
x))
+
C
sin(
µ
(b
−
x))
dj’d gthijraƒ
1
2
x
2
1
2
1
2
2
2


R (µC1 sin(µ(b − a)) − µC2 cos(µ(b − a))) −
−S (C1 cos(µ(b − a)) + C2 sin(µ(b − a))) = 0,

−µC = 0.
2
˜pi‚suzr~ l = b − a X „rtd~u eujsitrh•su geujsdsr‘
C1 (Rµ sin(µl) − S cos(µl)) = 0.
˜su r~dd sdsghdj’d ed–dsram dthr C
eujds sgh‘X edipeu‚gd~ doi f jryg
1
tg(µl) =
6= 0
S
.
Rµ
X xsuzr jiei€ ~si|rdh•
iteir~ oeurfr {euji€ r hdji€ zutr geujsdsraX
†‚ ertgsfu cXV jrysim zi geujsdsrd r~dd pdtfisdzsid ~si|dtji
{ihi|rdh•s’“ ed–dsr€ µ m k = 1, 2, . . . X Sha su“i|ydsra Ÿr“ ed–dsr€
thdygd rt{ih•‚iju• zrthdss’d ~diy’X
nufr~ ipeu‚i~m λ = µ (k = 1, 2, . . .) ¤ Ÿi tiptjdss’d zrthu i{deu}
ieu eutt~uerjud~i€ ‚uyuzr £ge~u¤lrgjrhha X
VW
k
k
2
k
ëìíî ïîð
ihi|r~ C = 1 m ioyu y (x) = cos(µ (b − x)) (k = 1, 2, . . .)¤ Ÿi tii}
jdtjg‘¢rd r~ tiptjdss’d gsf”rr X ˜sr ipeu‚g‘ {ihsg‘ ieioisuh•}
sg‘ trtd~g j {eiteustjd L [a, b; 1] X vu€yd~ fjuyeu’ sie~ tiptjdss’“
gsf”r€ƒ
1
k
k
2
kyk (x)k2 =
Zb
yk2 (x)dx =
Zb
cos2 (µk (b − x)dx =
a
a
Zb
b
1
1
1
=
(1 + cos(2µk (b − x))) dx =
x+
sin(2µk (b − x))
=
2
2
2µk
a
a
1
1
1
1
(b − a) −
sin(2µk (b − a)) =
l−
sin(2µk l) .
=
2
2µk
2
2µk
˜jdƒ V] tX zX λ = µ m oyd µ ¤ Ÿi {ihi|rdh•s’d fiesr geujsdsra
™ c] tX X y (x) = cos(µ (b − x)), k =
S m
tg(µl) =
k = 1, 2, . . . (l = b − a)
R™ µ
= 1, 2, . . .
`] L [a, b; 1] ™ Š] ky (x)k = 1 l − 1 sin(2µ l) X
2
2µ
­ N®Q® Sha i{deuieu L (y) = − d y m 0 < ϕ < 2π m i{edydhds}
dϕ {deriyrzdtfr~ feudj’~
sioi su ~si|dtjd gsf”r€m gyijhdjiea‘¢r“
gthijra~
2
k
k
k
k
2
k
2
k
k
k
ϕ
2
2
y 0 (0) = y 0 (2π)
su€r tiptjdss’d zrthu r tiptjdss’d gsf”rrm gfu‚u• {eiteustji j
fiiei~ tiptjdss’d gsf”rr ipeu‚g‘ {ihsg‘ ieioisuh•sg‘ trtd~g r
su€r fjuyeu’ sie~ tiptjdss’“ gsf”r€X
VZ
y(0) = y(2π),
d–r~ ‚uyuzg £ge~u¤lrgjrhhaƒ
(
−y 00 = λy, 0 < ϕ < 2π,
y(0) = y(2π), y 0 (0) = y 0 (2π).
Tuf r j {edy’yg¢r“ {er~deu“m yha i{deuieu −y tiptjdss’d zrthu
gyijhdjiea‘ gthijr‘ λ ≥ 0 X
utt~ier~ thgzu€ λ = 0 X ˜p¢r~ ed–dsrd~ yrdeds”ruh•sioi geuj}
sdsra pgyd gsf”ra y(ϕ) = C + C ϕ X iytujr~ dd j feudj’d gthijraƒ
00
1
2
(
C1 = C1 + C2 2π,
C2 = C2 .
d–dsrd Ÿi€ trtd~’
(
C1 = t, t ∈ R
C2 = 0.
ihi|r~ C = 1 m ioyu gsf”ra y (ϕ) = 1 pgyd tiptjdssi€ gsf”rd€
i{deuieum tiijdtjg‘¢d€ tiptjdssi~g zrthg λ = 0 X
gt• d{de• λ = µ > 0 X Tiesa~r “ueufdertrzdtfioi geujsdsra
pgyg zrthu p = µi m p = −µi X ˜p¢dd ed–dsrd yrdeds”ruh•}
−p
=
µ
sioi geujsdsra ‚u{r–d~ j jryd y(ϕ) = C cos(µϕ)) + C sin(µϕ) m {er Ÿi~
X _it{ih•‚gd~ta feudj’~r gthijra}
y~rƒ
(ϕ) = −µC sin(µϕ)) + µC cos(µϕ)
(
1
0
0
2
2
2
1
2
1
0
1
2
2
C1 = C1 cos(2πµ) + C2 sin(2πµ),
µC2 = −µC1 sin(2πµ) + µC2 cos(2πµ).
xu{r–d~ trtd~g j ~uerzsi~ jrydƒ
.
1 − cos(2πµ)
− sin(2πµ)
C1
0
=
.
µ sin(2πµ) µ − µ cos(2πµ) C2
0
˜ysieiysua trtd~u hrsd€s’“ geujsdsr€ r~dd sdsghdj’d ed–dsram dthr
i{edydhrdh• ~uer”’ fiŸr”rdsij eujds sgh‘ƒ
µ(1 − cos(2πµ))2 + µ sin2 (2πµ) = 0.
ˆi eujdstji {edipeu‚gdta f jryg
2µ(1 − cos(2πµ)) = 0.
itfih•fg µ 6= 0 m i cos(2πµ) = 1 X ihi|rdh•s’d fiesr geujsdsra ¤
Ÿi zrthu µ = k, k = 1, 2, . . . . xsuzr tiptjdss’~r zrthu~r i{deuieu
j thgzud {deriyrzdtfr“ feudj’“ gthijr€ uf|d pgyg zrthu
L (y) =m −y
X ˜{edydhr~ tiptjdss’d gsf”rrm tiijdtjg‘¢rd
λ = k k = 1, 2, . . .
VY
ϕ
k
2
k
00
Ÿr~ tiptjdss’~ zrthu~X itfih•fg yha h‘p’“ ‚suzdsr€ C r C gsf}
”rr
1
2
pgyg gyijhdjiea• feudj’~ gthijra~m i fu|yi~g tiptjdssi~g zrt}
hg λ = k tiijdtjg‘ yjd hrsd€si sd‚ujrtr~’d gsf”rr cos(kϕ) r
m k = 1, 2, . . . X Sha eutt~uerjud~i€ ‚uyuzr {ihsi€ ieioisuh•si€
sin(k
ϕ
)
{eiteustjd L [a, b; 1] pgyd trtd~u gsf”r€
yk (ϕ) = C1 cos(k ϕ) + C2 sin(k ϕ)
k
2
2
{1,
cos(k ϕ),
sin(k ϕ)},
vu€yd~ fjuyeu’ sie~ Ÿr“ gsf”r€ƒ
k1k2 =
Z2π
k = 1, 2, . . . .
12 (x)dx = 2π,
0
k cos(k ϕ)k2 =
Z2π
0
Z2π
k sin(k ϕ)k2 =
Z2π
Z2π
cos(k ϕ)2 (x)dx =
1
(1 + cos(2k ϕ)) dx = π,
2
1
(1 − cos(2k ϕ)) dx = π.
2
0
sin(k ϕ)2 (x)dx =
˜jdƒ V] tX zX λ = 0 m λ = k m k = 1, 2, . . . ™
c] tX X {1, cos(kϕ), sin(kϕ)}, k = 1, 2, . . . ™
`] L [0, 2π; 1] ™ Š] k1k = 2π m k cos(kϕ)k = π m k sin(kϕ)k = π X
­ N®°® Sha i{deuieu ^dttdha B (y) = − 1 d (ρ dy ) m
ρ dρ dρ
m
i{edydhdssioi
su
~si|dtjd
gsf”r€m
gyijhdjiea‘¢r“
iysi}
0eiys’~
<ρ<T
feudj’~ gthijra~ j izfu“ 0 r T
ioeusrzdsu {er ρ → 0 + 0, y(T ) = 0,
y(ρ)
su€r tiptjdss’d zrthu r tiptjdss’d gsf”rrm gfu‚u• {eiteustji j
fiiei~ tiptjdss’d gsf”rr ipeu‚g‘ {ihsg‘ ieioisuh•sg‘ trtd~g r
su€r fjuyeu’ sie~ tiptjdss’“ gsf”r€X
d–r~ ‚uyuzg £ge~u¤lrgjrhhaƒ
0
0
2
0
2
k
2
2
2
0

− 1 (ρy 0 )0 = λy,
ρ
y(ρ)
0 < ρ < T,
ioeusrzdsu {er ρ → 0 + 0, y(T ) = 0.
Sha i{deuieu ^dttdha B (y) gsf”ra q(ρ) eujsu sgh‘X iŸi~g jtd
tiptjdss’d zrthu sdier”udh•s’X
c\
0
utt~ier~ {ithdyijudh•si yju thgzuaƒ
V] gt• λ = 0 m ioyu {er 0 < ρ < T geujsdsrd
1
− (ρy 0 )0 = 0
ρ
−y 0 − ρy 00 = 0
⇔
ρ2 y 00 + ρy 0 = 0
⇔
ajhadta geujsdsrd~ ˆ€hdeuX
_ {er~ded VX¥ su€ydsi ip¢dd ed–dsrd Ÿioi geujsdsra
y(ρ) = C1 + C2 ln(x).
†‚ {dejioi feudjioi gthijra thdygdm zi C = 0 m u r‚ jieioi ¤ C = 0 X
itfih•fg gsf”ra y(ρ) ≡ 0 m i zrthi λ = 0 sd ajhadta tiptjdss’~
zrthi~ i{deuieu B (y) X
c] gt• λ = µ > 0 X utt~ier~ geujsdsrd
2
2
1
0
1
− (ρy 0 )0 = µ2 y
ρ
1
⇔ y 00 + y 0 + µ2 y = 0.
ρ
_’{ihsr~ ‚u~dsg {ded~dssi€ ρ = t X nioyu t{eujdyhrju ”d{izfu eujdstj
X itfih•fg v = y m v = y m i isitr}
y(
ρ
)
=
y(
t)
=
v(t)
=
v(
µρ
)
dh•si {ded~dssi€ t geujsdsrd {er~d jryƒ
1
µ
1
µ
1
µ
0
t
0
ρ
1
µ2
00
t
00
ρ
1
v 00 + v 0 + v = 0.
t
ˆi geujsdsrd ^dttdha V¡X ˜p¢r~ ed–dsrd~ Ÿioi iysieiysioi yrd}
eds”ruh•sioi geujsdsra ajhadta gsf”ra v(t) = C J (t) + C N (t) m oyd
r N (t) ¤ gsf”rr ^dttdha r vd€~usum tiijdtjdssiX er x → 0 + 0
J
(t)
gsf”ra N (t) sdioeusrzdssuX iŸi~g ioeusrzdssid j sghd ip¢dd ed–d}
srd Ÿioi geujsdsra r~dd jry v(t) = C J (t) X nioyu ed–dsrd~ rt“iysioi
geujsdsra pgyd gsf”ra y(ρ) = C J (µρ) X xuyuyr~ C = 1 X _it{ih•‚gd~}
ta jie’~ feudj’~ gthijrd~ƒ
1 0
0
2
0
0
0
1 0
1 0
1
J0 (µT ) = 0.
qrthu µ T = γ (k = 1, 2, . . .) m oyd γ ¤ Ÿi fiesr gsf”rr J (ρ) X †‚jdtsi
V¡m zi Ÿu gsf”ra r~dd pdtfisdzsid ~si|dtji {eit’“ fiesd€X xsuzr
zrthu
k
k
0
k
λk =
µ2k
=
γ 2
k
,
k = 1, 2, . . .
ajha‘ta tiptjdss’~r zrthu~rm u gsf”rr y (ρ) = J ( ρ) ¤ tiptjdss’}
~r gsf”ra~r i{deuieu ^dttdha B (y) m eutt~uerjud~ioi su ~si|dtjd
gsf”r€m gyijhdjiea‘¢r“ ‚uyuss’~ feudj’~ gthijra~X
„rtd~u su€ydss’“ tiptjdss’“ gsf”r€ pgyd {ihsu r ieioisuh•su
j {eiteustjd L [0, T ; ρ] X
cV
T
0
2
k
γk
0 T
Sha su“i|ydsra fjuyeuij sie~ tiptjdss’“ gsf”r€ jit{ih•‚gd~ta
rsdoeuh•s’~ i|ydtji~ V¡ƒ
ZT
α
T2
Jp2 ( x)xdx =
T
2
2
p
2
Jp0 (α) + 1 − 2 Jp2 (α) ,
α
_ eutt~uerjud~i~ thgzud p = 0 m {iŸi~g
0
γk
kyk (ρ)k2 = kJ0 ( ρ)k2 =
T
ZT
γk
J02 ( ρ)ρdρ
T
α 6= 0.
[cXc]
i
T2 h 0
2
2
=
(J0 (γk )) + J0 (γk ) .
2
Rzr’juam zi J (x) = −J (x) rm zi J (γ ) = 0 m {ihgzr~
0
0
0
2
0
1
k
T2
γk
2
(J1 (γk ))2 .
kyk (ρ)k = kJ0 ( ρ)k =
T
2
γ 2
k
λk =
γk
T
2
˜jdƒ V] tX zX
m oyd ¤ Ÿi Ÿi fiesr gsf”rr J (ρ) m
™
kc]=tX1,X2, . . .
™
y (ρ) = J ( ρ), k = 1, 2, . . .
`] L [0, T ; ρ] ™ Š] ky (ρ)k = T (J (γ )) X
2
­ N®±® Sha i{deuieu
m i{ed}
dy m
1 d
(r
) 0<ρ<T
L (y) = −
r dr dr
ydhdssioi su ~si|dtjd gsf”r€m gyijhdjiea‘¢r“
iysieiys’~ feudj’~
gthijra~ j izfu“ 0 r T jryuƒ
ioeusrzdsu {er r → 0 + 0, Ry (T ) + Sy(T ) = 0 (R 6= 0, S 6= 0)
y(r)
su€r tiptjdss’d zrthu r tiptjdss’d gsf”rrm gfu‚u• {eiteustji j
fiiei~ tiptjdss’d gsf”rr ipeu‚g‘ {ihsg‘ ieioisuh•sg‘ trtd~g r
su€r fjuyeu’ sie~ tiptjdss’“ gsf”r€X
d–r~ ‚uyuzg £ge~u¤lrgjrhhaƒ
k
2
γk
0 T
k
2
2
1
k
0
2
2
r
2
0

 1 2 0 0
− 2 (r y ) = λy,
r
y(r)
0 < ρ < T,
ioeusrzdsu {er r → 0 + 0, Ry (T ) + Sy(T ) = 0.
Sha eutt~uerjud~ioi i{deuieu L (y) gsf”ra q(r) = 0 X iŸi~g
jtd tiptjdss’d zrthu sdier”udh•s’X
gt• λ = 0 X edipeu‚gd~ geujsdsrd
−
0
r
1
1 2 0 0
(r
y
)
=
0
⇔
(2ry 0 + r2 y 00 ) = 0 ⇔ 2y 0 + ry 00 = 0.
2
2
r
r
cc
_jdyd~ sijg‘ gsf”r‘ v(r) = y(r)r X Sha sdd ¤ v = y r+y m v
nioyu isitrdh•si Ÿi€ gsf”rr geujsdsrd {er~d jryƒ
0
0
00
= y 00 r+2y 0
X
v 00 = 0
˜p¢r~ ed–dsrd~ Ÿioi geujsdsra pgyd gsf”ra v(r) = C + C r X xsuzr
ip¢dd ed–dsrd rt“iysioi geujsdsra ¤ y(r) = C 1 + C X _it{ih•‚gd~ta
r
feudj’~r gthijra~rX †‚ {dejioi gthijra thdygdm zi
X iytujhaa
C
=
0
ji jieid feudjid gthijrdm gpd|yud~ta j i~m zi {er S 6=
y(r) gsf”ra
= C
X iŸi~g λ = 0 sd ajhadta tiptjdss’~ zrthi~
6=
0
y(r)
=
0
i{deuieuX
gt• λ = µ > 0 X d–r~ geujsdsrd
1
2
2
1
1
2
2
−
1 2 0 0
1
2
(r
y
)
=
µ
y
⇔
(2ry 0 + r2 y 00 ) + µ2 y = 0 ⇔ 2y 0 + ry 00 + rµ2 y = 0,
2
2
r
r
v(r) = y(r)r
v(r)
v 00 + µ2 v = 0.
†t{ih•‚gam fuf r eusddm gsf”r‘
trdh•si ƒ
X xu{r–d~ geujsdsrd isi}
ˆi geujsdsrd t {itiass’~r fiŸr”rdsu~rX ¬oi “ueufdertrzdtfid
geujsdsrd p = −µ r~dd fi~{hdfts’d fiesr p = µi m p = −µi X ˜p¢r~
ed–dsrd~ geujsdsra pgyd gsf”ra v(r) = C cos(µr) + C sin(µr) X nioyu
ip¢dd ed–dsrd rt“iysioi geujsdsra ¤ y(r) = C cos(µr) + C sin(µr) X
itfih•fg y(r) ioeusrzdsu {er r → 0 + 0 m i Cr = 0 Xxsuzrr C 6= 0 X
xuyuyr~ C = 1 r {iytujr~ gsf”rr y(r) = sin(µr) r y (r) = µ cos(µr) −
r
r
sin(µr) ji jieid feudjid gthijrdƒ
−
2
2
1
2
1
2
1
2
1
2
0
2
r2
µ cos(µT ) sin(µT )
R
−
=0
T
T2
⇔
RT cos(µT ) − sin(µT )(R − ST ) = 0.
ihgzdssid geujsdsrd {edipeu‚gdta f jrygƒ
ctg(µT ) =
R − ST
.
µRT
iteir~ oeurfr gsf”r€m tia¢r“ j {euji€ r hdji€ zuta“ geujsdsraX
„usijrta atsim zi Ÿi geujsdsrd r~dd pdtzrthdssid ~si|dtji
fiesd€m {i{uesi tr~~derzs’“ isitrdh•si suzuhu fiieyrsu [ert cXc]X
Tiesr su“iyam rt{ih•‚ga zrthdss’d ~diy’m su{er~dem ~diy futudh•}
s’“X Tieds• µ ”dhdtiipeu‚si su“iyr•m {edipeu‚ga geujsdsrd f jrygƒ
1
µRT
tg(µT ) =
.
R − ST
c`
ëìíî ïîï
˜pi‚suzr~ µ (k = 1, 2, . . .) {ihi|rdh•s’d fiesr Ÿioi geujsdsraX
nioyu zrthu λ = µ (k = 1, 2, . . .) ¤ tiptjdss’d zrthum u gsf”rr y (r) =
sin(µ r) m tiijdtjg‘¢rd r~m tiptjdss’d gsf”rr i{deuieuX
=
r
—gsf”rr
ipeu‚g‘ {ihsg‘ ieioisuh•sg‘ trtd~g j {eiteus}
y
(r)
tjd
X
L [0,vu€yd~
T ; r ] fjuyeu’ sie~ tiptjdss’“ gsf”r€ƒ
k
k
2
k
k
k
k
2
2
kyk (r)k2 =
ZT
yk2 (r)r2 dr =
0
ZT
sin2 (µk r)dr =
0
0
1
1
r−
sin(2µk r)
=
2
2µk
˜jdƒ V] tX zX λ
ZT
T
1
(1 − cos(2µk r)) dr =
2
1
1
=
T−
sin(2µk T ) .
2
2µk
m oyd µ ¤ Ÿi {ihi|rdh•s’d fiesr geujsdsra
=
™ c] tX X y (r) = sin(µ r), k = 1, 2, . . . ™
R − ST m
RT ctg(µT ) =
k = 1, 2, . . .
µ
r
`] L [0, T ; r ] ™ Š] ky (r)k = 1 T − 1 sin(2µ T ) X
2
2µ
­ N®²® Sha i{deuieu L (y) = − 1 d (sin θ dy ) m
sin θ dgyijhdjiea‘¢r“
θ
dθ
m i{edydhdssioi su ~si|dtjd gsf”r€m
iysi}
0eiys’~
<θ<π
feudj’~ gthijra~ j izfu“ 0 r π jryuƒ
ioeusrzdsu {er θ → 0 + 0 r {er θ → π − 0
y(θ)
su€r tiptjdss’d zrthu r tiptjdss’d gsf”rrm gfu‚u• {eiteustji j
fiiei~ tiptjdss’d gsf”rr ipeu‚g‘ {ihsg‘ ieioisuh•sg‘ trtd~g r
su€r fjuyeu’ sie~ tiptjdss’“ gsf”r€X
cŠ
k
µ2k
0
k
k
k
2
2
k
2
k
k
θ
d–r~ ‚uyuzg £ge~u¤lrgjrhhaƒ

1

(sin θy 0 )0 = λy,
−
sin θ
y(θ)
0 < θ < π,
ioeusrzdsu {er θ → 0 + 0 r {er θ → π − 0.
itfih•fg yha i{deuieu L (y) gsf”ra q(θ) = 0 m i tiptjdss’d
zrthu Ÿioi i{deuieu sdier”udh•s’dX
utt~ier~ geujsdsrd
1
−
(sin θy 0 )0 = λy
sin θ
⇔
θ
1 d
1
d
−
− sin2 θ(−
) y = 0.
sin θ dθ
sin θ dθ
_jdyd~ sijg‘ {ded~dssg‘ t = cos θ (−1 < t < 1) m ioyu pgyg j’{ih}
sa•ta eujdstju y(θ) = y(arccos t) = v(t) = v(cos θ) X er Ÿi~ v (t) =
−1
−1 X Rzr’jua Ÿg j‚ur~itja‚• ~d|yg {eir‚jiys’~rm
=y √
y
=
sin
θ
1
−
t
u uf|d eujdstji sin θ = 1 − t ‚u{r–d~ Ÿi geujsdsrd isitrdh•si
gsf”rr v(t) ƒ
0
t
0
θ
2
0
θ
2
2
d
d
−
(1 − t2 ) v = λv.
dt
dt
0
L(v) = − (1 − t2 )v 0
˜{deuie
¤ Ÿi i{deuie ld|usyeum i{edydhds}
s’€ su ~si|dtjd gsf”r€ yju|y’ yrdeds”regd~’“ {er −1 < t < 1 X
†t“iysua ‚uyuzu £ge~u¤lrgjrhha {edipeu‚ijuhut• f ‚uyuzd su tip}
tjdss’d ‚suzdsra yha i{deuieu ld|usyeuƒ
(
0
− (1 − t2 )v 0 = λv,
v(t)
<t<1
ioeusrzdsu {er −1
r {er t → 1 − 0.
t → −1 + 0
_ {itiprr V¡ teX cY {iyeipsi i{rtusi ed–dsrd {ihgzdssi€ ‚uyuzrX
„iptjdss’~r gsf”ra~r i{deuieu ld|usyeu ajha‘ta ~sioizhds’ ld}
|usyeuƒ
vk (t) = Pk (t) =
1 2
k (k)
(t
−
1)
.
2k k!
ˆi ~sioizhds’ td{dsr k ƒ P (t) = 1, P (t) = t m P (t) = 1 (3t − 1) . . . X ˜sr
tiijdtjg‘ tiptjdss’~ zrthu~ λ = k(k + 1) (k = 0,2 1, . . .) X
„iptjdss’~r gsf”ra~r rt“iysioi i{deuieu L (y) pgyg gsf”rr
m tiijdtjg‘¢r~r tiptjdss’~ zrthu~ λ = k(k + 1)
y (θ) = P (cos
θ
)
X
(k = 0,ihsg‘
1, . . .) ieioisuh•sg‘ trtd~g gsf”rr
ipeu‚g‘ j {eiteus}
y
(
θ
)
tjd
X
L [0, π; sin θ]
cU
0
1
2
2
k
θ
k
k
k
k
2
vu€yd~ fjuyeu’ sie~ Ÿr“ gsf”r€ƒ
kyk (θ)k2 =
Zπ
yk2 (θ) sin θdθ =
Zπ
Pk2 (cos θ) sin θdθ.
er j’zrthdsrr rsdoeuhu j’{ihsr~ ‚u~dsg {ded~dssi€ t = cos θ m dt =
0
0
= − sin θdθ
kyk (θ)k2 =
Z1
Pk2 (t)dt =
2
.
2k + 1
xsuzdsrd {ithdysdoi rsdoeuhu su€ydsi j [1] teX `VX
˜jdƒ V] tX zX λ = k(k + 1) m k = 0, 1, 2, . . . ™
c] tX X y (θ) = P (cos θ), k = 0, 1, 2, . . . ™
`] L [0, π; sin θ] ™ Š] ky (θ)k = 2 X
2k + 1
³´µ¶· ¸¹¸º»
_ g{eu|sdsra“ cXV}cXVV yha ‚uyussioi i{deuieu L(y) m i{edydhdssioi
su ~si|dtjd gsf”r€m gyijhdjiea‘¢r“ iysieiys’~ feudj’~ gthijra~m
su€r tiptjdss’d zrthu r tiptjdss’d gsf”rrm gfu‚u• {eiteustji j
fiiei~ tiptjdss’d gsf”rr ipeu‚g‘ {ihsg‘ ieioisuh•sg‘ trtd~g r
su€r fjuyeu’ sie~ tiptjdss’“ gsf”r€X
(
L (y) = −y , a < x < b m
2.1.
−1
k
k
k
2
k
x
2
00
y(a) = 0, y 0 (b) = 0
(
Lx (y) = −y 00 , a < x < b
2.2.
y 0 (a) = 0, y(b) = 0
(
Lx (y) = −y 00 , a < x < b
2.3.
(R 6= 0, S 6= 0),
Ry 0 (a) − Sy(a) = 0, y(b) = 0
(
Lx (y) = −y 00 , a < x < b
2.4.
(R 6= 0, S 6= 0),
y 0 (a) = 0, Ry 0 (b) + Sy(b) = 0
(
Lx (y) = −y 00 , a < x < b
2.5.
R1 y 0 (a) − S1 y(a) = 0, R2 y 0 (b) + S2 y(b) = 0
(R1 6= 
0, S1 6= 0 R2 6= 0, S2 6= 0),
B (y) = − 1 d (ρ dy ), 0 < ρ < T
0
ρ dρ dρ
2.6.
,
y(ρ)
0
ρ → 0 + 0, y (T ) = 0

B (y) = − 1 d (ρ dy ), 0 < ρ < T
0
ρ dρ dρ
2.7.
y(ρ)
ρ → 0 + 0, Ry 0 (T ) + Sy(T ) = 0
m
ioeusrzdsu {er
ioeusrzdsu {er
c¥
(R 6= 0,
 S 6= 0),
1 d dy p2

Bp (y) = −
(ρ + y), 0 < ρ < T
2.8.
,
ρ dρ dρ ρ2

ρ → 0 + 0, y(T ) = 0
y(ρ)

1 d dy p2

Bp (y) = −
(ρ + y), 0 < % < T
2.9.
ρ dρ dρ ρ2

ρ → 0 + 0, Ry 0 (T ) + Sy(T ) = 0
y(ρ)
(R 6= 0, S 6= 0),
1 d
dy

Lr (y) = − 2 (r2 ), 0 < r < T
2.10.
,
r dr dr
y(ρ)
ρ → 0 + 0, y(T ) = 0

1 d
dy

Lr (y) = − 2 (r2 ), 0 < r < T
2.11.
.
r dr dr
y(ρ)
0
ρ → 0 + 0, y (T ) = 0

1 d
dy

Lr (y) = − 2 (r2 ), 0 < r < T
2.12.
.
r dr dr
y(ρ)
0
ρ → 0 + 0, y (T ) + hy(T ) = 0
ioeusrzdsu {er
ioeusrzdsu {er
ioeusrzdsu {er
ioeusrzdsu {er
ioeusrzdsu {er
¼½¾¹½¿
cXVX V] tX zX λ = + πk m l = b − a m k = 0, 1, . . . ™
l
c] tX X y (x) = sin (π + 2πk)(x − a), k = 0, 1, . . . ™
`] L [a, b; 1] ™ Š] ky (x)k = 2ll X
2 cXcX V] tX zX λ = + πk m l = b − a m k = 0, 1, . . . ™
l
c] tX X y (x) = cos (π + 2πk)(x − a), k = 0, 1, . . . ™
`] L [a, b; 1] ™ Š] ky (x)k = 2ll X
cX`X V] tX zX λ = µ m 2oyd µ ¤ Ÿi {ihi|rdh•s’d fiesr geujsdsra
m
™
S m
ctg(µl) = −
l = b − a k = 1, 2, . . .
c] tX X y (x)Rµ= sin(µ (b − x)), k = 1, 2, . . . ™
`] L [a, b; 1] ™ Š] ky (x)k = l − sin(2µ l) X
cXŠX V] tX zX λ = µ m 2oyd µ 4¤µ Ÿi {ihi|rdh•s’d fiesr geujsdsra
m
™
S m
tg(µl) =
l = b − a k = 1, 2, . . .
µ
c] tX X yR(x)
™
= cos(µ (x − a)), k = 1, 2, . . .
2
π
2
k
k
2
2
k
2
π
2
k
k
2
2
k
2
k
k
k
2
k
k
2
k
2
k
k
k
k
k
k
k
cW
`] L [a, b; 1] ™ Š] ky (x)k = l + sin(2µ l) X
4µ sdier”udh•s’d fiesr geujsdsra
cXUX V] tX zX λ = µ m oyd2 µ ¤ Ÿi
m
™
µ(R S + R S ) m
tg(µl) = −
l = b − a k = 0, 1, 2, . . .
SS
c] tX X y (x)R =R Rµ µ− cos(
™
µ (x − a)) + S sin(µ (x − a)), k = 0, 1, 2, . . .
`] L [a, b; 1] ™ Š] ky (x)k = l (R µ + S ) + sin(2µ l) + S R sin (µ l) X
2
4µ
cX¥X V] tX zX λ = γ m oyd γ ¤ Ÿi sdier”udh•s’d fiesr geujsdsra
T™
m
J (γ) = 0 k = 0, 1,2, . . .
c] tX X y (ρ) = J γ ρ , k = 0, 1, 2, . . . ™
`] L [0, T ; ρ] ™ Š] ky (ρT)k = T J (γ ) X
cXWX V] tX zX λ = γ m2oyd γ ¤ Ÿi {ihi|rdh•s’d fiesr geujsdsra
T
™
J (γ) ST m
=
k = 1, 2, . . .
c]J (tXγ)X yR(γρ) = J γ ρ , k = 1, 2, . . . ™
`] L [0, T ; ρ] ™ Š] ky (ρT)k = T J (γ ) + J (γ ) X
cXZX V] tX zX λ = γ m2oyd γ ¤ Ÿi {ihi|rdh•s’d fiesr geujsdsra
m
™T
J (γ) = 0 k = 1, 2,. . . c] tX X y (ρ) = J γ ρ , k = 1, 2, . . . ™
T
`] L [0, T ; ρ] ™ Š] ky (ρ)k = T p J (γ ) − J (γ ) X
2 γ
cXYX V] tX zX λ = γ m oyd γ ¤ Ÿi {ihi|rdh•s’d fiesr geujsdsra
T
J (γ)
™
ST m
k = 1, 2, . . .
=−
c]J (tXγ)X y (Rρ)γ= J γ ρ , k = 1, 2, . . . ™
`] L [0, T ; ρ] ™ Š] ky (ρT)k = T J (γ ) 1 + S T X
2
Rγ
cXV\X V] tX zX λ = µ m µ = πk m k = 1, 2, . . . ™
c] tX X y (r) = sin(µ r) , k = 1,T 2, . . . ™
2
2 1
2
1 2
1 k
2
k
2
k
1 2
k
k
k
2
k
k
1 2
1
k
2 2
1 k
2
k
k
k
k
2
1
1
k
2
k
k
1
k
k
0
2
2
0
2
k
k
k
2
k
k
1
0
k
k
0
2
2
k
k
2
1
2
0
k
k
2
k
k
p
k
k
p
2
2
2
k
k
p
k
p+1
k
2
k
k
0
p
p
k
k
2
p
2
k
2
k
k
2
p
2
k
2 2
k
k
k
k
2
r
cZ
k
1
2
k
`] L [0, T ; r ] ™ Š] ky (r)k = T X
2
cXVVX V] tX zX λ = µ m oyf
¤ Ÿi {ihi|rdh•s’d fiesr geujsdsra
µ
™
1 m
ctg(µT ) =
k = 1, 2, . . .
µT
c] tX X y (r) = sin(µ r) , k = 1, 2, . . . ™
`] L [0, T ; r ] ™ Š] ky r(r)k = T sin (µ T ) X
2
cXVcX V] tX zX λ = µ m oyf
¤ Ÿi {ihi|rdh•s’d fiesr geujsdsra
µ
™
1 − Thm
ctg(µT ) =
k = 1, 2, . . .
µT
c] tX X y (r) = sin(µ r) , k = 1, 2, . . . ™
`] L [0, T ; r ] ™ Š] ky r(r)k = T − sin(2µ T ) = T T µ + T h − h X
2
2
2
k
2
k
k
k
k
k
2
2
2
2
k
2
k
k
k
k
k
k
2
2
k
k
2
2
4µk
2
2
2
k
2
2
T µk +
2
(1 − T h)2
ñ ' ?A&15 C5>C 2C@( E@A E1 E= 5 EF2>
gt• eutt~uerjudta suoedj dhu Ω t fgtizsi}ohuyfi€ oeusr”d€ S X
˜pi‚suzr~ u(M, t) [M ∈ Ω r t > 0] gsf”r‘m i{rt’ju‘¢g‘ r‚~dsdsrd
d~{deuge’ Ÿioi dhuX _ ip¢d~ ed“~desi~ thgzud geujsdsrd d{hi{ei}
jiysitr t gzdi~ rtizsrfij d{hu ‚u{rt’judta j jryd
òóô (k grad u) + q,
oyd c ¤gydh•sua d{hid~fit•m ¤ {hisit•m k ¤fiŸr”rds d{hi{eijiy}
sitrm q ¤ gsf”ram i{rt’ju‘¢ua ipõd~sg‘ {hisit• rtizsrfij d{huX
_’jiy ed“~desioi geujsdsra {iyeipsi r‚hi|ds j Š¡X ¬thr dhi Ω iysi}
eiysid r r‚iei{sid r j ~iydhr ~i|si tzru•m zi fiŸr”rds’ c m ρ
r k ¤ Ÿi {itiass’d jdhrzrs’m i geujsdsrd d{hi{eijiysitr ip’zsi
{edipeu‚g‘ f jrygƒ
[`XV]
∂u
= a ∆u + f,
∂t
oyd zrthi a = k su‚’judta fiŸr”rdsi~ d~{deugei{eijiysitrm
gsf”ra f = qρcX ˜{deuie ∆ ¤ Ÿi i{deuie lu{hutuX _ry i{deuieu
ρc
‚ujrtr i j’peussi€
trtd~’ fiieyrsuƒ
∂ u ∂ u ∂ u [j ydfueiji€]m
∆u =
+
+
ρc
∂u
=
∂t
ρ
2
2
2
2
2
∂x2
∂y 2
∂z 2
cY
[j ”rhrsyerzdtfi€]m
1 ∂
∂u
1 ∂ 2u ∂ 2u
∆u =
ρ
+
+
ρ ∂ ρ ∂ ρ ρ2 ∂ ϕ2 ∂z 2
∂ 2u
∂
1 ∂
∂u
1
∂u
1
2
∆u = 2
ρ
+ 2
sin θ
+ 2 2
ρ ∂ρ
∂ρ
ρ sin θ ∂ θ
∂θ
ρ sin θ ∂ ϕ2
[j tderzdtfi€]X
T Ÿi~g geujsdsr‘ yipujhadta suzuh•sid gthijrd {er t = 0
oyd ϕ(M ) ¤ gsf”ram i{rt’ju‘¢ua suzuh•sg‘ d~{deugeg dhuX
vu oeusr”d iphutr S ‚uyu‘ta feudj’d gthijra j tiijdtjrr t gthi}
jra~r d{hiip~dsu dhu t ifeg|u‘¢d€ tedyi€X ˆi ~i|d p’• iysi r‚
ed“ gthijr€ jryuƒ
u(M, 0) = ϕ(M ),
u
S
= ν(M 0 , t),
−k
∂u
∂~n
= Q(M 0 , t),
S
−k
∂u
∂~n
S
= α(u − T0 ) S .
xydt• M ¤ izfu oeusr”’ iphutr S m ∂u ¤ {eir‚jiysua {i su{eujhdsr‘
∂~n
jsd–sd€ sie~uhr f oeusr”d S X
dejid gthijrd ‚uyudtam dthr su oeusr”d {iyyde|rjudta ‚uyussid
eut{edydhdsrd d~{deuge’ ν(M , t) X ˆi feudjid gthijrd su‚’judta gthi}
jrd~ Srer“hd rhr gthijrd~ V}oi eiyuX
_ieid su‚’judta gthijrd~ vd€~usu rhr gthijrd~ c}oi eiyuX xuyu}
dta j thgzudm fioyu su oeusr”d {iyyde|rjudta d{hiji€ {iif Q(M , t)
[k ¤ fiŸr”rds dhi{eijiysitr]X
ned•d gthijrd ‚uyudtam dthr su oeusr”d d{hiip~ds {eirt“iyr {i
‚ufisg v•‘isuX _ ie~ghd T ¤ Ÿi d~{deugeu ifeg|u‘¢d€ tedy’m
¤ fiŸr”rds dhi{eijiysitrm ¤ fiŸr”rds d{hiip~dsuX ˆi
kgthijrd su‚’judta ed•r~ feudj’~αgthijrd~X
vu eu‚s’“ zuta“ oeusr”’ S ~iog p’• ‚uyus’ oeusrzs’d gthijra
eu‚sioi eiyuX ˜ysi~desid geujsdsrd d{hi{eijiysitr {ihgzdsi j gzdp}
si~ {itiprr V¡X nu~ |d i{rtusu {itusijfu suzuh•sioi r feudj’“ gthijr€X
Reujsdsrd jryu [`XV] {iajhadta sd ih•fi {er ed–dsrr ‚uyuzm tja}
‚uss’“ t eut{iteusdsrd~ d{huX vu{er~dem Ÿi geujsdsrd i{rt’jud {ei}
”dtt’ yrg‚rr zutr” j sdfiiei€ tedydX bdsadta ih•fi t~’th fiŸ}
r”rdsijX Reujsdsrd yrg‚rr eutt~iedsi j Š¡X
˜ysr~ r‚ surpihdd zuti {er~dsad~’“ ~diyij ed–dsra suzuh•si}
feudj’“ ‚uyuz yha geujsdsra d{hi{eijiysitr ajhadta ~diy —ge•dX
khoier~ {er~dsdsra eayij —ge•d yha ed–dsra feudj’“ ‚uyuz i{rtus j
V¡ su {er~ded ed–dsra iysi~desi€ ‚uyuzrX
d–dsrd feudji€ ‚uyuzr su“iya j jryd eayu —ge•dm j’{ihsaa eu‚}
hi|dsrd rtfi~i€ gsf”rr {i tiptjdss’~ gsf”ra~ hrsd€sioi tr~~d}
erzsioi yrdeds”ruh•sioi i{deuieu £ge~u¤lrgjrhha L m tja‚ussioi
`\
0
0
0
0
t ed–ud~i€ ‚uyuzd€X ˜{deuie L eutt~uerjudta su ~si|dtjd gsf”r€m
gyijhdjiea‘¢r“ iysieiys’~ feudj’~ gthijra~X ¬thr feudj’d gthijra
rt“iysi€ feudji€ ‚uyuzr sdiysieiys’dm i tsuzuhu ‚uyuzg tjiya f ‚uyu}
zd t iysieiys’~r feudj’~r gthijra~rX Sha Ÿioi rtfi~g‘ gsf”r‘ u
{edytujha‘ j jryd tg~~’ yjg“ gsf”r€ u = v + w X —gsf”r‘ w [zuti
hrsd€sg‘m si sd jtdoyu] j’preu‘ ufm zip’ isu gyijhdjieahu ufr~ |d
feudj’~ gthijra~m zi r gsf”ra u X nioyu gsf”ra v pgyd gyijhdjiea•
iysieiys’~ oeusrzs’~ gthijra~X xud~ jt‘ feudjg‘ ‚uyuzg ‚u{rt’ju‘
isitrdh•si gsf”rr v X ihgzdssg‘ j ed‚gh•ud ‚uyuzg ed–u‘ ~di}
yi~ —ge•dX Sha Ÿioi
vu“iya tiptjdss’d zrthu λ r tiptjdss’d gsf”rr y hrsd€sioi tr~}
1)
~derzsioi yrdeds”ruh•sioi i{deuieu jieioi {ieayfu L m eutt~uer}
jud~ioi su ~si|dtjd gsf”r€m gyijhdjiea‘¢r“ iysieiys’~ feudj’~
gthijra~m tiijdtjg‘¢r~ feudj’~ gthijra~ ‚uyuzrX nX dX ed–u‘ tii}
jdtjg‘¢g‘ ‚uyuzg £ge~u¤lrgjrhhaX örtd~u tiptjdss’“ gsf”r€
ieioisuh•su r {ihsu j {eiteustjd L [a, b; ρ)] m oyd ρ ¤ Ÿi jdti}
{y
}
jua gsf”ra i{deuieu L X
—gsf”r‘ v su“iya j jryd eayu —ge•dm eu‚huoua dd {i su€ydss’~ tip}
2)
tjdss’~ gsf”ra~ y yrdeds”ruh•sioi i{deuieuX Sha i{edydhdsra
‚suzdsr€ fiŸr”rdsij eayu —ge•d Ÿi eay thdygd {iytujr• j geuj}
sdsrd d{hi{eijiysitr r j suzuh•sid gthijrdX ¬thr j geujsdsrr rhr j
suzuh•si~ gthijrr dt• ihrzs’d i sgha gsf”rrm i Ÿr gsf”rr uf|d
thdygd eu‚hi|r• j eay’ —ge•dX ithd {iytusijfr yha fiŸr”rdsij
rtfi~ioi eayu {ihgzu‘ta yrdeds”ruh•s’d geujsdsraX †“ ed–u‘ r
{ihgzu‘ j’eu|dsra yha fiŸr”rdsijX
†tfi~id ed–dsrd {edytujha‘ j jryd u = v + w m oyd gsf”ra v {edy}
3)
tujhdsu j jryd eayu —ge•dX
bdiy eu‚hi|dsra {i tiptjdss’~ gsf”ra~ dtsi tja‚us t ~diyi~
eu‚ydhdsra {ded~dss’“m fiie’€ {edysu‚suzds yha i{edydhdsra zuts’“
ed–dsr€ yrdeds”ruh•s’“ geujsdsr€ j zuts’“ {eir‚jiys’“X
er ed–dsrr sdfiie’“ feudj’“ ‚uyuz ”dhdtiipeu‚si rt{ih•‚iju•
ferjihrsd€sg‘ trtd~g fiieyrsum su{er~de ”rhrsyerzdtfg‘ rhr tder}
zdtfg‘X nioyu {er {er~dsdsrr ~diyu —ge•d {iajata t{d”ruh•s’d gsf}
”rrX iyeipsi tji€tju Ÿr“ gsf”r€ i{rtus’ j U¡m ¥¡m Š¡m V¡X
u‚pded~ sdtfih•fi {er~deijX
­ ¯®P® „de|ds• yhrsi€ l t d{hir‚ihreijussi€ {ijde“sit•‘
suoed yi d~{deuge’ T X vuzrsua t ~i~dsu jed~dsr t = 0 iyrs r‚ fis”ij
tde|sa i“hu|yu‘m {iyyde|rju‘ d~{deugeg su Ÿi~ fis”d eujsg‘
\ [su{er~dem fisd” tde|sa i{gtfu‘ j hdy]X vu yegoi~ fis”d tde|sa
d~{deugeg {iyyde|rju‘ eujsi€ T X vu€r d~{deugeg j tedysd€ izfd
tde|sa j ~i~ds jed~dsr t X
k
k
+∞
k k=1
2
k
0
0
0
`V
gt• gsf”ra u(x, t) i{rt’jud eut{edydhdsrd d~{deuge’ j tde|}
sdX ˜su gyijhdjiead geujsdsr‘
∂ u {er
∂u
=a
0 < x < l, t > 0,
∂t
∂x
suzuh•si~g gthijr‘ƒ u(x, 0) = T m
feudj’~ gthijra~ƒ u(0, t) = 0, u(l, t) = T .
bdiy —ge•d sd{itedytjdssi {er~dsadta f ‚uyuzu~m j fiie’“ rt}
fi~ua gsf”ra gyijhdjiead iysieiys’~ feudj’~ gthijra~X _ eutt~u}
erjud~i€ ‚uyuzd gthijrd su fis”d tde|sa x = l sdiysieiysidX iŸi~g
tsuzuhu tjdyd~ Ÿg ‚uyuzg f ‚uyuzd t iysieiys’~r feudj’~r gthijra}
~rX Sha Ÿioi gsf”r‘ u(x, t) {edytujr~ j jryd tg~~’ yjg“ gsf”r€
X xuyuyr~ w(x, t) = αx + β r {iypded~ α r β ufm
u(x,
t)
=
v(x,
t)
+
w(x,
t)
zip’ yha gsf”rr w(x, t) j’{ihsahrt• feudj’d gthijra
2
2
2
0
0
w(0, t) = β = 0,
w(l, t) = αl + β = T0 .
xsuzrm α = T m β = 0 r w(x, t) = T x X „hdyijudh•si
l
l
T X
u(x, t) = v(x, t) + x
˜sitrdh•si lgsf”rr v(x, t) {ihgzrta ‚uyuzu t iysieiys’~r oeu}
srzs’~r gthijra~rƒ

∂v
∂ v {er


=a
0 < x < l, t > 0,

0
0
0
2
2
∂x2
T0 x
,
v(x,
0)
=
T
−
0


l


v(0, t) = 0, v(l, t) = 0.
 ∂t
ˆg ‚uyuzg pgyd~ ed–u• ~diyi~ —ge•dX
V] —gsf”r‘ v(x, t) pgyd~ rtfu• j jryd eayu —ge•dm eu‚huoua dd {i
trtd~d tiptjdss’“ gsf”r€ hrsd€sioi yrdeds”ruh•sioi i{deuieu
∂ v X Sha su“i|ydsra tiptjdss’“ zrtdh r tiptjdss’“ gsf}
L (v) = −
∂x sdip“iyr~i ed–r• tiijdtjg‘¢g‘ ‚uyuzg £ge~u¤lr}
”r€ i{deuieu
gjrhha
2
x
2
(
−y 00 (x) = λy(x), 0 < x < l,
y(0) = 0, y(l) = 0.
ˆu ‚uyuzu p’hu eutt~iedsu j eu‚ydhd c [{er~de cXV] r yha i{deui}
eu L = −y m eutt~uerjud~ioi su ~si|dtjd gsf”r€m gyijhdjiea‘}
¢r“ feudj’~ gthijra~ Srer“hdm p’hr su€yds’ tiptjdss’d zrthu λ =
r tiijdtjg‘¢rd r~ tiptjdss’d gsf”rr y (x) =
πk m
=
k = 1, 2, . . .
00
x
k
2
l
k
`c
πkx
,
l
m ipeu‚g‘¢rd {ihsg‘ ieioisuh•sg‘ trtd~g j
{eiteustjd L [0, l; 1] X
c] —gsf”r‘ v(x, t) {edytujr~ j jryd eayu —ge•dm j’{ihsrj eu‚hi}
|dsrd {i su€ydssi€ trtd~d tiptjdss’“ gsf”r€ yrdeds”ruh•sioi
i{deuieuƒ
= sin
k = 1, 2, . . . .
2
v(x, t) =
+∞
X
ck (t)yk (x).
Teudj’d gthijram {er Ÿi~m pgyg j’{ihsa•taX Sha i{edydhdsra ‚suzdsr€
fiŸr”rdsij c (t) eay —ge•d {iytujr~ j geujsdsrd d{hi{eijiysitr r
j suzuh•sid gthijrdX —gsf”ra r‚ suzuh•sioi gthijra ϕ÷ (x) = T x ihrzsu
l
i sghaX ¬d uf|d eu‚hi|r~ j eay —ge•dƒ
k=1
k
0
÷
ϕ (x) =
+∞
X
ϕk yk (x).
k=1
TiŸr”rds’ —ge•d su€yd~ {i r‚jdtsi~g {eujrhg ϕ = (ϕ÷ , y ) X Tjuy}
ky k
eu’ sie~ tiptjdss’“ gsf”r€ ky k = l m k = 1, 2, . . . p’hr {ihgzds’ j
2
{er~ded cXVX _’zrthr~ (ϕ÷ , y ) ƒ
k
k
k
2
2
k
k
÷
(ϕ , yk ) =
Zl
0
πkx
T0 x
T0 x
) sin
dx = (T0 −
)
(T0 −
l
l
l
T0
−
πk
Zl
0
−l
πkx
cos
πk
l
πkx
T0 l
T0 l
πkx
cos
dx =
−
sin
l
πk
(πk)2
l
l
=
0
l
0
−
T0 l
.
πk
nioyu ϕ = 2T m k = 1, 2, ... .
πk d{de• eay’ —ge•d gsf”r€
iytujr~
r ϕ÷ (x) j geujsdsrd
v(x,
t)
d{hi{eijiysitr r j suzuh•sid gthijrdƒ
0
k
+∞
X
c 0k (t)yk (x)
2
=a
ck (t)y 00k (x),
k=0
k=1
+∞
X
+∞
X
ck (0)yk (x) =
k=1
+∞
X
k=0
``
ϕk yk (x).
itfih•fg j’{ihsa‘ta eujdstju y (x) = −λ y (x) r eu‚hi|dsrd
gsf”rr j eay —ge•d dyrstjdssim i yha fiŸr”rdsij c (t) {ihgzr~
‚uyuzr Ti–rƒ
00
k
k k
k
(
c 0k (t) = −a2 λk ck (t),
ck (0) = ϕk .
˜p¢dd ed–dsrd yrdeds”ruh•sioi geujsdsraƒ c (t) = A e
{ih•‚ga eujdstji c (0) = ϕ m {ihgzr~ A = ϕ m u ‚suzrm
k
k
k
k
k
−a2 λk t
X †t}
k
2T0 −a2 π22 k2 t
e l .
ck (t) =
πk
—gsf”ra v(x, t) eu‚huoudta j eay —ge•d
2 2 2t
v(x, t) =
+∞ −a π k
2T0 X e l2
π
sin
k
k=1
πkx
.
l
`] †tfi~ua gsf”ra u(x, t) = v(x, t) + T x m i{rt’ju‘¢ua eut{edyd}
hdsrd d~{deuge’ j tde|sdm {edytujr~u j ljryd
0
+∞
−a2 π2 k2 t
T0
2T0 X e l2
u(x, t) = x +
l
π
k
sin
k=1
πkx
.
l
iytujr~ j {ihgzdssid j’eu|dsrd x = l r t = t X ihgzrta zrthi}
2
ji€ eaym yha i{edydhdsra d~{deuge’ tde|sa
j tedysd€ izfd j ~i~ds
jed~dsr t X
˜jdƒ
0
0
+∞
l
T0 2T0 X e
u( , t0 ) =
+
2
2
π
−a2 π2 k2 t0
l2
k
sin
πkl
.
2
­ ¯®N® ˜ysu r‚ {ijde“sitd€ isfi€ {hutrs’ ih¢rsi€ l m su}
oedi€ yi d~{deuge’ T m d{hir‚ihreijusum u yegoua ti{erfutudta t
ji‚yg“i~ d~{deuge’ T (T < T ) m X dX i“hu|yudta {i ‚ufisg v•‘isuX
vu€r d~{deugeg {hutrs’ su Ÿi€ {ijde“sitr j ~i~ds jed~dsr t X
_ ~iydhr pgyd~ tzru•m zi d~{deugeu {hutrs’ t dzdsrd~ jed}
~dsr ~dsadta ih•fi {i ih¢rsdX ˜pi‚suzr~ u(x, t) gsf”r‘m fiieua
i{rt’jud r‚~dsdsrd d~{deuge’X ˆu gsf”ra gyijhdjiead geujsd}
sr‘ d{hi{eijiysitr
∂u
∂ u {er
0 < x < l, t > 0,
=a
k=1
1
0
0
1
0
2
∂t
2
∂x2
`Š
suzuh•si~g gthijr‘ƒ u(x, 0) = T m
feudj’~ gthijra~ƒ ∂u(0, t) = 0, − ∂u(l, t) = h(u(l, t) − T ).
˜ysi r‚ feudj’“ ∂xgthijr€ ‚uyuzr∂xsdiysieiysidX iŸi~g sd{itedy}
tjdssi {er~dsa• ~diy —ge•d f {ihgzdssi€ feudji€ ‚uyuzd sdh•‚aX edy}
tujr~ gsf”r‘ u(x, t) j jryd tg~~’ yjg“ gsf”r€ u(x, t) = v(x, t)+
X gt• w(x, t) = αx + β ¤ hrsd€sua {i {ded~dssi€ x gsf”raX
+w(x,
t)
iypded~ α r β ufm zip’ gsf”ra w(x, t) gyijhdjieahu d~ |d feu}
dj’~ gthijra~m zi r gsf”ra u(x, t) m X dX {iytujr~ w(x, t) j feudj’d
gthijraƒ
1
0
(
α = 0,
−α = h(αl + β) − T0 .
ihgzr~ α = 0 m β = T X xsuzr w(x, t) = T x.
h
h
—gsf”ra v(x, t) pgyd
gyijhdjiea• iysieiys’~
feudj’~ gthijra~X
xu{r–d~ feudjg‘ ‚uyuzg isitrdh•si Ÿi€ gsf”rr

∂ v {er
∂v


=
a
0 < x < l, t > 0,

0
0
2
2

∂x2
 ∂t
T0
v(x, 0) = T1 − x,

h



 ∂v(0, t) = 0, ∂v(l, t) + hv(l, t) = 0.
∂x
∂x
ˆg ‚uyuzg pgyd~ ed–u• ~diyi~ —ge•dX
d–r~ ‚uyuzg £ge~u¤lrgjrhha yha i{deuieu L (v) = − ∂ v ƒ
1)
2
x
∂x2
(
−y 00 (x) = λy(x), 0 < x < l,
y 0 (0) = 0, y 0 (l) + hy(l) = 0.
ˆi ‚uyuzu r‚ g{eu|sdsra cXŠX „iptjdss’d zrthu i{deuieu λ = µ m oyd
¤ Ÿi {ihi|rdh•s’d fiesr geujsdsra tg(µl) = h m k = 1, 2, . . . X †~
µ
tiijdtjg‘ tiptjdss’d gsf”rr y (x) = cos(µ x)µm k = 1, 2, . . . m ipeu}
‚g‘¢rd {ihsg‘ ieioisuh•sg‘ trtd~g j {eiteustjd L [0, l; 1] X Tjuy}
eu’ sie~ tiptjdss’“ gsf”r€ i{edydha‘ta {i {eujrhg ky (x)k =
l sin(2µ l) X Rzr’juam zi
2 tg(α) m fjuyeu’ sie~ ~i|}
= +
sin(2α) =
2
4µ
1 + tg (α)
si {edytujr• j jryd ky (x)k = l + h X
2 2(
) —ge•dm eu‚huoua dd {i
c] —gsf”r‘ v(x, t) pgyd~ rtfu•
j µjryd+ heayu
2
k
k
k
k
k
2
k
k
2
k
k
2
2
k
`U
2
2
trtd~d tiptjdss’“ gsf”rr€ yrdeds”ruh•sioi i{deuieuƒ
v(x, t) =
+∞
X
ck (t)yk (x).
Teudj’d gthijra j’{ihsa‘taX
—gsf”r‘ ϕ÷ (x) = T − T x r‚ suzuh•sioi gthijra eu‚hi|r~ j eay
h
—ge•d {i i€ |d trtd~d gsf”r€
k=1
0
1
+∞
X
÷
ϕ (x) =
{er Ÿi~ ϕ
k
=
ϕk yk (x),
k=1
÷
(ϕ , yk )
kyk k2
X _’zrthr~ tfuhaes’d {eir‚jdydsra
÷
(ϕ , yk ) =
Zl
0
(T1 −
T0
x) cos(µk x)dx =
h
T0
T0 l sin(µk l)
− 2 (cos(µk l) − 1)
= T1 −
h
µk
hµk
T0
2(µ2k + h2 )
T0 l sin(µk l)
− 2 (cos(µk l) − 1)
ϕk = 2
T1 −
lµk + lh2 + h
h
µk
hµk
k = 1, 2, ... .
v(x, t) ϕ (x)
ioyu
iytujr~ eay’ —ge•d gsf”r€
jiysitr r j suzuh•sid gthijrdƒ
+∞
X
c 0k (t)yk (x)
m
r ÷ j geujsdsrd d{hi{ei}
2
=a
+∞
X
ck (t)y 00k (x),
k=1
k=1
+∞
X
ck (0)yn (x) =
+∞
X
ϕk yk (x).
_it{ih•‚gd~ta eujdstju~r y (x) = −λ y (x) X _ trhg dyrstjdssi}
tr eu‚hi|dsra gsf”rr j eay —ge•dm yha fiŸr”rdsij c (t) {ihgzr~
‚uyuzr Ti–rƒ
k=1
k=1
00
k
k k
k
(
c 0k (t) = −a2 λk ck (t),
ck (0) = ϕk .
d–dsra Ÿr“ ‚uyuz {edytujha‘ta j jrydƒ c (t) = ϕ e
k
2(µ2 + h2 )
ck (t) = 2 k 2
lµk + lh + h
T0 l
T1 −
h
k
−a2 λk t
X nioyu
sin(µk l)
T0
2 2
− 2 (cos(µk l) − 1) e−a µk t .
µk
hµk
`¥
xu{r–d~ eay —ge•d gsf”rr v(x, t)
+∞
X
T0 l sin(µk l)
2(µ2k + h2 )
v(x, t) =
T1 −
−
lµ2k + lh2 + h
h
µk
k=1
T0
2 2
− 2 (cos(µk l) − 1) e−a µk t cos(µk x).
hµk
`] —gsf”ra u(x, t) = v(x, t) + w(x, t) ƒ
+∞
T0 l sin(µk l)
T0 x X 2(µ2k + h2 )
−
+
T1 −
u(x, t) =
h
lµ2k + lh2 + h
h
µk
k=1
T0
2 2
− 2 (cos(µk l) − 1) e−a µk t cos(µk x).
hµk
x =l
t = t0
x = l t = t0
Sha i{edydhdsra d~{deuge’ {hutrs’ su {ijde“sitr
jed~dsr
{iytujr~ j su€ydssid ed–dsrd
r
zrthiji€ eayX
˜jdƒ
j ~i~ds
X ihgzrta
+∞
T0 l X 2(µ2k + h2 )
T0 l sin(µk l)
−
+
u(l, t0 ) =
T1 −
h
lµ2k + lh2 + h
h
µk
k=1
T0
2 2
− 2 (cos(µk l) − 1) e−a µk t0 cos(µk l).
hµk
­ ¯®¯® nd~{deugeu su pifiji€ {ijde“sitr yhrssioi ”rhrs}
yerzdtfioi tde|sa euyrgtu R m suoedioi yi d~{deuge’ T m suzrsua t
~i~dsu jed~dsr t = 0 {iyyde|rjudta eujsi€ T [T > T ) X vu€r d~{d}
eugeg j ”dsed tde|sa j ~i~ds jed~dsr t X
Sha i{rtusra suoedju tde|sa gyipsi jjdtr ”rhrsyerzdtfg‘ trtd}
~g fiieyrsuX i~dtr~ suzuhi fiieyrsu j ”dsed {eir‚jih•sioi {i{d}
edzsioi tdzdsra tde|saX ˜t• OZ su{eujr~ jyih• itr tde|sam j {hit}
fitr tdzdsra jjdyd~ {ihaesg‘ trtd~g fiieyrsuX _ ~iydhr ~i|si tzr}
u• zi gsf”ra u m i{rt’ju‘¢ua r‚~dsdsrd d~{deuge’ j tde|sd sd
‚ujrtr i {ded~dss’“ z r ϕ m u ajhadta gsf”rd€ ih•fi {ihaesi€ fi}
ieyrsu’ ρ r jed~dsr t u = u(ρ, t) X nioyu gsf”ra u(ρ, t) gyijhdjiead
geujsdsr‘ d{hi{eijiysitr
{er 0 < ρ < R, t > 0,
∂u
1 ∂
∂u
=a
ρ
∂t
ρ ∂ρ
∂ρ
suzuh•si~g gthijr‘ƒ u(ρ, 0) = T m
feudj’~ gthijra~ƒ u(ρ, t) ioeusrzdsu {er ρ → 0 + 0 m u(R, t) = T .
`W
0
1
1
0
0
2
0
1
Rthijrd ioeusrzdssitr u(ρ, t) {er r → 0 + 0 ajhadta iysieiys’~X
Teudjid gthijrd {er ρ = R ¤ sdiysieiysidX iŸi~g tsuzuhu ‚u~dsi€
tjdyd~ eutt~uerjud~g‘ ‚uyuzg f ‚uyuzd t iysi}
u(
ρ
,
t)
=
v(
ρ
,
t)
+
w(
ρ
,
t)
eiys’~r feudj’~r gthijra~rX gt• w(ρ, t) = T X _ Ÿi~ thgzud w(ρ, t)
gyijhdjiead d~ |d feudj’~ gthijra~m zi r gsf”ra u(ρ, t) ƒ w(ρ, t)
ioeusrzdsu {er ρ → 0 + 0 m w(R, t) = T .
nioyu gsf”ra v(ρ, t) ajhadta ed–dsrd~ ‚uyuzr t iysieiys’~r feu}
dj’~r gthijra~rƒ

{er 0 < ρ < R, t > 0,
∂v
1
∂
∂v



=a
ρ
1
1
2
 ∂t
ρ ∂ρ
∂ρ
v(ρ, 0) = T0 − T1 ,



v(ρ, t)
ioeusrzdsu {er ρ → 0 + 0, v(R, t) = 0.
ˆg ‚uyuzg pgyd~ ed–u• ~diyi~ —ge•dX
V] d–r~ ‚uyuzg £ge~u¤lrgjrhha yha tr~~derzsioi hrsd€sioi
yrdeds”ruh•sioi i{deuieu ^dttdha B m eutt~uerjud~ioi su ~si|d}
tjd gsf”r€m gyijhdjiea‘¢r“ iysieiys’~ feudj’~ gthijra~ ed–ud~i€
‚uyuzrƒ
0

− 1 (ρy 0 (ρ))0 = λy(ρ), 0 < ρ < R,
ρ
y(ρ)
ρ → 0 + 0, y(R) = 0.
ioeusrzdsu {er
ˆu ‚uyuzu p’hu ed–dsu u j eu‚ydhd c [{er~de cXU]X „iptjdss’d zrthu
i{deuieuƒ λ = γ , k = 1, 2, . . . m oyd γ ¤ Ÿi {ihi|rdh•s’d ed}
–dsra geujsdsra JR(γ) = 0 X „iptjdss’d gsf”rr i{deuieuƒ y (ρ) =
X „rtd~u tiptjdss’“ gsf”r€ {ihsu r ieioi}
=
J
(
ρ
)
k
=
1,
2,
.
.
.
suh•su j {eiteustjd L [0, R; ρ] X Tjuyeu’ sie~ tiptjdss’“ gsf”r€
i{edydha‘ta {i ie~ghu~ ky (ρ)k = R (J (γ )) X
c] u‚hi|r~ gsf”r‘ v(ρ, t) j eay2 —ge•d {i trtd~d tiptjdss’“
gsf”r€ i{deuieu ^dttdhaƒ
k
2
k
γk
0 R
k
0
k
2
k
v(ρ, t) =
2
2
+∞
X
1
k
2
ck (t)yk (ρ).
Teudj’d gthijra {er Ÿi~ j’{ihsa‘taX
—gsf”ra r‚ suzuh•sioi gthijra ϕ÷ (ρ) = T
j eay —ge•dƒ
k=1
÷
ϕ (ρ) =
+∞
X
k=1
`Z
0
ϕk yk (ρ).
− T1
X ¬d uf|d eu‚hi|r~
TiŸr”rds’ —ge•d su€yd~ {i {eujrhg ϕ = (ϕ÷ , y ) X Tjuyeu’ sie~
tiptjdss’“ gsf”r€ g|d r‚jdts’X ˜{edydhr~ (kyϕ÷k, y ) rt{ih•‚ga ie}
~ghg (cXc) m tzruam zi α = γ m β = 0 r gzr’juam zi J (γ) = −J (γ) ƒ
k
k
k
0
0
k
ZR
÷
(ϕ , yk ) =
0
k
2
1
γk
R2
(T0 − T1 )J0 ( ρ)ρdρ = (T0 − T1 ) J1 (γk ).
R
γk
_ riod {ihgzr~ ϕ = 2(T − T ) m k = 1, 2, ... .
γ J (γ—ge•d
)
nd{de• {iytujr~ eay’
gsf”r€ v(ρ, t) r ϕ÷ (ρ) j geujsdsrd
d{hi{eijiysitr r j suzuh•sid gthijrdƒ
0
1
k
k 1
+∞
X
k
c0k (t)yk (ρ)
2
=a
k=1
+∞
X
k=1
+∞
X
ck (0)yk (ρ) =
Rzr’jua eujdstju − 1 (ρy
hgzr~ ‚uyuzr Ti–rƒ ρ
1
0
ck (t) (ρy 0k (ρ)) ,
ρ
+∞
X
ϕk yk (ρ).
k=1
k=1
0
0
k (ρ))
= λyk (ρ)
m yha fiŸr”rdsij c (t) {i}
k
(
c 0k (t) = −a2 λk ck (t),
ck (0) = ϕk .
d–dsra fiie’“ƒ c (t) = ϕ e
k
k
−a2 λk t
rhr
2(T0 − T1 ) −a22γ2k t
ck (t) =
e R .
γk J1 (γk )
—gsf”ra v(x, t) {edytujhadta j jryd eayuƒ
−a2 γ2
`] nioyu
k
+∞
X
e R2 t
γk
v(ρ, t) = 2(T0 − T1 )
J0 ( ρ).
γk J1 (γk )
R
k=1
−a2 γ2
k
+∞
X
e R2 t
γk
J0 ( ρ).
u(ρ, t) = T1 + 2(T0 − T1 )
γk J1 (γk )
R
vu€yd~ d~{deugeg j ”dsed tde|sa j ~i~ds jed~dsr t X Sha Ÿioi
{iytujr~ j ed–dsrd ρ = 0 r t = t X Rzr’juam zi J (0) = 1 m {ihgzr~
zrthiji€ eaym yha i{edydhdsra rtfi~i€ d~{deuge’X
`Y
k=1
0
0
0
˜jdƒ
−a2 γ2
k
+∞
X
e R 2 t0
u(0, t0 ) = T1 + 2(T0 − T1 )
.
γk J1 (γk )
­ ¯®Q® bduhhrzdtfr€ –uerf euyrgtu R m suoed’€ yi d~{deu}
ge’ T m {i~dtrhr j |ryfit• d~{deuge’ T (T < T ) X vu€r d~{deu}
geg su {ijde“sitr –uerfu j ~i~ds jed~dsr t m dthr d{hiip~ds –uerfu
t |ryfit•‘ {eirt“iyr {i ‚ufisg v•‘isuX
_ ~iydhr gyipsi jjdtr tderzdtfg‘ trtd~g fiieyrsuX i~dtr~
suzuhi fiieyrsu j ”dsed –uerfuX Rzr’jua gthijra ‚uyuzr ~i|si tzr}
u•m zi gsf”ra u m i{rt’ju‘¢ua r‚~dsdsrd d~{deuge’ –uerfu sd ‚u}
jrtr i gohij’“ fiieyrsu θ r ϕ m u ~dsadta ih•fi {i fiieyrsud r j
‚ujrtr~itr i jed~dsr t u = u(r, t) X —gsf”ra u(r, t) gyijhdjiead geuj}
sdsr‘ d{hi{eijiysitr
{er 0 < r < R, t > 0,
∂u
1 ∂
∂u
=a
r
∂t
r ∂r
∂r
suzuh•si~g gthijr‘ƒ u(ρ, 0) = T m
feudj’~ gthijra~ƒ
ioeusrzdsu {er r → 0 + 0 m − ∂u(R, t) = h(u(R, t) − T ).
u(r, t)
∂r
Teudjid gthijrd su oeusr”d r = R ajhadta
sdiysieiys’~X iŸi~g
tsuzuhu rt{ih•‚ga ‚u~dsg u(r, t) = v(r, t) + w(r, t) tjdyd~ ed–ud~g‘ ‚uyuzg
f ‚uyuzd t iysieiys’~r feudj’~r gthijra~rX xuyuyr~ w(r, t) = α X er
gsf”ra w(r, t) gyijhdjiead d~ |d feudj’~ gthijra~m zi r
α
=
T
gsf”ra u(r, t) ƒ
ioeusrzdsu {er r → 0 + 0 r ∂w(R, t) + h(w(R, t) − T ) = 0.
w(r, t)
∂r tiijdtjg‘¢r~ iysieiys’~
er Ÿi~ gsf”ra v(r, t) gyijhdjiead
feudj’~ gthijra~X „ie~ghregd~ ‚uyuzg yha gsf”rr v(r, t) X
—gsf”ra v(r, t) gyijhdjiead geujsdsr‘ d{hi{eijiysitr
{er 0 < r < R, t > 0,
∂v
∂v
1 ∂
r
=a
∂t
r ∂r
∂r
suzuh•si~g gthijr‘ƒ v(r, 0) = T − T m
feudj’~ gthijra~ƒ
ioeusrzdsu {er r → 0 + 0 m ∂v(R, t) + hu(R, t) = 0.
v(r, t)
ˆg ‚uyuzg pgyd~ ed–u• ~diyi~∂r—ge•dX
V] Sha tr~~derzsioi hrsd€sioi yrdeds”ruh•sioi i{deuieu L (y) m
eutt~uerjud~ioi su ~si|dtjd gsf”r€m gyijhdjiea‘¢r“ iysieiys’~
k=1
0
1
1
0
0
2
2
2
0
1
1
1
2
2
2
0
1
r
Š\
feudj’~ gthijra~ ed–ud~i€ ‚uyuzrm ed–r~ ‚uyuzg £ge~u¤lrgjrhhaƒ

 1 2 0 0
− 2 r y (r) = λy(r), 0 < r < R,
r
y(r)
r → 0 + 0, y 0 (R) + hy(R) = 0.
ioeusrzdsu {er
ˆi ‚uyuzu r‚ g{eu|sdsra cXVcX „iptjdss’d zrthu j Ÿi€ ‚uyuzdƒ λ =
m oyd µ ¤ Ÿi {ihi|rdh•s’d fiesr geujsdsra ctg(µR) = 1 − Rh m
= µ
µR
X „iptjdss’d gsf”rrƒ y (r) = sin(µ r) m k = 1, 2, . . . X „rtd~u
k = 1, 2, . . .
tiptjdss’“ gsf”r€ {ihsu r ieioisuh•su jr {eiteustjd L [0, R; r ] X
Tjuyeu’ sie~ tiptjdss’“ gsf”r€ su“iyata {i ie~ghu~ ky (r)k =
R sin(2µ R) R Rµ + Rh − h X
= −
=
2 c] —gsf”r‘
4µ
2 ‚u{r–d~
R µ + (1j−jryd
Rh)eayu —ge•dm eu‚huoua dd {i su€yds}
v(r, t) gsf”r€ƒ
si€ trtd~d tiptjdss’“
2
k
k
k
k
k
2
2
2
k
2 2
k
2
k
k
2
k
2
2
v(r, t) =
+∞
X
ck (t)yk (r).
Teudj’d gthijram {er Ÿi~m j’{ihsa‘taX
utt~ier~ gsf”r‘ ϕ÷ (r) = T − T r‚ suzuh•sioi gthijraX u‚hi}
|r~ dd j eay —ge•d {i i€ |d trtd~d gsf”r€
k=1
0
÷
ϕ (r) =
+∞
X
1
ϕk yk (r),
k=1
oyd ϕ = (ϕ÷ , y ) X Tjuyeu’ sie~ tiptjdss’“ gsf”r€ g|d su€yds’X
ky k
˜{edydhr~ tfuhaes’d
{eir‚jdydsra
k
k
ZR
k
2
÷
_it{ih•‚gd~ta eujdstji~m fiiei~g gyijhdjiea‘ zrthu µ ƒ ctg(µ R) =
1 − Rh m r {edipeu‚gd~ {ithdysdd {ihgzdssid j’eu|dsrdƒ
=
(ϕ , yk ) =
0
sin(µk R) R cos(µk R)
sin(µk r) 2
r dr = (T0 − T1 )
(T0 − T1 )
−
.
r
µ2k
µk
k
µk R
÷
(ϕ , yk ) =
vu€yd~ ϕ
k
=
(T0 − T1 )R2 h cos(µk R)
.
µk (1 − Rh)
(T0 − T1 )4R2 h cos(µk R)
k = 1, 2, . . . .
(1 − Rh)(2Rµk − sin(2µk R))
ŠV
k
ay’ —ge•d gsf”r€ v(x, t) r ϕ÷ (r) {iytujr~ j geujsdsrd d{hi{ei}
jiysitr r j suzuh•sid gthijrdƒ
+∞
X
c0k (t)yk (r)
2
=a
+∞
X
ck (t)
k=1
k=1
+∞
X
ck (0)yk (r) =
k=1
+∞
X
1 2 0 0
r y k (r) ,
r2
ϕk yk (r).
k=1
itfih•fg j’{ihsa‘ta eujdstju − 1 r y
r
r”rdsij c (t) {ihgzu‘ta ‚uyuzr Ti–rƒ
2
2
0
k (r)
k
0
= λyk (r)
m i yha fiŸ}
(
c 0k (t) = −a2 λk ck (t),
ck (0) = ϕk .
d–r~ r“ƒ
−a2 λk t
ck (t) = ϕk e
(T0 − T1 )4R2 h cos(µk R)
2 2
⇔ ck (t) =
e−a µk t .
(1 − Rh)(2Rµk − sin(2µk R))
—gsf”ra v(x, t) eu‚huoudta j eay —ge•dƒ
+∞
(T0 − T1 )4R2 h X
cos(µk R)
2 2 sin(µk r)
e−a µk t
.
v(r, t) =
(1 − Rh)
2Rµk − sin(2µk R)
r
`] †tfi~ua gsf”ra {edytujhadta j jrydƒ
k=1
+∞
(T0 − T1 )4R2 h X
cos(µk R)
2 2 sin(µk r)
u(r, t) = T1 +
e−a µk t
.
(1 − Rh)
2Rµk − sin(2µk R)
r
¬thr j Ÿg ie~ghg {iytujr• r = R r t = t m i {ihgzrta zrthiji€ eay
yha i{edydhdsra d~{deuge’ su {ijde“sitr –uerfu j ~i~ds jed~dsr
X
t
˜jdƒ
k=1
0
0
+∞
(T0 − T1 )4R2 h X
sin(µk r)
cos(µk R)
2 2
u(R, t0 ) = T1 +
e−a µk t0
.
(1 − Rh)
2Rµk − sin(2µk R)
r
³´µ¶· ¸¹¸º»
`XVX vuzuh•sua d~{deugeu iysieiysioi tde|sa yhrsi€ l t d{hir‚ihr}
eijussi€ {ijde“sit•‘ eujsu T X vuzrsua t ~i~dsu jed~dsr t = 0 iyrs r‚
fis”ij tde|sa {iyyde|rjudta {er d~{deuged T mu yegoi€ ¤ {er d~}
{deuged T X vu€r d~{deugeg tde|saX
Šc
k=1
0
0
1
`XcX vu€r d~{deugeg iysieiysioi tde|sa yhrsi€ l t d{hir‚ihreijus}
si€ {ijde“sit•‘m dthr doi suzuh•sua d~{deugeu i{rt’judta gsf”rd€
fisd” x = 0 d{hir‚ihregdtam u
T x r suzrsua t ~i~dsu jed~dsr
t=0
l
fisd”
{iyyde|rjudta {er d~{deugedm eujsi€ T X
x
=
l
`X`X ^dtfisdzsua {hutrsu ih¢rsi€ l suoedu yi d~{deuge’ T X vuzr}
sua t ~i~dsu jed~dsr t = 0 oeus• {hutrs’ x = 0 {iyyde|rjudta {er
d~{deuged T m u yegoua x = l i“hu|yudta |ryfit•‘m d~{deugeu fi}
iei€ T (T < T ) X nd{hiip~ds su Ÿi€ oeusr {hutrs’ {eirt“iyr {i
‚ufisg v•‘isuX vu€r d~{deugeg {hutrs’X
`XŠX ^dtfisdzsua {hutrsu ih¢rsi€ l suoedu yi d~{deuge’ T X vuzr}
sua t ~i~dsu jed~dsr t = 0 oeus• {hutrs’ x = 0 {iyyde|rjudta {er
d~{deuged T cos(mt) u yegoua x = l ¤ {er d~{deuged T X vu€r d~{d}
eugeg {hutrs’X
`XUX ^dtfisdzsua {hutrsu ih¢rsi€ l suoedu yi d~{deuge’ T X vuzr}
sua t ~i~dsu jed~dsr t = 0 su oeus• {hutrs’ x = 0 {iyudta d{hiji€
{iif {hisitr q m u yegoua oeus• x = l {iyyde|rjudta {er d~{deuged
eujsi€ T X vu€r d~{deugeg {hutrs’X
`X¥XShrss’€ ”rhrsyerzdtfr€ tde|ds• euyrgtu R suoed yi d~{deuge’
X vuzrsua t ~i~dsu jed~dsr t = 0 tde|ds• {i~d¢udta j |ryfit•m
T
d~{deugeu fiiei€ T (T < T ) X nd{hiip~ds tde|sa t |ryfit•‘ {ei}
rt“iyr {i ‚ufisg v•‘isuX nd~{deugeu j fu|yi~ {i{dedzsi~ tdzdsrr
tde|sa tzrudta iyrsufiji€X vu€r d~{deugeg tde|saX
`XWXørhrsyerzdtfr€ {eijiysrf euyrgtu R suoedjudta jthdytjrd {ei“i|}
ydsra {itiassioi ifuX ˜põd~sua {hisit• eut{edydhdsra d{hiji€
Ÿsdeorr ¤ Q X ^ifijua {ijde“sit• {eijiysrfu {iyyde|rjudta {er d~}
{deuged eujsi€ T X vuzuh•sua d~{deugeu {eijiysrfu uf|d eujsu T X
vu€r d~{deugeg {eijiysrfum dthr j fu|yi~ doi {i{dedzsi~ tdzdsrr
d~{deugeu tzrudta iyrsufiji€X
`XZX bduhhrzdtfr€ –uerf euyrgtu R m suoed yi d~{deuge’ T X vu€r
d~{deugeg –uerfum dthr suzrsua t ~i~dsu jed~dsr t = 0 d~{deugeu
su doi {ijde“sitr {iyyde|rjudta eujsi€ T X
`XYX vu€r d~{deugeg tderzdtfi€ ipihizfr R ≤ r ≤ R m dthr dd su}
zuh•sua d~{deugeu eujsi T m oeus• r = R {iyyde|rjudta {er d~{d}
euged T m u oeus• r = R ¤ {er d~{deuged T X
`XV\X Shrss’€ ~duhhrzdtfr€ tde|ds• t {ea~igoih•s’~ tdzdsrd~ (0 ≤
m 0 ≤ y ≤ B) suoed yi d~{deuge’ T X vuzrsua t ~i~dsu jed}
≤
x
≤
A
~dsr t = 0 oeusr tde|sa x = 0 m x = A r y = 0 d{hir‚ihreg‘tam u
oeus• y = B {iyyde|rjudta {er d~{deuged T X nd~{deugeu j fu|yi~
{i{dedzsi~ tdzdsrr tde|sa tzrudta iyrsufiji€X vu€r d~{deugeg
tde|saX
0
0
0
0
1
1
0
0
0
0
0
0
0
1
1
0
0
0
0
1
1
0
0
1
2
1
0
1
Š`
2
¼½¾¹½¿ `XVX u(x, t) = T
−
T
1
0
x + T0 +
l
−a2 π2 k2 t
+∞
2(T1 − T0 ) X (−1)k e l2
πkx
+
sin
.
π
k
l
k=1
`XcX u(x, t) = T
2
2t
+∞ −a (π+2πk)
4l2
8T0 X e
(π + 2πk)x
cos
.
0−
π2
(1 + 2k)2
2l
k=0
(T0 − T1 )h
x−
u(x, t) = T0 −
1 + hl
+∞
X
cos(µk l)(h2 + µ2k ) −a2 µ2k t
−2(T0 − T1 )
e
sin(µk x)
µk (h2 l + µ2k l + h)
k=1
h
µk
ctg(µl) =
µ
T0 (1 − cos(mt))
x + T0 cos(mt)+
u(x, t) =
l
+∞
2 2
π
kx
2T0 X
a
µ
m2
2 2
k
+
sin(mt) − cos(mt) sin
e−a µk t +
2
2
2
2
π
k(m + (a µk ) )
m
l
k=1
πk
µk =
k = 1, 2 . . .
l
2
2t
+∞ −a (π+2πk)
X
2
4l
e
(π + 2πk)x
8q
cos
u(x, t) = T0 + q(l − x) − 2
.
lπ
(1 + 2k)2
2l
k=0
+∞
γ ρ
X
a2 γ2 t
J1 (γk )
k
− R2k
u(ρ, t) = T1 + 2(T0 − T1 )
J0
e
2
2
γk (J0 (γk ) + J1 (γk ))
R
k=1
J1 (γ) hR
γk
=
J0 (γ)
γ
t
a2 γ2
k
+∞
2QR2 X (1 − e− R2 ) γk ρ J0
u(ρ, t) = T0 +
a2
γ3k J1 (γk )
R
k=1
γk
J0 (γ) = 0
−a2 π2 k2 t
+∞
2R(T0 − T1 ) X (−1)k e R2
πkr
u(r, t) = T1 +
.
sin
πr
k
R
k=1
T1 − T0
R 2 T0 − T1 R 1
u(r, t) =
r+
+
R2 − R1
R2 − R1
2 22
+∞ −a π k t
2(T1 − T0 ) X
R2 (−1)k 2((−1)k − 1)
(R2 −R1 )2
+
−
e
+
(R2 − R1 )r
πk
π2 k 2
`X`X
m
oyd ¤ Ÿi {ihi|rdh•s’d fiesr geujsdsra
`XŠX
oyd
m
X
m
X
`XUX
`X¥X
oyd ¤ Ÿi {ihi|rdh•s’d fiesr geujsdsra
`XWX
oyd ¤ Ÿi {ihi|rdh•s’d fiesr geujsdsra
`XZX
`XYX
k=1
ŠŠ
X
m
X
m
X
−a2 π2 k2 t
π
k(r
−
R
)
2(R2 − R1 )(1 − (−1)k )
1
sin
+
1 − e (R2 −R1 )2
a2 π3 k 3
R2 − R1
u(x, y, t) = u(y, t) =
−a2 (π+2πk)2 t
+∞
(π + 2πk)y
2B(T0 − T1 ) X (−1)k e 4B2
cos
.
= T1 +
π
1 + 2k
2B
`XV\X
k=0
ù ' 1 E(5 E1 EC ?A&15C5>C
er ed–dsrr ~sior“ r‚rzdtfr“ ‚uyuzm tja‚uss’“m su{er~dem t r‚gzd}
srd~ fihdpusr€m tegsm ~d~peusm tde|sd€m Ÿhdfei~uosrs’“ fihdpusr€m
fihdpusr€ ou‚um su“iya¢doita j ioeusrzdssi~ ipõd~dm ji‚srfu‘ geujsd}
sra jryuƒ
∂ 2u
= a2 ∆u − qu + f.
2
∂t
u = u(M, t) M ∈ Ω t > 0
S
a2 q
—gsf”ra
[
r
] i{rt’jud fihdpusra j iphutr
t oeusr”d€ X TiŸr”rds’ m i{edydha‘ta tji€tju~r tedy’ j
Ω
fiiei€ {eirt“iyr {ei”dtt fihdpusr€X —gsf”ra f (M, t) j’eu|ud rsds}
trjsit• jsd–sdoi ji‚yd€tjraX ˜{deuie ∆ ¤ Ÿi i{deuie lu{hutuX _ry
i{deuieu ‚ujrtr i j’peussi€ trtd~’ fiieyrsum fiieua i{edydhadta
ie~i€ iphutr Ω X ˜p’zsi tij~dtsi t geujsdsrd~ ‚uyu‘ta yju suzuh•}
s’“ gthijraƒ
du(M, 0)
= ψ(M ),
dt
u(M, 0) = ϕ(M ),
i{rt’ju‘¢r“ suzuh•sid {ihi|dsrd r suzuh•sg‘ tfieit• izdf dhum ti}
jde–u‘¢doi fihdpusraX vu oeusr”d S iphutr Ω ‚uyu‘ta feudj’d gthi}
jram tiijdtjg‘¢rd eutt~uerjud~i€ r‚rzdtfi€ ‚uyuzdX ˆi gthijra
Srer“hdm vd€~usu rhr `}oi eiyum tiijdtjdssiƒ
u
S
= ν(M 0 , t),
∂u
∂~n
= Ψ(M 0 , t),
S
∂u
+ σu
∂~n
S
= χ(M 0 , t).
xydt• M ¤ izfu oeusr”’ S m ∂u ¤ {eir‚jiysua {i su{eujhdsr‘ jsd–}
∂~n
sd€ sie~uhr f oeusr”d S X vu eu‚s’“
zuta“ oeusr”’ ~iog p’• ‚uyus’
oeusrzs’d gthijra eu‚sioi eiyuX ¬thr j ~iydhr tzrudtam zi iphut• Ω
sdioeusrzdssum i {edy{ihuou‘m zi gsf”ra u(M, t) ioeusrzdsu su pdt}
fisdzsitrX _’jiy jihsijioi geujsdsra i{rtus j Š¡X
_ {itiprr V¡ {ihgzdsi geujsdsrd {i{dedzs’“ fihdpusr€ tegs’ r {i}
yeipsi i{rtusu {itusijfu feudj’“ gthijr€X Reujsdsrd fihdpusr€ tegs’
0
ŠU
¤ Ÿi iysi~desid jihsijid geujsdsrd jryuƒ
[ŠXV]
iyips’d geujsdsra {iajha‘ta uf|dm su{er~dem {er rtthdyijusrr
yhrss’“ hrsr€ ¤ Ÿhdferzdtfr“ ”d{d€ t eut{edydhdss’~r {ueu~deu~rX
ˆi yjg“{eijiys’d hrsrr tja‚rm fiuftruh•s’d fupdh•s’d hrsrr r X {XX
vu{ea|dsrd u(x, t) r if i(x, t) j ufr“ hrsra“ tja‚us’ yrdeds”ruh•}
s’~r geujsdsra~rm fiie’d su‚’ju‘ta dhdoeus’~rƒ
2
∂ 2u
2∂ u
=a
+ f.
∂t2
∂x2

∂u
∂i


+ L + Ri = 0

∂x
∂t

∂u
∂i


+C
+ Gu = 0.
∂x
∂t
xydt• R ¤ ti{eirjhdsrdm L ¤ rsygfrjsit•m C ¤ d~fit•m G ¤ {eijiyr}
~it• ~d|yg {eijiyu~rm euttzruss’d su dyrsr”g yhrs’ {eijiyuX
eiyrdeds”regd~ {dejid geujsdsrd trtd~’ {i {ded~dssi€ x r
{iytujr~ j~dti ∂i j’eu|dsrd r‚ jieioi geujsdsram {ihgzrta iysi
geujsdsrd
∂x
∂ 2u
∂u
∂ 2u
LC 2 + (RC + LG)
+ RGu − 2 = 0,
∂t
∂t
∂x
fiieid uf|d su‚’judta dhdoeus’~X
¬thr ti{eirjhdsrd {eijiyij ~uhi r isr “iei–i r‚ihreijus’m i j
~iydhr ~i|si tzru•m zi R = G = 0 X nufua hrsra su‚’judta hrsrd€ pd‚
{ide•X ndhdoeusid geujsdsrd {er Ÿi~ tusijrta geujsdsrd~ fihdpusr€
tegs’ [ŠXV] [a = 1 m f = 0]X
¬thr {ueu~de’LChrsrr tja‚us’ tiisi–dsrd~ RC = LG m i ufua hr}
sra su‚’judta hrsrd€ pd‚ rtfu|dsraX d–dsrd dhdoeusioi geujsdsra j
Ÿi~ thgzud su“iya j jryd u(x, t) = e v(x, t) m oyd v(x, t) ¤ jt{i~ioudh•}
sua gsf”raX ¬thr gsf”r‘ u(x, t) m {edytujhdssg‘ j gfu‚ussi~ jrydm
{iytujr• j dhdoeusid geujsdsrdm i sdegysi {ifu‚u•m zi gsf”ra
uf|d pgyd gyijhdjiea• geujsdsr‘ fihdpusr€ tegs’ [ŠXV] {er
v(x, t)
X
1 m
a =
f =0
LC
iyeipsi diera dhdoeus’“ geujsdsr€ r‚hi|dsum su{er~dem j W¡X
˜ysr~ r‚ ~diyij ed–dsra feudji€ ‚uyuzr yha jihsijioi geujsdsra
ajhadta ~diy —ge•dX er~dsadta is uf|dm fuf r yha geujsdsra d{}
hi{eijiysitrX _ {itiprr V¡ eu‚ipeus’ {er~de’ {er~dsdsra ~diyu —g}
e•d yha ed–dsra iysi~desioi jihsijioi geujsdsra t eu‚s’~r suzuh•s’~r
r feudj’~r gthijra~rX utt~ier~ d¢d sdtfih•fi {er~deij {er~dsdsra
Š¥
2
−R
L
2
Ÿioi ~diyu yha ed–dsra feudj’“ ‚uyuzm tja‚uss’“ t r‚gzdsrd~ fihdpu}
sr€ tegs r ~d~peusX
­ Q®P® vu€r {i{dedzs’d fihdpusra suasgi€ tegs’ t ‚u}
fed{hdss’~r fis”u~r x = 0 r x = l m dthr j suzuh•s’€ ~i~ds jed~dsr
tegsg j yjg“ ~dtu“ iasghr r {eryuhr ie~gm i{rt’jud~g‘ gsf”rd€
2πx [ertgsif]X vuzuh•sua tfieit• izdf tegs’ eujsu sgh‘X
U sin
l
˜pi‚suzr~
gsf”r‘m fiieua
u(x,
t)
i{rt’jud {i{dedzs’d fihdpusra tegs’X
ˆu gsf”ra gyijhdjiead jihsiji~g
geujsdsr‘ƒ
∂ u
∂ u {er
0 < x < l, t > 0,
=a
∂t
∂x
suzuh•s’~ gthijra~ƒ u(x, 0) = U sin 2πx m du(x, 0) = 0 m
feudj’~ gthijra~ƒ u(0, t) = 0, u(l, t) =l 0. dt
Teudj’d gthijra j Ÿi€ ‚uyuzd ¤ iysieiys’dX —gsf”r‘ u(x, t) pgyd~
rtfu• j jryd eayu —ge•dm j’{ihsaa eu‚hi|dsrd {i tiptjdss’~ gsf”r}
a~ i{deuieu L (u) = − ∂ u X
∂x
V] d–r~ ‚uyuzg £ge~u}lrgjrhhaƒ
0
2
2
2
2
2
0
2
x
2
(
−y 00 (x) = λy(x), 0 < x < l,
y(0) = 0, y(l) = 0.
ˆu ‚uyuzu eutt~uerjuhut• j eu‚ydhd c [{er~de cXV]X „iptjdss’d zrthu
i{deuieu L ƒ λ = πk m k = 1, 2, . . . m tiptjdss’d gsf”rrƒ y (x) =
l
X ˜sr ipeu‚g‘ {ihsg‘ ieioisuh•sg‘ trtd~g j
πkx
, k = 1, 2, . . .
= sin
l
{eiteustjd
X er Ÿi~ ky (x)k = l X
L [0, l; 1]
2 —ge•dƒ
c] edytujr~ gsf”r‘ u(x, t) j jryd eayu
2
x
k
k
2
k
u(x, t) =
+∞
X
2
ck (t)yk (x).
k=1
Teudj’d gthijram {er Ÿi~m j’{ihsa‘taX —gsf”r‘ ϕ(x) = U sin 2πx m
l
i{rt’ju‘¢g‘ suzuh•sid {ihi|dsrd izdf tegs’m eu‚hi|r~ j eay {i i€
|d trtd~d gsf”r€ƒ
0
ϕ(x) =
+∞
X
ŠW
k=1
ϕk yk (x).
vu€yd~ ‚suzdsra fiŸr”rdsij —ge•d ϕ
s’d {eir‚jdydsra
(ϕ, yk ) =
Zl
k
=
(ϕ, yk )
||yk ||2
X _’zrthr~ tfuhae}
πkx
2πx
sin
dx.
l
l
U0 sin
0
itfih•fg eroisi~derzdtfrd gsf”rr sin πkx r sin πmx ieioisuh•s’
l
l
{er k 6= 2 X
su {ei~d|gfd [0, l] m i tfuhaes’d {eir‚jdydsra
(
ϕ
,
y
)
=
0
˜hrzs’~ i sgha pgyd ih•fi (ϕ, y ) ƒ
k
2
(ϕ, y2 ) =
Zl
U0 sin2
l
2πx
dx = U0 .
l
2
nioyu ϕ = U m u {er k 6= 2 ϕ = 0 X
Sha i{edydhdsra ‚suzdsr€ fiŸr”rdsij c (t) {iytujr~ eay’ —g}
e•d gsf”r€ u(x, t) r ϕ(x) j jihsijid geujsdsrd r j suzuh•s’d gthijraƒ
0
2
0
k
k
+∞
X
c00k (t)yk (x)
2
=a
ck (0)yk (x) =
ck (t)yk00 (x),
k=0
k=1
+∞
X
+∞
X
+∞
X
ϕk yk (x),
+∞
X
c0k (0)yk (x) = 0.
†t{ih•‚ga eujdstji y (x) = −λ y (x) m u uf|d tji€tji hrsd€si€ sd‚u}
jrtr~itr gsf”r€ y (x) m yha fiŸr”rdsij c (t) {ihgzr~ thdyg‘¢rd
‚uyuzr Ti–rƒ
k=1
k=0
00
k
k=1
k k
k
k
(
c00k (t) = −a2 λk ck (t),
ck (0) = ϕk ,
c0k (0) = 0.
˜p¢rd ed–dsra Ÿr“ geujsdsr€ {edytujha‘ta j jryd
ck (t) = Ak cos
aπkt
aπkt
+ Bk sin
.
l
l
Ak = ϕ k B k = 0
Rzr’jua suzuh•s’d gthijram {ihgzr~
m
X usdd p’hi
gtusijhdsim zi {er k 6= 2 ϕ = 0 r ϕ = U X iŸi~g c (t) = U cos a2πt
l
r c (t) = 0 {er k 6= 2 X xsuzr eay —ge•d gsf”rr u(x, t) tiyde|r ih•fi
iysi ihrzsid i sgha thuoud~idX
†tfi~id ed–dsrdm i{rt’ju‘¢dd {i{dedzs’d fihdpusra tegs’X ¤ Ÿi
gsf”ra jryuƒ
2
k
0
k
u(x, t) = U0 cos
ŠZ
a2πt
2πx
sin
.
l
l
2
0
˜jdƒ u(x, t) = U cos a2πt sin 2πx .
l
l euyrgtu ‚ufed{hdsu {i {der~degX
­ Q®N® Tegohua ~d~peusu
vu€r {i{dedzs’d fihdpusra ~d~peus’m dthrRj suzuh•s’€ ~i~ds jed}
~dsr dd ifhisrhr i {ihi|dsra eujsijdtra r {eryuhr ie~gm tr~~d}
erzsg‘ isitrdh•si ”dseu ~d~peus’m i{rt’jud~g‘ ie~ghi€ ϕ(ρ) =
ρ X vuzuh•sua tfieit• izdf ~d~peus’ eujsu sgh‘X
=h 1−
R
itfih•fg
eutt~uerjudta fegohua ~d~peusum i yha dd i{rtusra
gyipsi {ded€r f {ihaesi€ trtd~d fiieyrsuX gt• gsf”ra u i{rt’}
jud {i{dedzs’d fihdpusra ~d~peus’X nioyu tiohutsi gthijra~ ‚uyuzr
Ÿu gsf”ra sd ‚ujrtr i gohiji€ fiieyrsu’ ϕ r ajhadta gsf”rd€
fiieyrsu’ ρ r jed~dsr t ƒ u = u(ρ, t) X
—gsf”ra u gyijhdjiead jihsiji~g geujsdsr‘ƒ
{er 0 < ρ < R, t > 0,
∂ u
∂u
1 ∂
ρ
=a
0
2
2
2
2
∂t2
ρ ∂ρ
∂ρ
suzuh•s’~ gthijra~ƒ u(ρ, 0) = h 1 − ρ m du(ρ, 0) = 0 m
R
dt m
feudj’~ gthijra~ƒ u(ρ, t) ioeusrzdsu {er
ρ →ajha‘ta
0+0
u(R,
t) = 0.
Teudj’d gthijra eutt~uerjud~i€ ‚uyuzr
iysieiys’~rX
i}
Ÿi~g ~diy —ge•d sd{itedytjdssi {er~dsr~ f Ÿi€ ‚uyuzdX
V] d–r~ ‚uyuzg £ge~u¤lrgjrhha yha tr~~derzsioi hrsd€sioi
yrdeds”ruh•sioi i{deuieu ^dttdha B m euttt~uerjud~ioi su ~si|d}
tjd gsf”r€m gyijhdjiea‘¢r“ iysieiys’~ feudj’~ gthijra~ ed–ud~i€
‚uyuzrƒ
2
2
0

− 1 (ρy 0 (ρ))0 = λy(ρ), 0 < ρ < R,
ρ
y(ρ)
ρ → 0 + 0, y(R) = 0.
ioeusrzdsu {er
ˆu ‚uyuzu p’hu ed–dsu u j eu‚ydhd c [{er~de VXU]X „iptjdss’d zrthu
γ m oyd γ ¤ Ÿi {ihi|rdh•s’d ed–dsra geujsd}
, k = 1, 2, . . .
λ =
sra J (γR) = 0 X „iptjdss’d gsf”rr y (ρ) = J ( ρ) k = 1, 2, . . . X „rtd}
~u tiptjdss’“ gsf”r€ {ihsu r ieioisuh•su j {eiteustjd L [0, R; ρ] X
Tjuyeu’ sie~ tiptjdss’“ gsf”r€ i{edydha‘ta {i ie~ghu~
X
R
ky (ρ)k =
J (γ )
2
c] —gsf”r‘
eu‚hi|r~ j eay —ge•d {i trtd~d tiptjdss’“
u(
ρ
,
t)
gsf”r€ƒ
k
2
k
k
0
k
γk
0 R
2
k
2
2
2
1
k
u(ρ, t) =
+∞
X
ŠY
k=1
ck (t)yk (ρ).
Teudj’d gthijram {er Ÿi~m j’{ihsa‘taX Sha su“i|ydsra ‚suzdsr€ fiŸ}
r”rdsij c (t) eay thdygd {iytujr• j jihsijid geujsdsrd r j suzuh•}
s’d gthijraX edyjuerdh•si gsf”r‘ ϕ(ρ) = h 1 − ρ r‚ suzuh•sioi
R
gthijra eu‚hi|r~ j eay —ge•d {i i€ |d trtd~d tiptjdss’“
gsf”r€
k
2
2
ϕ(ρ) =
+∞
X
ϕk yk (ρ).
k=1
TiŸr”rds’ —ge•d i{edydhr~ {i {eujrhg ϕ = (ϕ, y ) X Tjuyeu’ sie~
k jit{ih•‚gd~ta rs}
tiptjdss’“ gsf”r€ su€yds’X Sha j’zrthdsra (ϕky
,y )
doeuh•s’~r i|ydtju~rƒ
k
k
k
2
k
ZT
J0 (x)xdx = T J1 (T ),
0
ZT
J0 (x)x3 dx = T 3 J1 (T )T 3 − 2T 2 J2 (T ).
dejid i|ydtji {ihgzudtam dthr jit{ih•‚iju•ta eujdstji~ [cXc]m ‚u}
yuj α = T m β = 0 X _ieid i|ydtji thdygd r‚ {dejioim dthr {eirsdoer}
eiju• iyrs eu‚ {i zuta~X vu€yd~
0
ZR γk
ρ2
(ϕ, yk ) = h 1 − 2 J0 ( ρ)ρdρ.
R
R
0
_’{ihsr~ ‚u~dsg {ded~dssi€ t = γ ρ m ioyu
k
R
Zγk t2
hR2
1 − 2 J0 (t)tdt =
(ϕ, yk ) = 2
γk
γk
0

hR2 
= 2
γk
_ riod {ihgzr~ ϕ
Zγk
0
k
1
J0 (t)tdt − 2
γk
=
m
Zγk
0
2J2 (γk )hR2

J0 (t)t dt =
.
γ2k
3
4J2 (γk )h
k = 1, 2, ... .
γ2k J12 (γk )
U\

ay —ge•d gsf”rr u(ρ, t) {iytujr~ j jihsijid geujsdsrd r j su}
zuh•s’d gthijraƒ
+∞
X
c00k (t)yk (ρ)
2
=a
k=1
k=1
+∞
X
+∞
X
ck (0)yk (ρ) =
k=1
+∞
X
1
0
ck (t) (ρyk0 (ρ)) ,
ρ
+∞
X
ϕk yk (ρ),
k=1
c0k (0)yk (ρ) = 0.
k=1
_it{ih•‚gd~ta eujdstju~r − 1 (ρy (ρ)) = λy (ρ) ƒ itfih•fg eu‚hi|dsrd
ρ
gsf”rr j eay —ge•d dyrstjdssim
i yha fiŸr”rdsij c (t) {ihgzr~
‚uyuzr Ti–rƒ
0
k
0
k
k
(
c00k (t) = −a2 λk ck (t),
ck (0) = ϕk , c0k (0) = 0.
d–r~ yrdeds”ruh•sid geujsdsrdm rt{ih•‚ga ~diy “ueufdertr}
zdtfioi geujsdsra r gzr’jua eujdstji λ = γ X xud~ {iytujr~
{ihgzdssid ed–dsrd j suzuh•s’d gthijraX _ riod R{ihgzr~
k
2
k
aγ aγ 4J (γ )h
k
k
2 k
cos
ck (t) = ϕk cos
t = 2 2
t .
R
γk J1 (γk )
R
nioyu eay —ge•d rtfi~i€ gsf”rr u(ρ, t) m i{rt’ju‘¢d€ {i{dedzs’d fi}
hdpusra ~d~peus’m r~dd jryƒ
+∞
aγ X
J2 (γk )
γk
k
u(ρ, t) = 4h
cos
t J0 ( ρ).
2
2
γk J1 (γk )
R
R
k=1
˜jdƒ u(ρ, t) = 4h X J (γ ) cos aγ t J ( γ ρ).
(γ )
R
R
­ Q®¯® bd~peusumγ Jr~d‘¢ua
ie~g
{ea~igoih•srfu
ti tieisu}
~r A r B m ‚ufed{hdsu {i {der~degX vu€r {i{dedzs’d fihdpusra ~d~peu}
s’m dthr j suzuh•s’€ ~i~ds jed~dsr izfu~ ~d~peus’ {eryuhr tfieit•
X vuzuh•sid ifhisdsrd izdf ~d~peus’ eujsi sgh‘X
V
gt• gsf”ra u = u(x, y, t) (0 ≤ x ≤ A, 0 ≤ y ≤ B, t ≥ 0)
i{rt’jud {i{dedzs’d fihdpusra ~d~peus’X ˆu gsf”ra gyijhdjiead
jihsiji~g geujsdsr‘ƒ
{er 0 < x < A, 0 < y < B, t > 0,
∂ u
∂ u ∂ u
=a
+
+∞
k=1
2
∂t2
2
2
2
∂x2
∂y 2
2 k
2 2
k
k 1
k
k
0
UV
suzuh•s’~ gthijra~ƒ u(x, y, 0) = 0 m du(x, y, 0) = V m
feudj’~ gthijra~ƒ u(0, y, t) = 0, u(A, y, t) =dt 0, u(x, 0, t) = 0, u(x, B, t) = 0.
†tfi~ua gsf”ra u = u(x, y, t) gyijhdjiead iysieiys’~ feudj’~
gthijra~X iŸi~g ~diy —ge•d sd{itedytjdssi {er~dsr~ f Ÿi€ ‚uyuzdX
V] —gsf”r‘ u(x, y, t) eu‚hi|r~ j eay —ge•d {i tiptjdss’~ gsf”r}
a~ hrsd€sioi yrdeds”ruh•sioi i{deuieu L (u) = − ∂ u X Sha su“i|}
∂x ed–r~ ‚uyuzg
ydsra tiptjdss’“ zrtdh r tiptjdss’“ gsf”r€ i{deuieu
£ge~u¤lrgjrhha
2
x
2
(
−y 00 (x) = λy(x), 0 < x < A,
y(0) = 0, y(A) = 0.
ˆu ‚uyuzu p’hu ed–dsu j eu‚ydhd c [{er~de cXV]X „iptjdss’d zrthu
m tiijdtjg‘¢rd r~ tiptjdss’d gsf”rr
πk m
λ =
k = 1, 2, . . .
A
X —gsf”rr ipeu‚g‘ {ihsg‘ ieioisuh•sg‘
πkxm
y (x) = sin
k = 1, 2, . . . .
A
trtd~g j {eiteustjd
X Tjuyeu’ sie~ tiptjdss’“ gsf”r€
L [0, A; 1]
X
Am
ky k =
k = 1, 2, . . .
2
c]_’{ihsr~
eu‚hi|dsrd {i su€ydssi€ trtd~d tiptjdss’“ gsf”r€
yrdeds”ruh•sioi i{deuieuƒ
2
k
k
2
k
2
u(x, y, t) =
+∞
X
ck (y, t)yk (x).
Teudj’d gthijra {i {ded~dssi€ x pgyg j’{ihsa•taX ˜ysu r‚ gsf”r€
r‚ suzuh•s’“ gthijr€ ihrzsu i sghaX u‚hi|r~ dd j eay —ge•d {i i€
|s trtd~d tiptjdss’“ gsf”r€
k=1
V =
+∞
X
vk yk (x).
k=1
TiŸr”rds’ —ge•d j’zrthr~ {i {eujrhg v
g|d su€yds’X ˜{edydhr~ d{de•
ky k
k
k
=
2
(V, yk ) =
ZA
πkx
−V A
πkx
V sin
dx =
cos
A
πk
A
0
Uc
A
0
(V, yk )
kyk k2
X Tjuyeu’ sie~
V A(1 − (−1)k )
=
.
πk
nioyu v = 2V (1 − (−1) ) m k = 1, 2, ... .
πk jtd eay’ —ge•d j jihsijid geujsdsrdm suzuh•s’d r
iytujr~ d{de•
feudj’d gthijraƒ
k
k
+∞ 2
X
∂ ck (y, t)
∂t2
k=1
+∞
X
yk (x) = a2
+∞ X
k=0
2
∂
c
(y,
t)
k
ck (y, t)yk00 (x) +
yk (x) ,
∂y 2
+∞
X
ck (y, 0)yk (x) = 0,
k=1
ck (y, 0)yk (x) =
k=1
+∞
X
ck (0, t)yk (x) = 0,
+∞
X
vk yk (x),
k=0
+∞
X
ck (B, t)yk (x) = 0.
itfih•fg t{eujdyhrji eujdstji y (x) = −λ y (x) m i
k=1
k=1
00
k
+∞ 2
X
∂ ck (y, t)
∂t2
k k
+∞ X
∂ 2 ck (y, t)
−λk ck (y, t) +
yk (x).
yk (x) = a
∂y 2
2
u‚hi|dsrd gsf”rr j eay —ge•d dyrstjdssim {iŸi~g fu|yua gsf”ra
gyijhdjiead geujsdsr‘ƒ
c (y, t)
k=1
k=0
k
∂ 2 ck
∂ 2 ck
2
,
= a −λk ck +
∂t2
∂y 2
suzuh•s’~ gthijra~ƒ c (y, 0) = 0, dc (y, 0) = v m
feudj’~ gthijra~ƒ c (0, t) = 0, c (B, t)dt= 0 X
ihgzdss’d yha gsf”r€ c (y, t) suzuh•si}feudj’d ‚uyuzr tsiju ~i|}
si ed–r• ~diyi~ —ge•dm rt{ih•‚ga eu‚hi|dsrd rtfi~’“ gsf”r€ j eay
{i trtd~d tiptjdss’“ gsf”r€ hrsd€sioi yrdeds”ruh•sioi i{deu}
ieu L (c ) = − ∂ c X Sha Ÿioi suyi ed–r• tiijdtjg‘¢g‘ ‚uyuzg
£ge~u¤lrgjrhha∂y
k
k
k
k
k
k
2
y
k
k
2
(
−z 00 (y) = µz(y), 0 < y < B,
z(0) = 0, z(B) = 0.
iyipsua ‚uyuzu g|d p’hu ed–dsuX „iptjdss’d zrthu µ = πm m
B
m
tiijdtjg‘¢rd
r~
tiptjdss’d
gsf”rr
πmy m
m = 1, 2, . . .
z (x) = sin
B
X —gsf”rr ipeu‚g‘ {ihsg‘ ieioisuh•sg‘ trtd~g j {ei}
m = 1, 2, . . .
teustjd L [0, B; 1] X Tjuyeu’ sie~ tiptjdss’“ gsf”r€ kz k = B m
2
X
m = 1, 2, . . .
U`
2
m
m
2
m
2
Tu|yua —gsf”ra c (y, t) {edytujhadta j jryd eayu
k
ck (y, t) =
+∞
X
αkm (t)zm (y).
er Ÿi~ gsf”ra v r‚ suzuh•sioi gthijra uf |d eu‚huoudta j eay {i
i€ |d trtd~d gsf”r€
m=1
k
vk (y, t) =
+∞
X
wkm (t)zm (y).
m=1
TiŸr”rds’ w = (v , z ) j’zrtha‘ta {i usuhiorr t fiŸr”r}
kz k
dsu~r v X iŸi~g w = 2v (1 − (−1) ) = 4V (1 − (−1) )(1 − (−1) ) m
πm
π km
k = 1, ihgzdss’d
2, ... , m = 1,
2,
...
.
eay’ {iytujhad~ j yrdeds”ruh•sid geujsdsrd r su}
zuh•s’d gthijraX ithd {iytusijfr r rt{ih•‚ijusra eujdstj −z (y) =
m u uf|d tji€tju dyrstjdssitr eu‚hi|dsra gsf”rr j eay
=
µ
z
(y)
—ge•dm yha fiŸr”rdsij α (t) {ihgzu‘ta thdyg‘¢rd ‚uyuzr Ti–rƒ
k
km
m
m
2
k
k
m
km
k
m
2
00
m
m m
km
(
α00km (t) = −a2 (λk + µm )αkm (t),
αkm (0) = 0, α0km (0) = wkm .
†t{ih•‚ga “ueufdertrzdtfid geujsdsrd sdegysi {ihgzr• ip¢dd ed–d}
srd yrdeds”ruh•sioi geujsdsra
p
p
αkm (t) = C1km cos(a λk + µm t) + C2km sin(a λk + µm t).
ithd {iytusijfr j suzuh•s’d gthijram {ihgzr~
xsuzr
C1km = 0,
wkm
C2km = √
.
a λk + µm
p
wkm
αkm (t) = √
sin(a λk + µm t).
a λk + µm
ck (y, t)
_ riod fu|yua gsf”ra
ck (y, t) =
+∞
X
eu‚huoudta j eay
p
w
πmy
√ km
.
sin(a λk + µm t) sin
B
a
λ
+
µ
k
m
m=1
_it{ih•‚gd~ta Ÿr~ eu‚hi|dsrd~ r {ihgzr~ eay —ge•d rtfi~i€ gsf”rr
X
u(x, y, t)
UŠ
˜jdƒ
u(x, y, t) =
oyd w
+∞
+∞ X
X
p
w
πkx
πmy
√ km
sin(a λk + µm t) sin
sin
.
B
A
a
λ
+
µ
k
m
k=1 m=1
m
4V (1 − (−1)k )(1 − (−1)m )
k = 1, 2, ... , m = 1, 2, ... ,
km =
π2 km
2
πm 2
πk
k = 1, 2, . . . µm =
m = 1, 2, . . .
λk =
A
B
m
m
m
X
³´µ¶· ¸¹¸º»
ŠXVX vu€r {i{dedzs’d fihdpusra suasgi€ tegs’m fisd” x = 0 fiiei€
‚ufed{hdsm u fisd” x = l ~i|d tjipiysi pd‚ edsra {ded~d¢u•ta jyih•
jderfuh•si€ hrsrr [ ∂u(l, t) = 0]X _ suzuh•s’€ ~i~ds jed~dsr ie~u
∂x
πx m suzuh•sua tfieit• i}
tegs’ i{rt’judta gsf”rd€
u(x, 0) = sin
zdf tegs’ eujsu sgh‘X
l
ŠXcXvu€r {i{dedzs’d fihdpusra suasgi€ tegs’m fisd” x = 0 fiiei€
‚ufed{hdsm u fisd” x = l ~i|d tjipiysi pd‚ edsra {ded~d¢u•ta jyih•
jderfuh•si€ hrsrr [ ∂u(l, t) = 0]X _ suzuh•s’€ ~i~ds jed~dsr {i tegsd
∂x
gyuerhr r j’jdhr r‚ {ihi|dsra
eujsijdtraX vuzuh•sua tfieit• jtd“ i}
zdf tegs’ eujsu V X
ŠX`X vu€r {i{dedzs’d fihdpusra suasgi€ tegs’m j’‚juss’d jsd–sd€
trhi€ t {hisit•‘ f (x, t) = F X _ izfu“ x = 0 r x = l tegsu ‚ufed{hdsuX
_ suzuh•s’€ ~i~ds jed~dsr isu su“iyrta j {ifidX
ŠXŠX vu€r {i{dedzs’d fihdpusra suasgi€ tegs’m fis”’ x = 0 r x = l
fiiei€ ‚ufed{hds’X _ suzuh•s’€ ~i~ds jed~dsr {i tegsd gyuerhr r
j’jdhr r‚ {ihi|dsra eujsijdtraX vuzuh•sua tfieit• izdf tegs’ i{r}
t’judta gsf”rd€ ∂u(x, 0) = V (1 − x ) X
l
ŠXUX vu€r {i{dedzs’d ∂tfihdpusra suasgi€
tegs’m fisd” x = 0 fiiei€
‚ufed{hdsm u fisd” x = l tijde–ud fihdpusram i{rt’jud~’d gsf”rd€
X _ suzuh•s’€ ~i~ds jed~dsr tegsu su“iyrta j {ifidX [ er
U
sin(
ω
t)
{ded“iyd f ‚uyuzd t iysieiys’~r feudj’~r gthijra~r gsf”r‘ w(x, t)
rtfu• j jryd w(x, t) = sin(ωt)Φ(x)]X
ŠX¥ vu€r su{ea|dsrd j fupdhd yhrs’ l m dthr j suzuh•s’€ ~i~ds jed~dsr
fisd” x = 0 {iyfh‘zrhr f rtizsrfg {ded~dssi€ ŸXyXtX E sin(ωt) m su fis”d
fupdh• eu‚i~fsg [if i(l, t) = 0]X edy{ihuoudtam zi {idea~r j
x
=
l
fupdhd ~i|si {edsdpedz• [R = G = 0]X _ ~i~ds jfh‘zdsra su{ea|dsrd r
if j fupdhd eujs’ sgh‘X [ er {ded“iyd f ‚uyuzd t iysieiys’~r feudj’~r
gthijra~r gsf”r‘ w(x, t) rtfu• j jryd w(x, t) = sin(ωt)Φ(x)]X
ŠXW vu€r su{ea|dsrd j fupdhd yhrs’ l m dthr j suzuh•s’€ ~i~ds jed}
UU
0
~dsr fisd” x = 0 {iyfh‘zrhr f rtizsrfg {ded~dssi€ ŸXyXtX E sin(ωt) m
su fis”d x = l fieifid ‚u~’fusrd [su{ea|dsrd u(l, t) = 0]X edy{ihu}
oudtam zi {idea~r j fupdhd ~i|si {edsdpedz• [R = G = 0]X _ ~i~ds
jfh‘zdsra su{ea|dsrd r if j fupdhd eujs’ sgh‘X [ er {ded“iyd f ‚u}
yuzd t iysieiys’~r feudj’~r gthijra~r gsf”r‘ w(x, t) rtfu• j jryd
]X
w(x,
t)
=
sin(
ω
t)Φ(x)
ŠXZX Tegohua ~d~peusu euyrgtu R ‚ufed{hdsu {i {der~degX vu€r {i{d}
edzs’d fihdpusra ~d~peus’m dthr j suzuh•s’€ ~i~ds jed~dsr dd j’jdhr
r‚ {ihi|dsra eujsijdtra gyuerj {i sd€X vuzuh•sua tfieit• jtd“ izdf
~d~peus’ eujsu V X
ŠXYX Tegohua ~d~peusu euyrgtu R ‚ufed{hdsu {i {der~degX vu€r {i}
{dedzs’d fihdpusra ~d~peus’m j’‚juss’d jsd–sd€ trhi€ t {hisit•‘
X _ suzuh•s’€ ~i~ds jed~dsr isu su“iyrta j {ifidX
fŠXV\X
(ρ, t)bd~peusum
=F
r~d‘¢ua ie~g {ea~igoih•srfu ti tieisu~r A r B m
‚ufed{hdsu {i {der~degX vu€r {i{dedzs’d fihdpusra ~d~peus’m dthr j
suzuh•s’€ ~i~ds jed~dsr ~d~peusd {eryuhr ie~gm i{rt’jud~g‘ gsf}
”rd€ u(x, y, 0) = x(A − x)y(B − y) X vuzuh•sua tfieit• izdf ~d~peus’
eujsu sgh‘X
ŠXVVX bd~peusum r~d‘¢ua ie~g {ea~igoih•srfu ti tieisu~r A r B m ‚u}
fed{hdsu {i {der~degX vu€r {i{dedzs’d fihdpusra ~d~peus’m j’‚jus}
s’d jsd–sd€ trhi€ t {hisit•‘ f (x, y, t) = F X _ suzuh•s’€ ~i~ds jed}
~dsr isu su“iyrta j {ifidX
ŠXVcX vu€r {i{dedzs’d fihdpusra ~d~peus’m r~d‘¢d€ ie~g {ea~igoih•}
srfum j’‚juss’d jsd–sd€ trhi€ t {hisit•‘ f (x, y, t) = F X „ieis’ ~d~}
peus’ x = 0 r x = A ‚ufed{hds’m tieis’ y = 0 r y = B ~iog tjipiysi
pd‚ edsra {ded~d¢u•ta jyih• jderfuh•si€ hrsrrX _ suzuh•s’€ ~i~ds
jed~dsr ~d~peusu su“iyrta j {ifidX
¼½¾¹½¿
ŠXVX u(x, t) = 8 X (−1) cos(aµ t) sin(µ x), oyd µ = + πk .
+∞
ŠXcX
ŠX`X
ŠXŠX
ŠXUX
k+1
k
k
π2
(−1 + 2k)(3 + 2k)
k=0
+∞
8V l X cos(aµk t)
u(x, t) =
µk =
sin(µk x),
aπ2
(1 + 2k)2
k=0
+∞
2 X
(1 − cos( aπlkt ))
πkx
2F l
sin
u(x, t) = 2 3
aπ
k3
l
k=1
+∞
2V l X sin( aπlkt )
πkx
u(x, t) =
sin
aπ2
k2
l
k=1
ωx
sin( a )
u(x, t) = U0 sin(ωt)
+
sin( ωal )
oyd
X
X
U¥
k
π
2
π
2
+ πk
.
l
l
oyd µ
+∞
2U0 ωa X (−1)k sin(µk at) sin(µk x)
+
,
l
µ2k a2 − ω2
k
k=1
ŠX¥X u(x, t) = E sin(ωt) cos(
=
πk
l
ω(l−x)
a )
−
cos( ωal )
+∞
oyd µ
2E ωa X sin(µk at) sin(µk x)
−
,
l
µ2k a2 − ω2
k
k=0
ŠXWX u(x, t) = E sin(ωt) sin(
=
π
2
+ πk
l
ω(l−x)
a )
−
sin( ωal )
oyd
+∞
X
X
X
2E ωa X sin(µk at) sin(µk x)
πk
,
−
µ
=
k
2
l
µk a2 − ω2
l
k=0
+∞
2V R X sin( aγRk t ) γk ρ J0
u(ρ, t) =
a
γ2k J1 (γk )
R
k=1
+∞
2F R2 X (1 − cos( aγRk t ) γk ρ J0
u(ρ, t) =
a2
γ3k J1 (γk )
R
k=1
+∞ +∞
16A2 B 2 X X (1 − (−1)k )(1 − (−1)m )
×
u(x, y, t) =
3 m3
π6
k
q
k=1 m=1
πkx
πmy
πk 2
πm 2
sin
.
× cos a
t
sin
+
A
B
B
A
+∞ +∞
F AB X X (1 − (−1)k )(1 − (−1)m )
u(x, y, t) = 2 2
×
aπ
km(λ
+
µ
)
k
m
k=1 m=1
√
πmy
πkx
× 1 − cos(a λk + µm )t sin
sin
,
B
A
2
πm 2
πk
λk =
µm =
A
B
+∞ +∞
F AB X X (1 − (−1)k )(1 − (−1)m )
×
u(x, y, t) = 2 2
aπ
km(λ
+
µ
)
k
m
k=1 m=0
√
πmy
πkx
× 1 − cos(a λk + µm )t cos
sin
,
B
A
2
πm 2
πk
λk =
µm =
A
B
ŠXZX
X
ŠXZX
X
ŠXV\X
ŠXVVX
oyd
m
X
m
X
ŠXVcX
oyd
UW
ú ' ) C2E= = &( &) ¨CA&
utt~ier~ pdtfisdzsg‘ tegsgm fiieg‘ j suzuh•s’€ ~i~ds jed}
~dsr j’jdhr r‚ {ihi|dsra eujsijdtraX gt• gsf”ra ϕ(x) i{rt’jud
suzuh•sid {ihi|dsrdm u gsf”ra ψ(x) ¤ suzuh•sg‘ tfieit• izdf teg}
s’X ¬thr jsd–srd trh’ itgtjg‘m ioyu gsf”ra u(x, t) m i{rt’ju‘¢ua
tjipiys’d {i{dedzs’d fihdpusra tegs’m gyijhdjiead jihsiji~g geuj}
sdsr‘ r suzuh•s’~ gthijra~ jryuƒ
 2
2

 ∂ u = a2 ∂ u , −∞ < x < +∞, t > 0,
∂t2
∂x2

 u(x, 0) = ϕ(x), ∂u(x, 0) = ψ(x).
∂t
d–dsrd Ÿi€ ‚uyuzr u(x, t) ~i|d p’• su€ydsi ~diyi~ Suhu~pdeu
r {edytujhadta j jrydƒ
u(x, t) =
1
ϕ(x − at) + ϕ(x + at)
+
2
2a
x+at
Z
ψ(ξ).
_ {itiprr V¡ {iyeipsi i{rtus uhoier~ t {i~i¢•‘ fiieioi {ihgzdsu Ÿu
ie~ghum su‚’jud~ua ie~ghi€ Suhu~pdeuX
¬thr r‚gzudta {ei”dtt fihdpusr€ {ihgpdtfisdzsi€ tegs’ [x ≥ 0]m
i eutt~uerjud~ua ‚uyuzu tjiyrta f ed–dsr‘ ‚uyuzr i fihdpusrr pdt}
fisdzsi€ tegs’X er Ÿi~ gsf”rr ϕ(x) r ψ(x) r‚ suzuh•s’“ gthijr€
{eiyih|u‘ su jt‘ it• OX sdzds’~ ipeu‚i~m dthr ‚uyusi feudjid gthi}
jrd u(x, 0) = 0 r zds’~ ipeu‚i~ yha feudjioi gthijra ∂u(x, 0) = 0 X
­ °®P® vu€r {i{dedzs’d fihdpusra pdtfisdzsi€∂t tegs’m dthr
r‚jdtsim zi tegsu tijde–ud fihdpusra ‚u tzd suzuh•sioi ifhisdsram
i{rt’jud~ioi gsf”rd€ ϕ(x) X vuzuh•sua tfieit• izdf tegs’ eujsu sg}
h‘X
†tfi~ua gsf”ra u(x, t) m ajhadta ed–dsrd~ thdyg‘¢d€ ‚uyuzrƒ
x−at
 2
2

 ∂ u = ∂ u − ∞ < x < +∞, t > 0,
∂t2
∂x2

 u(x, 0) = ϕ(x), ∂u(x, 0) = 0.
∂t
„iohutsi ie~ghd Suhu~pdeu ed–dsrd r~dd jryƒ
u(x, t) =
ϕ(x − at) + ϕ(x + at)
.
2
UZ
—gsf”ra u(x, t) tfhuy’judta r‚ yjg“ jihsƒ {dejua 1 ϕ(x −at) eut{ei}
teusadta j{euji r su‚’judta {ea~i€ jihsi€m jieua 21 ϕ(x + at) ¤ jhdji
2
r su‚’judta ipeusi€ jihsi€X
gt• gsf”ra ϕ(x) ‚uyusu {i {eujrhg [ert UXV]ƒ
ϕ(x) =
(
−x2 + l2 x ∈ [−l, l],
0 x 6∈ [−l, l].
vu ertX UXc thdju |resi€ hrsrd€
r‚ipeu|ds oeurf gsf”rr u(x, t) j ~i}
~ds’ jed~dsr t = 0 m t = l m t =
2a
{ihgzds
fuf
lm
2l X ‡eurf
=
t =
u(x, t)
2
a
tg~~u
oeurfij
gsf”r€ 1 ϕ(x + at) r
2
[r‚ipeu|ds’
{gsfresi€
hrsrd€]X
1
ϕ(x − at)
0
1
u
l
2
3
−l
o
ëìíî ûîðî
l
x
2
u
−l
l
u(x,t 0)
o
u
l
x
l
u(x,t 1 )
−l ou
l
x
l
u(x,t 2)
−l
o
u
l
x
l
u(x,t 3 )
−2l
−l
o
ëìíî ûîï
l
2l x
˜jdƒ u(x, t) = ϕ(x − at) + ϕ(x + at) .
­ °®N® utt~ied• 2{ei”dtt {i{dedzs’“ fihdpusr€ tegs’ dthr
UY
j suzuh•s’€ ~i~ds jed~dsr {i suasgi€ tegsd gyuerhr r izfr tegs’
{ihgzrhr suzuh•sg‘ tfieit•m i{rt’jud~g‘ gsf”rd€ƒ
ψ(x) =

 v x ∈ [−l, l],
 0 x 6∈ [−l, l].
gt• gsf”ra u(x, t) i{rt’jud {i{dedzs’d fihdpusra tegs’X ˜su gyi}
jhdjiead jihsiji~g geujsdsr‘ r suzuh•s’~ gthijra~ jryuƒ

∂ 2u
∂ 2u

2


=a


∂x2
 ∂t2



u(x, 0) = 0,



− ∞ < x < +∞, t > 0,

 v x ∈ [−l, l],
∂u(x, 0)
=
 0 x 6∈ [−l, l].
∂t
˜pi‚suzr~ zded‚ Ψ(x) = 1 Z
a
pdeum
x
ψ(t)dt
m ioyu tiohutsi ie~ghd Suhu~}
0
_ yussi~ thgzud
1
u(x, t) = (Ψ(x + at) − Ψ(x − at)).
2
1
Ψ(x) =
a
Zx
vdt =
x
vx
=h ,
a
l
−l ≤ x ≤ l,
Zl
vdt =
vl
= h,
a
x>l
Z−l
vdt = −
0
1
Ψ(x) =
a
0
1
Ψ(x) =
a
0
vl
= −h, x < −l,
a
oyd h = vl X nioyu Ψ(x) {edytujhadta j jrydƒ
a

x < −l,

−h,
x
Ψ(x) = h , −l ≤ x ≤ l,

 l
h,
x > l.
iteir~ oeurf gsf”rr u(x, t) m ieu|u‘¢r€ {eirh• tegs’m j
¥\
~i~ds’ jed~dsr t = 0 m t = l m t = l m t = 2l [ertX UX`]X ‡eurf
teirta fuf tg~~u oeurfij yjg“2a gsf”r€2 1 Ψ(x +aat) r − 1 Ψ(x − at) X
0
1
2
3
2
2
u
h
o
ll
−l 0
u
0.5 Ψ ( x )
u(x , t 0)
x
−0.5Ψ ( x )
h
0.5 Ψ ( x + 0.5 l )
u (x , t 1)
x
l
−0.5 Ψ ( x − 0.5 l )
h
0.5 Ψ ( x + l )
u (x , t 2)
x
l
−0.5 Ψ ( x − l )
−l 0
u
−l 0
u
h
u (x , t 3)
−2l −l 0
0.5 Ψ ( x + 2 l )
l
x
2l
−0.5 Ψ ( x −2 l )
ëìíî ûîü
iyips’€ {er~de eu‚ipeus j {itiprr V¡X
˜jdƒ u(x, t) = 1 (Ψ(x + at) − Ψ(x − at)) X
2
­ °®¯® vu€r
{i{dedzs’d fihdpusra {ihgpdtfisdzsi€ tegs’m
fisd” x = 0 fiiei€ sd{iyjr|si ‚ufed{hdsX vuzuh•sid ifhisdsrd izdf
tegs’ i{rt’judta gsf”rd€ U x {er x ≥ 0 X vuzuh•sua tfieit• izdf
eujsu sgh‘X
˜pi‚suzr~ rtfi~g‘ gsf”r‘ u (x, t) X ˜su gyijhdjiead geujsdsr‘ƒ
∂ u
∂ u {er
x > 0, t > 0,
=a
2
0
1
2
1
2
∂t
2
2
1
2
∂x
suzuh•s’~ gthijra~ƒ u (x, 0) = U x , ∂u (x, 0) = 0,
∂t
feudji~g gthijr‘ƒ u (0, t) = 0 X
eiyih|r~ gsf”r‘ U x su ier”udh•sg‘ {ihgit• sdzds’~ ip}
1
0
1
0
2
2
¥V
1
eu‚i~ƒ
ϕ1 (x) =
(
U0 x2 , x ≥ 0,
−U0 x2 , x < 0.
^gyd~ ed–u• ‚uyuzg su jtd€ itr tzruam zi ϕ(x) = ϕ (x) r ψ(x) = 0 X _
Ÿi~ thgzudm tiohutsi ie~ghd Suhu~pdeu
1
u(x, t) =
1
(ϕ(x − at) + ϕ(x + at)) .
2
†tfi~ua gsf”ra u (x, t) = u(x, t) {er x ≥ 0 X —rftregd~ sdfiieid ‚su}
zdsrd x ≥ 0 m ioyu
1
ϕ(x − at) =
(
U0 (x − at)2 x − at ≥ 0,
−U0 (x − at)2 x − at < 0.
Rzr’jua Ÿim {ihgzr~ ed–dsrd ‚uyuzrX
˜jdƒ

U0


(x + at)2 + (x − at)2 ,

2
u1 (x, t) =


 U0 (x + at)2 − (x − at)2 ,
2
0≤t≤
t>
x
,
a
x
.
a
³´µ¶· ¸¹¸º»
UXV vu€r {i{dedzs’d fihdpusra pdtfisdzsi€ tegs’m dthr j suzuh•s’€
~i~ds jed~dsr tegsu r~dd ie~gm i{rt’jud~g‘ gsf”rd€ U cos x X vu}
zuh•sua tfieit• izdf tegs’ eujsu 0 X
UXcX vu€r {i{dedzs’d fihdpusra pdtfisdzsi€ tegs’m dthr suzuh•sua tfi}
eit• izdf tegs’ ‚uyudta ie~ghi€ƒ
0



 0, | x| > h,
∂u(x, 0)
= −v, −h ≤ x ≤ 0,

∂t

 v, 0 < x ≤ h,
u suzuh•sid ifhisdsrd izdf eujsi 0 X
UX`X vu€r {i{dedzs’d fihdpusra {ihgpdtfisdzsi€ tegs’m dthr j suzuh•}
s’€ ~i~ds tfieit• jtd“ izdf tegs’ eujsu 0 m fisd” tegs’ x = 0
‚ufed{hdsm u suzuh•sua ie~u tegs’ i{rt’judta gsf”rd€

πx
 sin
, 0 ≤ x ≤ l,
l
u(x, 0) =

0, x > l.
¥c
UXŠX ihgpdtfisdzsua tegsu t ‚ufed{hdss’~ fis”i~ x = 0 j suzuh•s’€
~i~ds r~dd ie~g u(x, 0) = 0 r suzuh•sg‘ tfieit•
∂u(x, 0)
=
∂t
(
v,
0,
0 ≤ x ≤ l,
x > l.
vu€r ie~g tegs’ yha ~i~dsij jed~dsr t = l r t = 5l X
a tegs’m
a
UXUX vu€r {i{dedzs’d fihdpusra {ihgpdtfisdzsi€
dthr dd fi}
sd” x = 0 ‚ufed{hdsm suzuh•sua ie~u tegs’ i{rt’judta gsf”rd€
x m u suzuh•sua tfieit• jtd“ izdf tegs’ eujsu X
u(x, 0) =
+x
UX¥X vu€r1 {i{dedzs’d
fihdpusra {ihgpdtfisdzsi€ tegs’m dthr 0j suzuh•}
s’€ ~i~ds jed~dsr t~d¢dsra izdf tegs’ eujs’ 0 m tfieit• izdf i{r}
t’judta gsf”rd€ ∂u(x, 0) = sin x {er Ÿi~ fisd” x = 0 tjipiysi pd‚
∂t
edsra {ded~d¢udta jyih•
jderfuh•si€ hrsrr ∂u(0, t) = 0 X
∂x
¼½¾¹½¿
UXV u(x, t) = U (cos(x − at) + cos(x + at)) X
2
0
2
UXc u(x, t) = 1 (Ψ(x+at)−Ψ(x−at)) m oyd
2
UX` u(x, t) = ϕ(x − at) + ϕ(x + at) , oyd
2


 v|x| , x ∈ [−h, h]
a
Ψ(x) = vh

 , x∈
/ [−h, h].
( a
sin πlx x ∈ [−l, l]
ϕ(x) =
0, x ∈
/ [−l, l].
UXŠ u(x, t) = 1 (Ψ(x + at) − Ψ(x − at)) m oyd Ψ(x) ¤ Ÿi gsf”ra r‚
2
g{eu|sdsra UXc™ u(x,
™
l
1
) = (Ψ(x + l) − Ψ(x − l))
a
2
X
1
5l
u(x, ) = (Ψ(x + 5l) − Ψ(x − 5l))
a
2
UXU
X
1
x − at
x + at
u(x, t) =
+
UX¥
2 1 + (x − at)2 1 + (x + at)2
(
1
1 − cos x,
u(x, t) = (Ψ(x+at)−Ψ(x−at))
Ψ(x) =
2
cos x − 1,
m oyd
ý ' ?A&15C5> D (&@(&F& > @?&FFE5&
Reujsdsrd jryuƒ
∆u = f,
¥`
x≥0
x < 0.
oyd ∆ ¤ Ÿi i{deuie lu{hutum su‚’judta geujsdsrd~ guttisuX ¬thr
gsf”ra f ≡ 0 m i geujsdsrd su‚’judta geujsdsrd~ lu{hutuX
nufrd geujsdsra ji‚srfu‘m su{er~dem {er su“i|ydsrr tu”risues’“
eut{edydhdsr€ d~{deuge’ j jdey’“ dhu“m {er rtthdyijusrr tu”risue}
s’“ Ÿhdfei~uosrs’“ {ihd€ r ji ~sior“ yegor“ {erfhuys’“ ‚uyuzu“X
¬thr rtfi~ua gsf”ra u eutt~uerjudta j ioeusrzdssi€ iphutr Ω t
oeusr”d€ S m i yi{ihsrdh•si su oeusr”d ‚uyu‘ta feudj’d gthijraX ˆi
~iog p’• gthijra jryuƒ
u
S
∂u
∂~n
= ν(M 0 , t),
= Ψ(M 0 , t),
∂u
+ σu
∂n
S
= χ(M 0 , t),
ajha‘¢r~rta gthijra~r Srer“hdm vd€~usu r ed•r~ feudj’~ gthijrd~m
tiijdtjdssiX xydt• M ¤ izfu oeusr”’ iphutr S m ∂u ¤ {eir‚jiysua {i
su{eujhdsr‘ jsd–sd€ sie~uhr f oeusr”d S X vu eu‚s’“∂~nzuta“ oeusr”’ S
~iog p’• ‚uyus’ oeusrzs’d gthijra eu‚sioi eiyuX
˜ysr~ r‚ ~diyij ed–dsra geujsdsr€ lu{hutu r guttisu ajhadta
~diy —ge•dX u‚pded~ sdtfih•fi {er~deij {er~dsdsra Ÿioi ~diyu yha
ed–dsra ‚uyuz i{edydhdsra tu”risues’“ eut{edydhdsr€ d~{deuge’ j
eu‚hrzs’“ dhu“X
­ ±®P® vu€r tu”risuesid eut{edydhdsrd d~{deuge’ j yhrs}
si~ ~duhhrzdtfi~ tde|sd t {ea~igoih•s’~ tdzdsrd~ (0 ≤ x ≤ A m
{er gthijrrm zi j tde|sd j’ydhada d{hi t ipõd~si€ {hi}
0sit•‘
≤ y ≤ eut{edydhdsra
B)
d{hiji€ Ÿsdeorr Q X ‡eusr tde|sa x = 0 r y = 0
d{hir‚ihreijus’m u oeussr x = A r y = B {iyyde|rju‘ta {er d~{d}
euged T X nd~{deugeu j fu|yi~ {i{dedzsi~ tdzdsrr tde|sa tzrudta
iyrsufiji€X
gt• u(x, y) ¤ Ÿi gsf”ram fiieua i{rt’jud eut{edydhdsrd d~{d}
euge’ j tdzdsrr tde|saX ˜su gyijhdjiead geujsdsr‘
{er 0 < x < A, 0 < y < B,
∂ u ∂ u
+
= −Q
S
0
0
2
2
∂x2
∂y 2
feudj’~ gthijra~ƒ
0
∂u(0, y)
= 0,
∂x
u(A, y) = T0 ,
∂u(x, 0)
= 0,
∂y
u(x, B) = T0 ,
oyd Q = Q [K ¤ fiŸr”rds d{hi{eijiysitr]X
K gthijra {i ipdr~ {ded~dss’~ sdiysieiys’dX iŸi~g tsuzu}
Teudj’d
hum rt{ih•‚ga ‚u~dsg u(x, y) = v(x, y) + w(x, y) m tjdyd~ Ÿg feudjg‘ ‚uyuzg
0
¥Š
f ‚uyuzd t iysieiys’~r feudj’~r gthijra~r {i {ded~dssi€ x X Sha Ÿi}
oi ‚uyuyr~ w(x, y) = αx + β X qrthu α r β {iypded~ ufm zip’ w(x, y)
gyijhdjieahu {i {ded~dssi€ x d~ |d feudj’~ gthijra~m zi r gsf”ra
ƒ
u(x, y)
(
α = 0,
⇔
β = T0 .
(
α = 0,
α A + β = T0
xsuzr w(x, y) = T X Rzr’juam zi u(x, y) = v(x, y) + T m ‚u{r–d~
feudjg‘ ‚uyuzg isitrdh•si gsf”rr v(x, y) ƒ

{er 0 < x < A, 0 < y < B,
∂ v ∂ v


+
=
−Q


∂y

 ∂x
[¥XV]
∂v(0, y)
= 0, v(A, y) = 0,
0
2
0
2
2
0
2

∂x



∂v(x,
0)


= 0,
∂y
v(x, B) = 0.
—gsf”ra v(x, y) gyijhdjiead iysieiys’~ feudj’~ gthijra~ su jtd€
oeusr”dX
ihgzdssg‘ feudjg‘ ‚uyuzg pgyd~ ed–u• ~diyi~ —ge•dm j’{ihsaa
eu‚hi|dsrd j eay {i trtd~d tiptjdss’“ gsf”r€ i{deuieu L = − d y X
dx
V] xu{r–d~ ‚uyuzg £ge~u}lrgjrhhaƒ
2
x
2
(
−y 00 = λy, 0 < x < A,
y 0 (0) = 0, y(A) = 0.
d–dsrd Ÿi€ ‚uyuzr su€ydsi [g{eu|sdsrd cXc]X „iptjdss’d zrthuƒ
m oyd µ = + πk m k = 0, 1, . . . ™ tiptjdss’d gsf”rrƒ y (x) =
λ = µ
A X ˜sr ipeu‚g‘ {ihsg‘ ieioisuh•sg‘ trtd~g j
m
= cos(µ x) k = 0, 1, . . .
{eiteustjd L [0, A; 1] X er Ÿi~ ky (x)k = A X
2 —ge•dm j’{ihsrj eu‚hi}
c] —gsf”r‘ v(x, y) pgyd~ rtfu• j jryd eayu
|dsrd {i trtd~d gsf”r€ y (x) ƒ
k
2
k
π
2
k
k
k
2
2
k
k
v(x, y) =
+∞
X
ck (y)yk (x).
—gsf”r‘ Q r‚ {euji€ zutr geujsdsra guttisu uf|d eu‚hi|r~ j
eay —ge•d {i i€ |d trtd~d gsf”r€ƒ
k=0
0
Q0 =
+∞
X
qk yk (x).
¥U
k=0
TiŸr”rds’ —ge•d su€yd~ {i {eujrhg q
zi ky (x)k = A m {ihgzr~ƒ
k
=
2
k
(Q0 , yk (x))
kyk (x)k2
X Rzr’juam
2
2
qk =
A
ZA
2 2AQ0
(π + 2πk)x
(π + 2πk)x
dx =
sin
Q0 cos
2A
A π + 2πk
2A
0
A
=
0
4Q0 (−1)k
=
.
π(1 + 2k)
Sha i{edydhdsra ‚suzdsr€ fiŸr”rdsij c (y) {iytujr~ eay’ —g}
e•d gsf”r€ v(x, y) r Q j geujsdsrd guttisu r j feudj’d gthijra {i
{ded~dssi€ y X Rzr’jua eujdstju −y (x) = λy (x) m yha fiŸr”rdsij
{ihgzr~ feudj’d ‚uyuzrƒ
c (y)
k
0
k
00
k
k
(
−λk ck + c00k = −qk 0 < y < B,
c0k (0) = 0, ck (B) = 0.
˜p¢dd ed–dsrd sdiysieiysioi yrdeds”ruh•sioi geujsdsra r~dd jryƒ
ck (y) = ck,0 (y) + c̃k (y),
oyd c (y) ¤ Ÿi ip¢dd ed–dsrd iysieiysioi geujsdsra −λ c + c = 0 m
¤ zutsid ed–dsrd sdiysieiysioi geujsdsraX
c̃ (y)vu€yd~ tsuzuhu
X Rzr’juam zi λ = µ m ‚u{r–d~ yha iy}
c
(y)
sieiysioi yrdeds”ruh•sioi geujsdsra “ueufdertrzdtfid geujsdsrdƒ
X ¬oi fiesrƒ p = µ r p = −µ X xsuzr
−µ + p = 0
k,0
k k
k
k,0
2
k
2
2
k
k
1
k
2
00
k
k
ck,0 (y) = C1 ch(µk y) + C2 sh(µk y).
bdiyi~ sdi{edydhdss’“ fiŸr”rdsij `¡ su“iyr~ c̃ (y) = q . ˜p}
λ
¢dd ed–dsrd sdiysieiysioi yrdeds”ruh•sioi geujsdsra {edytujhad}
ta j jrydƒ
k
k
k
qk
.
λk
ck (y)
ck (y) = C1 ch(µk y) + C2 sh(µk y) +
vu€yd~ ‚suzdsra C r C {er fiie’“ gsf”rr
dj’~ gthijra~ƒ
1
(
µk C2 = 0
2
C1 ch(µk B) + C2 sh(µk B) +
qk
=0
λk
¥¥
⇔
gyijhdjiea‘ feu}

C2 = 0
C1 =
−qk
λk ch(µk B)
Rzr’juam zi
16Q0 A2 (−1)k
qk
= 3
λk
π (1 + 2k)3
m {ihgzr~
16Q0 A2 (−1)k
ck (y) = 3
π (1 + 2k)3
nioyu
+∞
16Q0 A2 X (−1)k
v(x, y) =
π3
(1 + 2k)3
ch(µk y)
.
1−
ch(µk B)
ch(µk y)
1−
cos(µk x).
ch(µk B)
`] †tfi~ua gsf”ra {edytujhadta j jrydƒ u(x, y) = T + v(x, y) X
Teudjg‘ ‚uyuzg [¥XV] ~i|si ed–r• yegor~ t{itipi~X —gsf”ra v(x, y)
gyijhdjiead iysieiys’~ feudj’~ gthijra~ su jtd€ oeusr”dX iŸi~g
~i|si rtfu• j jryd eayu —ge•dm j’{ihsrj eu‚hi|dsrd {i tip}
v(x,
y)
tjdss’~ gsf”ra~ i{deuieu lu{hutuX
V] utt~ier~ ‚uyuzg su tiptjdss’d ‚suzdsra yha i{deuieu lu{hu}
tuƒ
 {er 0 < x < A, 0 < y < B,
∂ U ∂ U


+
= λU
−
k=0
0
2
2
∂x2
∂y 2
∂U (0, y)
= 0, U (A, y) = 0,

∂x



∂U (x, 0)


= 0, U (x, B) = 0.
∂y



d–dsrd Ÿi€ ‚uyuzr pgyd~ rtfu• j jryd {eir‚jdydsra yjg“ gsf”r€
m tzruam zi X(x) gyijhdjiead feudj’~ gthijra~
U
(x,
y)
=
X(x)Y
(y)
‚uyuzr {i {ded~dssi€ x m u Y (y) ¤ feudj’~ gthijra~ {i {ded~dssi€ y ƒ
X 0 (0) = 0,
X(A) = 0,
Y 0 (0) = 0,
Y (B) = 0.
nioyu gsf”ra U (x, y) pgyd gyijhdjiea• ‚uyuss’~ feudj’~ gthijra~X
iytujr~ gsf”r‘ U (x, y) = X(x)Y (y) j yrdeds”ruh•sid geuj}
sdsrdƒ
−(X 00 (x)Y (y) + X(x)Y 00 (y)) = λX(x)Y (y).
u‚ydhr~ ipd zutr Ÿioi eujdstju su X(x)Y (y) ƒ
X 00 (x) Y 00 (y)
−
+
X(x)
Y (y)
= λ.
ihgzdssid eujdstji j’{ihsadta ih•fi j i~ thgzudm dthr
r − Y (y) = λ ,
X (x)
−
=λ
X(x)
Y (y)
oyd λ r λ ¤ fistus’X er Ÿi~ λ = λ + λ X
¥W
00
00
1
1
2
2
1
2
„hdyijudh•sim gsf”rr X(x) r Y (y) ajha‘ta ed–dsra~r yjg“ ‚uyuz
£ge~u¤lrgjrhhaƒ
r
(
−X 00 = λ1 X, 0 < x < A,
X 0 (0) = 0, X(A) = 0
(
−Y 00 = λ2 Y, 0 < y < B,
Y 0 (0) = 0, Y (B) = 0.
ˆr ‚uyuzr g|d ed–ds’ [g{eu|sdsrd cXc]X „iptjdss’d zrthu r tiijd}
tjg‘¢rd r~ tiptjdss’d gsf”rr yha {deji€ r jiei€ ‚uyuz r~d‘ jryƒ
oyd µ = + πk , X (x) = cos(µ x), k = 0, 1, . . . ;
λ =µ ,
2
k
1,k
π
2
k
k
A
π
+ πm
2
λ2,m = νm ,
νm = 2
,
2B
λkm = µ2k + ν2m
k = 0, 1, . . . ; m = 0, 1, . . .
k
oyd
Y (y) = cos(ν y), m = 0, 1, . . . .
nioyu zrthu
r gsf”rr U (x, y) = cos(µ x) cos(ν y)
pgyg tiptjdss’~r zrthu~r r tiptjdss’~r
gsf”ra~r i{deuieu lu{hutu yha ed–ud~i€ ‚uyuzr j {ea~igoih•srfdX
er Ÿi~
kUkm (x, y)k2 =
ZA ZB
m
m
km
2
Ukm
(x, y)dxdy =
ZA
cos2 (µk x)dx
k
ZB
cos2 (νm y)dy =
m
AB
.
4
c] —gsf”r‘ v(x, y) {edytujr~ j jryd eayu —ge•d {i su€ydssi€ tr}
td~d tiptjdss’“ gsf”r€ i{deuieu lu{hutuƒ
0
0
0
v(x, y) =
+∞
+∞ X
X
0
ckm Ukm (x, y).
—gsf”r‘ Q eu‚hi|r~ j eay —ge•d {i i€ |d trtd~d gsf”r€ƒ
k=0 m=0
0
Q0 =
oyd q
+∞ X
+∞
X
qkm Ukm (x, y),
k=0 m=0
km
=
(Q0 , Ukm (x, y))
kUkm (x, y)k2
(Q0 , Ukm (x, y)) =
ZAZB
X _’zrthr~ tfuhaes’d {eir‚jdydsra
Q0 Ukm (x, y)dxdy = Q0
0 0
ZA
cos(µk x)dx
0
(−1)k (−1)m
(−1)k+m 4AB
=
= 2
.
µk νm
π (1 + 2k)(1 + 2m)
¥Z
ZB
0
cos(νm y)dy =
Rzr’juam zi kU (x, y)k = AB m su€yd~ q = (−1) 16 .
π (1j+jryd
2k)(1
+ 2m)
iytujr~ gsf”rr v(x, y) 4r Q m {edytujhdss’d
eayijm
j geuj}
sdsrd feudji€ ‚uyuzr [¥XV]X
k+m
2
km
km
2
0
+∞
+∞ X
X
ckm
∂ 2 Ukm ∂ 2 Ukm
+
∂x2
∂y 2
=−
+∞
+∞ X
X
qkm Ukm (x, y).
itfih•fg eu‚hi|dsrd j eay —ge•d dyrstjdssi r yha gsf”r€ U
j’{ihsa‘ta eujdstju
k=0 m=0
∂ 2 Ukm ∂ 2 Ukm
−
+
∂x2
∂y 2
i c = q yha jtd“ k = 0, 1, . . .
‚uyuzr [¥XV]λ {edytujhadta j jrydƒ
km
km
;
k=0 m=0
km (x, y)
= λkm Ukm ,
m = 0, 1, . . .
r ed–dsrd feudji€
km
+∞ X
+∞
X
qkm
Ukm (x, y) =
v(x, y) =
λ
km
m=0
k=0
+∞ +∞
(−1)k+m
64Q0 X X
=
cos(µk x) cos(νm y).
1+2m 2
1+2k 2
π4
)
+
(
)
)
(1
+
2k)(1
+
2m)((
A
B
m=0
k=0
+∞
16Q0 A2 X (−1)k
u(x, y) = T0 +
π3
(1 + 2k)3
rhr
k=0
ch(µk y)
1−
cos(µk x)
ch(µk B)
u(x, y) = T0 +
+∞ +∞
64Q0 X X
(−1)k+m
cos(µk x) cos(νm y)
+ 4
1+2k 2
1+2m 2
π
(1
+
2k)(1
+
2m)((
)
+
(
)
)
A
B
k=0 m=0
π
π
+ πk
+ πm
Q
µk = 2
νm = 2
Q0 =
K
A
B
K
X
xydt•
m
m
[ ¤ fiŸr”rds d{hi{ei}
jiysitr ~uderuhu]X
­ ±®N® vu€r tu”risuesid eut{edydhdsrd d~{deuge’ j ”r}
hrsyed (0 ≤ z ≤ H, 0 ≤ ρ ≤ R) m jde“sdd itsijusrd fiieioi {iyyde|r}
judta {er d~{deuged T m u sr|sdd itsijusrd r pifijua tieisu ¤ {er
d~{deuged T X
_jdyd~ ”rhrsyerzdtfg‘ trtd~g fiieyrsum {i~dtrj dd suzuhi j ”ds}
ed sr|sdoi itsijusra ”rhrsyeuX Rthijra ‚uyuzr ufij’m zi gsf”ram
i{rt’ju‘¢ua tu”risuesid eut{edydhdsrd d~{deuge’ j ”rhrsyed sd ‚u}
0
1
¥Y
jrtr i gohiji€ fiieyrsu’ ϕ m X dX u = u(ρ, z) X ˜su gyijhdjiead geuj}
sdsr‘ lu{hutuƒ
{er 0 < ρ < R, 0 < z < H
1 ∂
∂u
∂ u
=0
ρ
+
ρ ∂ρ
∂ρ
∂z
r feudj’~ gthijra~ƒ u(ρ, z) ioeusrzdsu {er ρ → 0 + 0 m u(R, z) = T m
m u(ρ, H) = T X
u(ρ, 0)
=
T
d–r~ Ÿg ‚uyuzg rt{ih•‚ga eu‚hi|dsrd gsf”rr j eay —ge•d {i tip}
tjdss’~ gsf”ra~ i{deuieu £ge~u}lrgjrhha L (u) = 1 ∂ ρ ∂u X
ρ ∂ ρ rtfi~g‘
∂ρ
itfih•fg feudj’d gthijra {i {ded~dssi€ ρ sdiysieiys’dm
gsf”r‘ {edytujr~ j jryd u(ρ, z) = v(ρ, z) + w(ρ, z) X xuyuyr~ w(ρ, z) =
m ioyu gsf”ra v(ρ, z) pgyd gyijhdjiea• iysieiys’~ feudj’~
=
T
gthijra~ {i ρ X xu{r–d~ feudjg‘ ‚uyuzg isitrdh•si Ÿi€ gsf”rrƒ


{er 0 < ρ < R, 0 < z < H,
1
∂v
∂ v
∂


 ρ ∂ ρ ρ ∂ ρ + ∂z = 0
ioeusrzdsu {er ρ → 0 + 0, v(R, z) = 0,
v(ρ, z)
2
2
1
1
0
ρ
1
2
2


v(ρ, 0) = 0,
u(ρ, H) = T0 − T1 .
ˆg ‚uyuzg pgyd~ ed–u• ~diyi~ —ge•dX
V] utt~ier~ ‚uyuzg £ge~u}lrgjrhhaƒ

− 1 (ρy 0 (ρ))0 = λy(ρ), 0 < ρ < R,
ρ
y(ρ)
ρ → 0 + 0, y(R) = 0.
ioeusrzdsu {er
ˆu ‚uyuzu ed–dsu u j eu‚ydhd c [{er~de cXU]X „iptjdss’d zrthu i{deui}
euƒ λ = γ , k = 1, 2, . . . m oyd γ ¤ Ÿi {ihi|rdh•s’d ed–dsra geuj}
sdsra J (γR) = 0 X „iptjdss’d gsf”rr i{deuieum tiijdtjg‘¢rd su€}
ydss’~ tiptjdss’~ zrthu~ƒ y (ρ) = J ( ρ) k = 1, 2, . . . X „rtd~u tip}
tjdss’“ gsf”r€ {ihsu r ieioisuh•su j {eiteustjd L [0, R; ρ] X Tjuy}
eu’ sie~ tiptjdss’“ gsf”r€ i{edydha‘ta {i ie~ghu~ ky (ρ)k =
X
R
(J (γ ))
2 c] —gsf”r‘
eu‚hi|r~ j eay —ge•dƒ
v(ρ, z)
k
2
k
k
0
γk
0 R
k
2
k
2
1
k
2
v(ρ, z) =
+∞
X
ck (z)yk (ρ).
Teudj’d gthijram {er Ÿi~m j’{ihsa‘taX
k=1
W\
2
—gsf”r‘ T − T r‚ feudjioi gthijra {er z = H uf|d eu‚hi|r~ j
eay —ge•d {i trtd~d su€ydss’“ tiptjdss’“ gsf”r€ƒ
0
1
+∞
X
T0 − T1 =
ϕk yk (ρ).
k=1
TiŸr”rds’ ϕ = 2(T − T ) m k = 1, 2, ... . p’hr su€yds’ j {er~ded `X`X
γ J (γ ) fiŸr”rdsij
Sha i{edydhdsra ‚suzdsr€
Ÿr eay’ {iytujr~
c
(z)
j geujsdsrd lu{hutu r j feudj’d gthijra su {ijde“sita“ ”rhrsyeu z =
r z = H X ithd {iytusijfr r rt{ih•‚ijusra tji€tj gsf”r€ y (ρ)
=
0
{ihgzr~ thdyg‘¢rd feudj’d ‚uyuzrƒ
(
{er 0 < z < H,
−λ c (z) + c (z) = 0
0
1
k
k 1
k
k
k
00
k
k k
ck (0) = 0,
γ 2
ck (H) = ϕk .
itfih•fg λ =
m i ip¢dd ed–dsrd yrdeds”ruh•sioi geujsdsra
{edytujhadta j jryd
R
k
k
ck (z) = Ak ch
γ
k
R
z + Bk sh
†t{ih•‚ga feudj’d gthijram {ihgzr~ A
m
γ
k
R
z .
ϕk
sh( γRk H)
γ 2(T0 − T1 )
k
z .
sh
ck (z) =
γk
γk J1 (γk )sh( R H)
R
k
= 0 Bk =
X xsuzr
—gsf”ra v(ρ, z) {edytujhadta j jryd eayu —ge•dƒ
v(ρ, z) =
+∞
X
γ 2(T0 − T1 )
γk
k
sh
z J0 ( ρ).
γk
γk J1 (γk )sh( R H)
R
R
`] ihgzr~ rtfi~g‘ gsf”r‘ƒ
k=1
u(ρ, z) = T1 +
+∞
X
k=1
γ 2(T0 − T1 )
γk
k
sh
z J0 ( ρ).
γk
γk J1 (γk )sh( R H)
R
R
˜jdƒ u(ρ, z) = T + X 2(T − T ) sh γ z J ( γ ρ).
γ J (γ )sh( H)
R
R
­ ±®¯® vu€r tu”risuesid
eut{edydhdsrd
d~{deuge’
j –ued
euyrgtu R m zut• {ijde“sitr S = {(r, θ, ϕ) : r = R, 0 ≤ Θ ≤ α,
fiieioi {iyyde|rjudta {er d~{deuged T m u ituh•sua
0zut•
≤ ϕ{ijde“sitr
< 2π}
¤ {er d~{deuged eujsi€ sgh‘X
WV
+∞
0
1
k=1
k 1
k
1
γk
R
k
k
0
0
_ ~iydhr pgyd~ rt{ih•‚iju• tderzdtfg‘ trtd~g fiieyrsuX Rthi}
jra ‚uyuzr ufij’m zi gsf”ra u = u(r, θ) m i{rt’ju‘¢ua tu”risuesid
eut{edydhdsrd d~{deuge’ j –ued sd ‚ujrtr i gohiji€ fiieyrsu’ ϕ X
˜su gyijhdjiead geujsdsr‘ lu{hutuƒ
1 ∂
r2 ∂r
r
2 ∂u
∂r
∂
1
∂u
+ 2
sin θ
= 0,
r sin θ ∂ θ
∂θ
feudj’~ gthijra~ƒ u(r, θ) ioeusrzdsu {er θ → 0 + 0 r {er θ → π − 0 m
ioeusrzdsu
{er
m
T ,
0≤θ≤α X
u(r, θ)
r → 0 + 0 u(R, θ) =
0 ,
α<θ≤π
i {ded~dssi€ θ gsf”ra u = u(r, θ) gyijhdjiead
dtdtjdss’~
iysieiys’~ feudj’~ gthijra~X iŸi~g ed–dsrd ‚uyuzr pgyd~ rtfu• j
jryd eayu —ge•dm j’{ihsrj eu‚hi|dsrd {i tiptjdss’~ gsf”ra~ i{deu}
ieu L (y) = − 1 d (sin θ dy ) X
sin θ dθ
dθ
V] xuyuzu £ge~u}lrgjrhha
yha Ÿioi i{deuieuƒ
0
θ

1

(sin θy 0 )0 = λy,
−
sin θ
y(θ)
0<θ<π
ioeusrzdsu {er θ → 0 + 0 r {er θ → π − 0
p’hu ed–dsu j eu‚ydhd c [{er~de cXW]X
„iptjdss’d zrthu i{deuieuƒ λ = k(k + 1) m k = 0, 1, 2, . . . m tiptjds}
s’d gsf”rrƒ y (θ) = P (cos θ), k = 0, 1, 2, . . . m oyd P (x) ¤ Ÿi ~sioizhd}
s’ ld|usyeu (P (x) = 1, P (x) = x, P (x) = (3x − 1) . . .) m fjuyeu’
sie~ tiptjdss’“ gsf”r€ƒ ky (θ)k = 2 X „iptjdss’d gsf”rr ip}
+1
eu‚g‘ {ihsg‘ ieioisuh•sg‘ trtd~g j2k{eiteustjd
X
L
[0,
π
;
sin
θ
]
c] —gsf”r‘ u(r, θ) pgyd~ rtfu• j jryd eayu —ge•dƒ
k
k
k
0
1
2
1
2
k
2
2
k
2
u(r, θ) =
+∞
X
ck (r)yk (θ).
Teudj’d gthijra {i {ded~dssi€ θ {er Ÿi~ j’{ihsa‘taX
—gsf”r‘
k=0
h(θ) =
T0 ,
0 ,
0≤θ≤α
α<θ≤π
r‚ feudjioi gthijra uf|d eu‚hi|r~ j eay —ge•d {i i€ |d trtd~d gsf}
”r€ƒ
h(θ) =
+∞
X
k=0
Wc
hk yk (θ),
{er Ÿi~ h
k
=
(h, yk )
||yk ||2
(h, yk ) =
X _’zrthr~ tfuhaes’d {eir‚jdydsra
Zπ
h(θ)yk (θ) sin θdθ =
Zα
T0 Pk (cos θ) sin θdθ.
_’{ihsrj ‚u~dsg {ded~dssi€ x = cos θ m {ihgzr~
0
0
(h, yk ) = T0
Z1
Pk (x)dx.
Sha ~sioizhdsij ld|usyeu j’{ihsadta edfgeedssid tiisi–dsrdƒ
cos α
0
0
(2k + 1)Pk (x) = Pk+1
(x) − Pk−1
(x),
k = 1, 2, . . .
r t{eujdyhrj’ eujdstju P (1) = 1 r P (−1) = (−1) k = 0, 1, . . . X iyeip}
si tji€tju ~sioizhdsij ld|usyeu i{rtus’ j V¡X _it{ih•‚gd~ta Ÿr~r
tiisi–dsra~r yha j’zrthdsra rsdoeuhu
k
T0
(h, yk ) =
2k + 1
Z1
cos α
=
k
k
0
0
(Pk+1
(x) − Pk−1
(x))dx =
T0
(Pk−1 (cos α) − Pk+1 (cos α)),
2k + 1
k = 1, 2, . . . .
˜ydh•si su€yd~
(h, y0 ) = T0
Z1
P0 (x)dx = T0
Z1
ihgzr~ d{de• fiŸr”rds’ —ge•d h ƒ
cos α
cos α
1dx = T0 (1 − cos α).
k
h0 =
T0
(1 − cos α),
2
hk =
T0
(Pk−1 (cos α) − Pk+1 (cos α)),
2
k = 1, 2, . . . .
Sha i{edydhdsra ‚suzdsr€ fiŸr”rdsij c (r) {iytujr~ eay’ —g}
e•d gsf”r€ u(r, θ) r h(θ) j geujsdsrd lu{hutu r j feudj’d gthijram ‚u}
yuss’d {i {ded~dssi€ r X ithd {iytusijfr yha fiŸr”rdsij c (r) {i}
hgzuta feudj’d ‚uyuzrƒ
k
k
(
(r2 c0k (r))0 = k(k + 1)ck (r),
r → 0 + 0,
ck (r)
ioeusrzdsu {er
W`
ck (R) = hk .
Reujsdsrd (r c (r)) = k(k + 1)c (r) ⇔ r c (r) + 2rc (r) − k(k + 1)c (r) = 0
ajhadta geujsdsrd~ ˆ€hdeuX Sha doi ed–dsra jjdyd~ sijg‘ {ded~ds}
sg‘ t m rt{ih•‚ga {iytusijfg r = e X nioyu t{eujdyhrju ”d{izfu eujdstj
X vdegysi {ihgzr• geujsdsrdm fiiei~g
cgyijhdjiead
(r) = c (e ) =gsf”ra
z (t) = z (lnƒ r)
z (t)
2 0
k
0
2 00
k
k
0
k
k
t
k
k
t
k
k
k
zk00 (t) + zk0 (t) − k(k + 1)zk (t) = 0.
xu{r–d~ yha sdoi “ueufdertrzdtfid geujsdsrd λ + λ − k(k + 1) = 0 X
Tiesr “ueufdertrzdtfioi geujsdsra ¤ Ÿi zrthu λ = −(k + 1) m λ = k X
iŸi~g ip¢r~ ed–dsrd~ yrdeds”ruh•sioi geujsdsra pgyd gsf”ra
X xsuzr
z (t) = A e
+B e
2
1
k
k
−(k+1)
k
2
k
ck (r) = Ak r−(k+1) + Bk rk ,
k = 1, 2, . . . .
†t{ih•‚ga {dejid feudjid gthijrdm {ihgzr~ A = 0 (k = 1, 2, . . .) X †‚ ji}
eioi gthijra thdygdm zi B R = h m ioyu B = h X _ riod
k
k
k
k
k
Rk
k
r k
T0
ck (r) = (Pk−1 (cos α) − Pk+1 (cos α))
,
2
R
c0 (t)
k = 1, 2, . . . .
xsuzdsrd fiŸr”rdsu su€yd~ iydh•siX utt~ier~ {er k = 0 yr}
deds”ruh•sid geujsdsrdƒ
(r2 c00 (r))0 = 0 ⇔ r2 c00 (r) = A0 .
Z
A
A0
A0
0
dr
=
−
+B0
c00 (r) = 2 c0 (t) =
r
r2
r
T0
A0 = 0 B0 = (1 − cos α)
2
T0
c0 (r) = (1 − cos α).
2
u(r, θ)
nioyu
{ihgzr~
m
X †t{ih•‚ga feudj’d gthijram
m
X xsuzr
_td fiŸr”rds’ su€yds’X †tfi~ua gsf”ra
eayu —ge•dX
˜jdƒ
{edytujr~u j jryd
+∞
r k
T0
T0 X
u(r, θ) = (1 − cos α) +
(Pk−1 (cos α) − Pk+1 (cos α))
Pk (cos θ).
2
2
R
³´µ¶· ¸¹¸º»
¥XVX vu€r tu”risuesid eut{edydhdsrd d~{deuge’ j yhrssi~ ~duhhr}
zdtfi~ tde|sd t {ea~igoih•s’~ tdzdsrd~ (0 ≤ x ≤ A m 0 ≤ y ≤ B) m dthr
oeusr tde|sa x = 0 r x = A {iyyde|rju‘ta {er d~{deuged eujsi€
mu oeusr y = 0 r y = B ¤ {er d~{deuged T X nd~{deugeu j fu|yi~
T
WŠ
k=1
1
2
{i{dedzsi~ tdzdsrr tde|sa tzrudta iyrsufiji€X
¥XcX vu€r tu”risuesid eut{edydhdsrd d~{deuge’ j yhrssi~ ~duhhr}
zdtfi~ tde|sd t {ea~igoih•s’~ tdzdsrd~ (0 ≤ x ≤ A m 0 ≤ y ≤ B)
{er gthijrrm zi j tde|sd j’ydhadta d{hi t ipõd~si€ {hisit•‘ eut}
{edydhdsra d{hiji€ Ÿsdeorr Q X nd~{deugeu su oeusa“ tde|sa {iyyde}
|rjudta eujsi€ T X _ fu|yi~ {i{dedzsi~ tdzdsrr tde|sa d~{deugeu
tzrudta iyrsufiji€X
¥X`X vu€r tu”risuesid eut{edydhdsrd d~{deuge’ j {ea~igoih•si~ {u}
euhhdhd{r{dyd (0 ≤ x ≤ A m 0 ≤ y ≤ B m 0 ≤ z ≤ C) m oeus• fiieioi z = C
{iyyde|rjudta {er d~{deuged T m u ituh•s’d oeusr {er d~{deuged
X
T
¥XŠX vu€r tu”risuesid eut{edydhdsrd d~{deuge’ j yhrssi~ ~duhhr}
zdtfi~ tde|sd t fegoh’~ tdzdsrd~ (0 ≤ ρ ≤ R, 0 ≤ ϕ ≤ 2π) m dthr
zut• {ijde“sitr tde|sa ρ = R, 0 ≤ ϕ ≤ π {iyyde|rjudta {er
d~{deuged T m u yegoua zut• ρ = R, π < ϕ < 2π ¤ {er d~{deuged
X nd~{deugeu j fu|yi~ {i{dedzsi~ tdzdsrr tde|sa tzrudta iyrsu}
T
fiji€X
¥XUX vu€r tu”risuesid eut{edydhdsrd d~{deuge’ j yhrssi~ ~duhhr}
zdtfi~ tde|sd t fegoh’~ tdzdsrd~ (0 ≤ ρ ≤ R, 0 ≤ ϕ ≤ 2π) m dthr {i}
jde“sit• tde|sa ρ = R {iyyde|rjudta {er d~{deuged T (1 + sin ϕ) X
nd~{deugeu j fu|yi~ {i{dedzsi~ tdzdsrr tde|sa tzrudta iyrsufi}
ji€X
¥X¥X vu€r tu”risuesid eut{edydhdsrd d~{deuge’ j ”rhrsyed
m su jde“sdd itsijusrd fiieioi {iyudta d{hi}
(0
≤
z
≤
H,
0
≤
ρ
≤
R)
ji€ {iif {hisitr q m u sr|sdd itsijusrd r pifijua tieisu {iyyde|r}
ju‘ta {er d~{deuged T X
¥XWX vu€r tu”risuesid eut{edydhdsrd d~{deuge’ j ”rhrsyed
m sr|sdd itsijusrd fiieioi {iyyde|rjudta
(0
≤
z
≤
H,
0
≤
ρ
≤
R)
{er d~{deuged T m pifijua tieisu ¤ {er d~{deuged T m u jde“sdd it}
sijusrd d{hir‚ihreijusiX
¥XZX vu€r tu”risuesid eut{edydhdsrd d~{deuge’ j ”rhrsyed
m jde“sdd itsijusrd fiieioi {iyyde|rjudta
(0
≤
z
≤
H,
0
≤
ρ
≤
R)
{er d~{deuged T m u sr|sdd itsijusrd r pifijua tieisu i“hu|yu‘ta
ji‚yg“i~ d~{deugeu fiieioi T X nd{hiip~ds su Ÿi€ zutr {ijde“sitr
”rhrsyeu {eirt“iyr {i ‚ufisg v•‘isuX
¥XYX vu€r tu”risuesid eut{edydhdsrd d~{deuge’ j –ued
m {ijde“sit• fiieioi r = R
(0
≤
r
≤
R,
0
≤
θ
≤
π
,
0
≤
ϕ
<
2
π
)
{iyyde|rjudta {er d~{deuged T sin θ (0 ≤ θ ≤ π) yha h‘pioi ϕ X
¥XV\X vu€r tu”risuesid eut{edydhdsrd d~{deuge’ j –ued
m {ijde“sit• fiieioi r = R
(0
≤
r
≤
R,
0
≤
θ
≤
π
,
0
≤
ϕ
<
2
π
)
{iyyde|rjudta {er d~{deuged T sin θ (0 ≤ θ ≤ π) yha h‘pioi ϕ r
0
1
0
1
0
0
0
1
0
1
0
0
0
WU
2
2
j –ued j’ydhadta d{hi t ipõd~si€ {hisit•‘ eut{edydhdsra d{hiji€
Ÿsdeorr Q(1 − r ) X
¼½¾¹½¿ R
¥XVX u(x, y) = T +
m
2(T − T ) X 1 − (−1)
1 − ch(µ B)
sh(µ y) sin(µ x)
+
ch(µ y) +
1
+∞
2
k
1
π
oyd µ = πk X
A
¥XcX u(x,
y) = T
k
k=1
k
k
sh(µk B)
k
k
k
+
(1 − (−1)k
1 − ch(µk B)
2Q0 A
ch(µk y) +
sh(µk y) sin(µk x)
+
π3
k3
sh(µk B)
k=1
πk
Q
µk =
Q0 =
K
A
K
+∞ +∞
4Q0 X X (1 − (−1)k )(1 − (−1)m )
u(x, y) = T0 + 4
sin(µk x) sin(νm y)
k 2
m 2
π
km((
)
+
(
)
))
A
B
k=1 m=1
πk
πm
µk =
νm =
A
B
Q
Q0 =
K
K
u(x, y, z) = T0 +
p
+∞
+∞ X
k
m sh( µ2 + ν2 z)
X
(1 − (−1) )(1 − (−1) )
4(T1 − T0 )
m
p k
+
×
2
2 + ν2 C)
π
km
sh(
µ
m
k
k=1 m=1
πk
πm
µk =
νm =
× sin(µk x) sin(νm y)
A
B
+∞
X
sin(k ϕ) ρ k
T1 + T0 2(T1 − T0 )
+
u(ρ, ϕ) =
2
π
k
R
k=1
ρ
u(ρ, ϕ) = T0 (1 + sin ϕ)
R
+∞
X
sh( γRk z )
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