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ÄäìëíàóÖëäàâ ÜìêçÄã, 2004, ÚÓÏ 50, ‹ 4, Ò. 1–15
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íÂıÌÓÎӄ˘ÂÒÍËÈ ËÌÒÚËÚÛÚ ÅÎÂÍËÌ„Â
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*îËÁ˘ÂÒÍËÈ Ù‡ÍÛθÚÂÚ åÓÒÍÓ‚ÒÍÓ„Ó „ÓÒÛ‰‡ÒÚ‚ÂÌÌÓ„Ó ÛÌË‚ÂÒËÚÂÚ‡ ËÏ. å.Ç. ãÓÏÓÌÓÒÓ‚‡
119992 åÓÒÍ‚‡, ãÂÌËÌÒÍË „Ó˚
E-mail:[email protected]
èÓÒÚÛÔË· ‚ ‰‡ÍˆË˛ 10.02.2004 „.
è‰·„‡ÂÚÒfl ÔË̈ËÔ ‡ÔËÓÌÓ„Ó ËÒÔÓθÁÓ‚‡ÌËfl ÒËÏÏÂÚËÈ Í‡Í ÌÓ‚˚È ÔÓ‰ıÓ‰ Í ¯ÂÌ˲ ÌÂÎËÌÂÈÌ˚ı Á‡‰‡˜, ÓÒÌÓ‚‡ÌÌ˚È Ì‡ ‡ÁÛÏÌÓÏ ÛÒÎÓÊÌÂÌËË Ï‡ÚÂχÚ˘ÂÒÍÓÈ ÏÓ‰ÂÎË. í‡ÍÓ ÛÒÎÓÊÌÂÌË ˜‡ÒÚÓ ÔË‚Ó‰ËÚ Í ‚ÓÁÌËÍÌÓ‚ÂÌ˲ ‰ÓÔÓÎÌËÚÂθÌÓÈ ÒËÏÏÂÚËË Ë, ÒΉӂ‡ÚÂθÌÓ, ÓÚÍ˚‚‡ÂÚ ‚ÓÁÏÓÊÌÓÒÚË ‰Îfl ÔÓËÒ͇ ÌÓ‚˚ı ‡Ì‡ÎËÚ˘ÂÒÍËı ¯ÂÌËÈ. ä‡ÚÍÓ ËÁÎÓÊÂÌ˚ ÓÒÌÓ‚˚ ÏÂÚÓ‰Ó‚ „ÛÔÔÓ‚Ó„Ó ‡Ì‡ÎËÁ‡ Á‡‰‡˜ ÌÂÎËÌÂÈÌÓÈ ‡ÍÛÒÚËÍË. ÇÓÁÏÓÊÌÓÒÚË ÔÓ‰ıÓ‰‡ ËÎβÒÚËÓ‚‡Ì˚ ÚÓ˜Ì˚ÏË ¯ÂÌËflÏË,
Ëϲ˘ËÏË Ò‡ÏÓÒÚÓflÚÂθÌÓ Á̇˜ÂÌË ‰Îfl ÚÂÓËË ‚ÓÎÌ.
1. ǂ‰ÂÌËÂ
ÔÓ˘Â. àÁ‚ÂÒÚÌ˚Ï ‚ ‡ÍÛÒÚËÍ ÔËÏÂÓÏ Ú‡ÍÓ„Ó
“ÔÓÎÂÁÌÓ„Ó ÛÒÎÓÊÌÂÌËfl” ÒÎÛÊËÚ Û‡‚ÌÂÌËÂ Å˛„ÂÒ‡ (ÒÏ., ̇ÔËÏÂ, [4] Ë ÔËÌflÚ˚Â Ú‡Ï Ó·ÓÁ̇˜ÂÌËfl):
å‡ÚÂχÚ˘ÂÒÍË ÏÓ‰ÂÎË ÙËÁ˘ÂÒÍËı ÔÓˆÂÒÒÓ‚, ÔÓÎÛ˜ÂÌÌ˚ ̇ ÓÒÌÓ‚Â Ó·˘Ëı Ô‰ÒÚ‡‚ÎÂÌËÈ
(“ËÁ ÔÂ‚˚ı ÔË̈ËÔÓ‚”), ˜‡ÒÚÓ ËÏÂ˛Ú Ó˜Â̸
ÒÎÓÊÌ˚È ‚ˉ. èËÏÂ‡ÏË ÏÓ„ÛÚ ÒÎÛÊËÚ¸ ÒËÒÚÂÏ˚ Û‡‚ÌÂÌËÈ ÏÂı‡ÌËÍË ÒÔÎÓ¯Ì˚ı Ò‰ ËÎË ˝ÎÂÍÚÓ‰Ë̇ÏËÍË, ÍÓÚÓ˚ ӘÂ̸ ÚÛ‰ÌÓ ¯‡Ú¸ ‚
Ó·˘ÂÏ ‚Ë‰Â Ò ÔÓÏÓ˘¸˛ ËÁ‚ÂÒÚÌ˚ı ‡Ì‡ÎËÚ˘ÂÒÍËı
Ë ˜ËÒÎÂÌÌ˚ı ÏÂÚÓ‰Ó‚, ÓÒÓ·ÂÌÌÓ ÍÓ„‰‡ ÌÛÊÌÓ
Û˜ÂÒÚ¸ ÌÂÎËÌÂÈÌÓÒÚ¸, ÌÂÓ‰ÌÓÓ‰ÌÓÒÚ¸, ̇ÒΉÒÚ‚ÂÌÌ˚Â Ë ‰Û„Ë ҂ÓÈÒÚ‚‡ ‡θÌÓÈ Ò‰˚. ÖÒÚÂÒÚ‚ÂÌÌÓ ÔÓ˝ÚÓÏÛ Ï‡ÍÒËχθÌÓ ÛÔÓ˘‡Ú¸ ÏÓ‰ÂθÌ˚ Û‡‚ÌÂÌËfl, Û˜ËÚ˚‚‡fl ÒÔˆËÙËÍÛ ÍÓÌÍÂÚÌÓÈ
Á‡‰‡˜Ë. ä‡ÊÂÚÒfl Ә‚ˉÌ˚Ï, ˜ÚÓ ‰Îfl ·ÓΠÔÓÒÚ˚ı ÏÓ‰ÂÎÂÈ Î„˜Â ÔÓ‰˚Ò͇ڸ ÔÓ‰ıÓ‰fl˘ËÈ ÏÂÚÓ‰ ¯ÂÌËfl. ìÔÓ˘ÂÌË ‰ÓÒÚË„‡ÂÚÒfl Á‡ Ò˜ÂÚ ‡·ÒÚ‡„ËÓ‚‡ÌËfl ÓÚ ÏÂÌ ÒÛ˘ÂÒÚ‚ÂÌÌ˚ı ‰ÂÚ‡ÎÂÈ
fl‚ÎÂÌËfl Ë ÙÓÍÛÒËÓ‚ÍË ‚ÌËχÌËfl ̇ ̇˷ÓÎÂÂ
‚‡ÊÌ˚ı Â„Ó Ò‚ÓÈÒÚ‚‡ı. ÉÎÛ·ÓÍÓ ÓÔËÒ‡ÌË ˉÂÈ,
ÎÂʇ˘Ëı ‚ ÓÒÌÓ‚Â Ú‡ÍÓ„Ó ÔÓ‰ıÓ‰‡, ‰‡ÌÓ Ä.Ä. Ä̉ÓÌÓ‚˚Ï, Ä.Ä. ÇËÚÚÓÏ Ë ë.ù. ï‡ÈÍËÌ˚Ï ‚ Ëı
Í·ÒÒ˘ÂÒÍÓÈ ÏÓÌÓ„‡ÙËË ÔÓ ÚÂÓËË ÍÓη‡ÌËÈ
[1]. àÒÚÓ˘ÂÒÍË ‚‡ÊÌ˚Ï ÔËÏÂÓÏ ËÒÔÓθÁÓ‚‡ÌËfl ˝ÚËı ˉÂÈ ‚ ÚÂÓËË ‚ÓÎÌ ÒÎÛÊËÚ ‡Á‚ËÚË ê.Ç.
ïÓıÎÓ‚˚Ï [2] ÏÂÚÓ‰‡ ωÎÂÌÌÓ ËÁÏÂÌfl˛˘Â„ÓÒfl
ÔÓÙËÎfl [3] , ÔÓÁ‚ÓÎË‚¯Â„Ó Á̇˜ËÚÂθÌÓ ‡Ò¯ËËÚ¸ ‚ÓÁÏÓÊÌÓÒÚË ‡Ì‡ÎËÚ˘ÂÒÍÓ„Ó ¯ÂÌËfl ÌÂÎËÌÂÈÌ˚ı ‚ÓÎÌÓ‚˚ı Á‡‰‡˜ [4] .
∂ V
∂V
∂V
------- – V ------- = Γ --------2- .
∂θ
∂z
∂θ
2
ÑÓ·‡‚ÎÂÌË ‚ Ô‡‚Û˛ ˜‡ÒÚ¸ “‚flÁÍÓ„Ó” ˜ÎÂ̇, ÔÓÔÓˆËÓ̇θÌÓ„Ó ‚˚Ò¯ÂÈ (‚ÚÓÓÈ) ÔÓËÁ‚Ó‰ÌÓÈ,
ۉ˂ËÚÂθÌ˚Ï Ó·‡ÁÓÏ Ì ÛÒÎÓÊÌflÂÚ ËÒıÓ‰ÌÓÂ
Û‡‚ÌÂÌË ÔÂ‚Ó„Ó ÔÓfl‰Í‡, ‡ ̇ÔÓÚË‚, ‰‡ÂÚ ‚ÓÁÏÓÊÌÓÒÚ¸ ¯ËÚ¸ Á‡‰‡˜Û. ä‡Í ËÁ‚ÂÒÚÌÓ, Û‡‚ÌÂÌËÂ Å˛„ÂÒ‡ Ò ÔÓÏÓ˘¸˛ ÔÂÓ·‡ÁÓ‚‡ÌËfl
∂
V = 2Γ ------ ln U
∂θ
Û‰‡ÂÚÒfl ÎË̇ËÁÓ‚‡Ú¸ Ë Ò‚ÂÒÚË Í Û‡‚ÌÂÌ˲ ÚÂÔÎÓÔÓ‚Ó‰ÌÓÒÚË ‰Îfl U. á‡ÚÂÏ ÏÓÊÌÓ ÒÓ‚Â¯ËÚ¸
Ô‰ÂθÌ˚È ÔÂÂıÓ‰ Γ
0 Ë ÔÓÎÛ˜ËÚ¸ ÙËÁ˘ÂÒÍË ÍÓÂÍÚÌÓ ¯ÂÌË ÒÓÓÚ‚ÂÚÒÚ‚Û˛˘ÂÈ Á‡‰‡˜Ë
‰Îfl “Ì‚flÁÍÓÈ” Ò‰˚.
ùÚÓÚ ÔËÏÂ, ÍÓ̘ÌÓ, ÒÎÛ˜‡ÂÌ Ë Ì ‰‡ÂÚ Ó·˘Â„Ó ÔÓ‰ıÓ‰‡. é‰Ì‡ÍÓ ÓÔ˚Ú „ÛÔÔÓ‚ÓÈ Í·ÒÒËÙË͇ˆËË ‰ËÙÙÂÂ̈ˇθÌ˚ı Û‡‚ÌÂÌËÈ ÔÓ͇Á˚‚‡ÂÚ,
˜ÚÓ Ó˜Â̸ ˜‡ÒÚÓ ÛÒÎÓÊÌÂÌË ÏÓ‰ÂÎË Ó·Ó„‡˘‡ÂÚ
 ÌÓ‚˚ÏË ÒËÏÏÂÚËflÏË. ç‡ÔÓÚË‚, ÛÔÓ˘ÂÌËÂ
ÏÓÊÂÚ ÔË‚ÂÒÚË Í ÔÓÚÂ fl‰‡ Ò‚ÓÈÒÚ‚ ÒËÏÏÂÚËË, ÔËÒÛ˘Ëı ·ÓΠӷ˘ÂÈ ÏÓ‰ÂÎË, Ë Í ÛÚ‡ÚÂ
ÙËÁ˘ÂÒÍË ‚‡ÊÌ˚ı ¯ÂÌËÈ.
ä‡Í ËÁ‚ÂÒÚÌÓ, „ÛÎflÌ˚Ï Ë ˝ÙÙÂÍÚË‚Ì˚Ï
ÏÂÚÓ‰ÓÏ Ì‡ıÓʉÂÌËfl ÒËÏÏÂÚËÈ fl‚ÎflÂÚÒfl ÚÂÓËfl
ÌÂÔÂ˚‚Ì˚ı „ÛÔÔ ãË [5, 6]. í‡‰ËˆËÓÌÌ˚È ÔÛÚ¸
ä‡Á‡ÎÓÒ¸ ·˚, ËÌÓÈ ÔÛÚ¸ – ÛÒÎÓÊÌÂÌË ÏÓ‰ÂÎË Ò
ˆÂθ˛ ÓÚ˚Ò͇ÌËfl ¯ÂÌËÈ ËÌÚÂÂÒÛ˛˘ÂÈ Ì‡Ò ·ÓΠÔÓÒÚÓÈ Á‡‰‡˜Ë – Îӄ˘ÂÒÍË ‡·ÒÛ‰ÂÌ. íÂÏ ÌÂ
ÏÂÌÂÂ, ËÌÓ„‰‡ ÒÎÓÊ̇fl ÏÓ‰Âθ ‡Ì‡ÎËÁËÛÂÚÒfl
1
à·‡„ËÏÓ‚, êÛ‰ÂÌÍÓ*
2
ËÒÔÓθÁÓ‚‡ÌËfl „ÛÔÔ ãË ‰Îfl ÓÚ˚Ò͇ÌËfl ‡Ì‡ÎËÚ˘ÂÒÍËı ¯ÂÌËÈ ‰ËÙÙÂÂ̈ˇθÌ˚ı Û‡‚ÌÂÌËÈ
ÒÓÒÚÓËÚ ‚ ÒÎÂ‰Û˛˘ÂÏ. ê‡ÒÒχÚË‚‡˛ÚÒfl ÍÓÌÍÂÚÌ˚ Û‡‚ÌÂÌËfl ËÎË ÒËÒÚÂÏ˚ Û‡‚ÌÂÌËÈ. ÑÎfl ÌËı
‚˚˜ËÒÎfl˛ÚÒfl „ÛÔÔ˚ ÒËÏÏÂÚËË. á‡ÚÂÏ ˝ÚË „ÛÔÔ˚ ËÒÔÓθÁÛ˛ÚÒfl ‰Îfl ÔÓÒÚÓÂÌËfl ˜‡ÒÚÌ˚ı ÚÓ˜Ì˚ı ¯ÂÌËÈ, Á‡ÍÓÌÓ‚ ÒÓı‡ÌÂÌËfl, ËÌ‚‡ˇÌÚÓ‚.
ÅÓΠÒÎÓÊÌÓÈ Ë ËÌÚÂÂÒÌÓÈ Ô‰ÒÚ‡‚ÎflÂÚÒfl ÔÓ·ÎÂχ “„ÛÔÔÓ‚ÓÈ Í·ÒÒËÙË͇ˆËË” Û‡‚ÌÂÌËÈ, ÒÓ‰Âʇ˘Ëı ÌÂËÁ‚ÂÒÚÌ˚ ԇ‡ÏÂÚ˚ ËÎË ÙÛÌ͈ËË.
èÓ‰ „ÛÔÔÓ‚ÓÈ Í·ÒÒËÙË͇ˆËÂÈ ÔÓÌËχÂÚÒfl ‚˚·Ó Ú‡ÍËı Ô‡‡ÏÂÚÓ‚ (ÙÛÌ͈ËÈ), ÔË ÍÓÚÓÓÏ
‰ÓÔÛÒ͇Âχfl „ÛÔÔ‡ ·ÓΠ¯ËÓ͇, ˜ÂÏ „ÛÔÔ‡
ÒËÏÏÂÚËË ËÒıÓ‰ÌÓ„Ó Ó·˘Â„Ó Û‡‚ÌÂÌËfl. ç‡ ˝ÚÓÏ
ÔÛÚË ÛÊ ÔÓÎÛ˜ÂÌÓ ÏÌÓÊÂÒÚ‚Ó Û‡‚ÌÂÌËÈ, ӷ·‰‡˛˘Ëı ÙËÁ˘ÂÒÍË ËÌÚÂÂÒÌ˚ÏË ¯ÂÌËflÏË
(ÒÏ., ̇ÔËÏÂ, [7–10]. éÔËÒ‡ÌÌ˚È ‚˚¯Â ÔÓ‰ıÓ‰
ÏÓÊÂÚ ·˚Ú¸ ̇Á‚‡Ì ‡ÔÓÒÚÂËÓÌ˚Ï, ÔÓÒÍÓθÍÛ
‡Ì‡ÎËÁËÛ˛ÚÒfl ‰‡ÌÌ˚ ÒËÒÚÂÏ˚ Û‡‚ÌÂÌËÈ.
Ç Ì‡ÒÚÓfl˘ÂÈ ‡·ÓÚ Ô‰·„‡ÂÚÒfl ÔË̈ËÔˇθÌÓ ËÌÓÈ Ë ‚ÂҸχ ÔÓÒÚÓÈ ÔÓ‰ıÓ‰ Í ‚˚fl‚ÎÂÌ˲
ÌÓ‚˚ı ÒËÏÏÂÚ˘Ì˚ı ÏÓ‰ÂÎÂÈ, ÓÒÌÓ‚‡ÌÌ˚È Ì‡
‡ÁÛÏÌÓÏ ÛÒÎÓÊÌÂÌËË ‰‡ÌÌÓÈ ÏÓ‰ÂÎË ·ÂÁ ÔÓÚÂË
 ÙËÁ˘ÂÒÍÓ„Ó ÒÓ‰ÂʇÌËfl. éÒÌÓ‚ÓÈ ˝ÚÓ„Ó ÔÓ‰ıÓ‰‡, ÍÓÚÓ˚È Ï˚ ̇Á˚‚‡ÂÏ ÔË̈ËÔÓÏ ‡ÔËÓÌÓ„Ó ËÒÔÓθÁÓ‚‡ÌËfl ÒËÏÏÂÚËÈ, fl‚ÎflÂÚÒfl ÚÂÓÂχ
ç.ï.à·‡„ËÏÓ‚‡ Ó ÔÓÂ͈Ëflı „ÛÔÔ ˝Í‚Ë‚‡ÎÂÌÚÌÓÒÚË (ÒÏ. ‡Á‰ÂÎ 4.2).
ÑÎfl ÎÛ˜¯Â„Ó ÔÓÌËχÌËfl ÔÓ·ÎÂÏ˚ ̇˜ÌÂÏ Ò
ËÁÎÓÊÂÌËfl Òڇ̉‡ÚÌ˚ı ÏÂÚÓ‰Ó‚ „ÛÔÔÓ‚Ó„Ó ‡Ì‡ÎËÁ‡ ‰ËÙÙÂÂ̈ˇθÌ˚ı Û‡‚ÌÂÌËÈ, ‡‰‡ÔÚËÓ‚‡‚
Â„Ó Í ÏÓ‰ÂÎflÏ ÚÂÓËË ÌÂÎËÌÂÈÌ˚ı ‚ÓÎÌ Ë ÌÂÎËÌÂÈÌÓÈ ‡ÍÛÒÚËÍË.
2.1. Ç˚˜ËÒÎÂÌË ÒËÏÏÂÚËÈ ‰ËÙÙÂÂ̈ˇθÌ˚ı
Û‡‚ÌÂÌËÈ
ê‡ÒÒÏÓÚËÏ ˝‚ÓβˆËÓÌÌÓ Û‡‚ÌÂÌË ‚ ˜‡ÒÚÌ˚ı ÔÓËÁ‚Ó‰Ì˚ı ‚ÚÓÓ„Ó ÔÓfl‰Í‡:
∂F
---------- ≠ 0.
∂u xx
(1)
éÔ‰ÂÎÂÌËÂ. ëÓ‚ÓÍÛÔÌÓÒÚ¸ G Ó·‡ÚËÏ˚ı
ÔÂÓ·‡ÁÓ‚‡ÌËÈ ÔÂÂÏÂÌÌ˚ı t, x, u
t = f ( t, x, u, a ), x = g ( t, x, u, a ),
u = h ( t, x, u, a ),
t = t,
x = x,
u = u,
Ë ÍÓÏÔÓÁˈËË Î˛·˚ı ‰‚Ûı ÔÓÒΉӂ‡ÚÂθÌ˚ı
ÔÂÓ·‡ÁÓ‚‡ÌËÈ, ÚÓ ÂÒÚ¸
=
t ≡ f ( t , x, u, b ) = f ( t, x, u, a + b ),
x ≡ g ( t , x, u, b ) = g ( t, x, u, a + b ),
=
u ≡ h ( t , x, u, b ) = h ( t, x, u, a + b ).
=
í‡ÍËÏ Ó·‡ÁÓÏ, „ÛÔÔ‡ G ‰ÓÔÛÒ͇ÂÚÒfl Û‡‚ÌÂÌËÂÏ (1), ÂÒÎË ÔÂÓ·‡ÁÓ‚‡ÌË (2) „ÛÔÔ˚ G ÔÂ‚ӉËÚ Î˛·Ó ¯ÂÌË u = u(t, x) Û‡‚ÌÂÌËfl (1) ‚ ¯ÂÌË u = u ( t , x ) Û‡‚ÌÂÌËfl
u t = F ( t , x, u, u x, u x x ),
(3)
„‰Â F – Ú‡ Ê ҇χfl ÙÛÌ͈Ëfl, ˜ÚÓ Ë ‚ Û‡‚ÌÂÌËË
(1).
ëӄ·ÒÌÓ ÚÂÓËË ãË, ̇ıÓʉÂÌË „ÛÔÔ˚ ÒËÏÏÂÚËË G ˝Í‚Ë‚‡ÎÂÌÚÌÓ ‚˚˜ËÒÎÂÌ˲  ËÌÙËÌËÚÂÁËχθÌÓ„Ó ÔÂÓ·‡ÁÓ‚‡ÌËfl
t ≈ t + aτ ( t, x, u ), x ≈ x + aξ ( t, x, u ),
u ≈ u + aη ( t, x, u ),
(4)
ÔÓÎÛ˜‡ÂÏÓ„Ó ËÁ Ó·˘Ëı ÙÓÏÛÎ (2) Ëı ‡ÁÎÓÊÂÌËÂÏ ‚ fl‰ íÂÈÎÓ‡ ÔÓ Ô‡‡ÏÂÚÛ a Ò Û‰ÂʇÌËÂÏ
ÚÓθÍÓ ÎËÌÂÈÌ˚ı ˜ÎÂÌÓ‚. ëΉÛfl ãË, Û‰Ó·ÌÓ ‚‚ÂÒÚË ÒËÏ‚ÓÎ ÔÂÓ·‡ÁÓ‚‡ÌËfl (4) – ‰ËÙÙÂÂ̈ˇθÌ˚È ÓÔÂ‡ÚÓ
∂
∂
∂
X = τ ( t, x, u ) ----- + ξ ( t, x, u ) ------ + η ( t, x, u ) ------ ,
∂t
∂x
∂u
(5)
‰ÂÈÒÚ‚Ë ÍÓÚÓÓ„Ó Ì‡ β·Û˛ ‰ËÙÙÂÂ̈ËÛÂÏÛ˛ ÙÛÌÍˆË˛ J(t, x, u) ‰‡ÂÚÒfl ÙÓÏÛÎÓÈ
2. ä‡ÚÍÓ ËÁÎÓÊÂÌË ÏÂÚÓ‰Ó‚
„ÛÔÔÓ‚Ó„Ó ‡Ì‡ÎËÁ‡
u t = F ( t, x, u, u x, u xx ),
‰ÂÊËÚ ‚Ò ӷ‡ÚÌ˚ ÔÂÓ·‡ÁÓ‚‡ÌËfl, ÚÓʉÂÒÚ‚ÂÌÌÓ ÔÂÓ·‡ÁÓ‚‡ÌËÂ
(2)
Á‡‚ËÒfl˘‡fl ÓÚ ÌÂÔÂ˚‚ÌÓ„Ó Ô‡‡ÏÂÚ‡ a, ̇Á˚‚‡ÂÚÒfl Ó‰ÌÓÔ‡‡ÏÂÚ˘ÂÒÍÓÈ „ÛÔÔÓÈ, ‰ÓÔÛÒ͇ÂÏÓÈ
Û‡‚ÌÂÌËÂÏ (1), ËÎË „ÛÔÔÓÈ ÒËÏÏÂÚËË Û‡‚ÌÂÌËfl (1), ÂÒÎË Û‡‚ÌÂÌË (1) ÒÓı‡ÌflÂÚ Ò‚Ó˛ ÙÓÏÛ
‚ ÌÓ‚˚ı ÔÂÂÏÂÌÌ˚ı t , x , u , ‡ Ú‡ÍÊ ÂÒÎË G ÒÓ-
∂J
∂J
∂J
X ( J ) = τ ( t, x, u ) ------ + ξ ( t, x, u ) ------ + η ( t, x, u ) ------ . (6)
∂t
∂x
∂u
éÔÂ‡ÚÓ (5) ̇Á˚‚‡˛Ú Ú‡ÍÊ ËÌÙËÌËÚÂÁËχθÌ˚Ï ÓÔÂ‡ÚÓÓÏ ËÎË „ÂÌÂ‡ÚÓÓÏ „ÛÔÔ˚ G.
èÂÓ·‡ÁÓ‚‡ÌË (2), Óڂ˜‡˛˘Â ÓÔÂ‡ÚÓÛ
(5), ̇ıÓ‰ËÚÒfl Í‡Í ¯ÂÌË Û‡‚ÌÂÌËÈ ãË
dx
dt
------ = τ ( t , x, u ), ------ = ξ ( t , x, u ),
da
da
du
------ = η ( t , x, u )
da
(7)
Ò Ì‡˜‡Î¸Ì˚ÏË ÛÒÎÓ‚ËflÏË
t
a=0
= t,
x
a=0
= x,
u
a=0
= u.
é·‡ÚËÏÒfl ÚÂÔÂ¸ ÒÌÓ‚‡ Í Û‡‚ÌÂÌ˲ (3). Ç˚‡ÊÂÌËfl ‰Îfl ÔÓËÁ‚Ó‰Ì˚ı u t , u x , u x x ‚ Û‡‚ÌÂÌËË
(3) ÔÓÎÛ˜‡˛ÚÒfl Ò ÔÓÏÓ˘¸˛ ÔÂÓ·‡ÁÓ‚‡ÌËfl (2),
‡ÒÒχÚË‚‡ÂÏÓ„Ó Í‡Í Ó·˚˜Ì‡fl Á‡ÏÂ̇ ÔÂÂÏÂÌÌ˚ı. á‡ÚÂÏ, ‡Á·„‡fl ‚˚‡ÊÂÌËfl ‰Îfl ˝ÚËı ÔÓËÁÄäìëíàóÖëäàâ ÜìêçÄã ÚÓÏ 50 ‹ 4 2004
èË̈ËÔ ‡ÔËÓÌÓ„Ó ËÒÔÓθÁÓ‚‡ÌËfl ÒËÏÏÂÚËÈ
‚Ó‰Ì˚ı ‚ fl‰ ÔÓ Ô‡‡ÏÂÚÛ a, ÔÓÎÛ˜‡ÂÏ ËÌÙËÌËÚÂÁËχθÌÛ˛ ÙÓÏÛ ÔÂÓ·‡ÁÓ‚‡ÌËÈ
u t ≈ u t + aς 0 ( t, x, u, u t, u x ),
(8)
u x x ≈ u xx + aς 2 ( t, x, u, u t, u x, u tx, u xx ),
„‰Â ÙÛÌ͈ËË ς0, ς1, ς2 ‰‡˛ÚÒfl ÒÎÂ‰Û˛˘ËÏË “ÙÓÏÛ·ÏË ÔÓ‰ÓÎÊÂÌËfl”
ς 0 = D t ( η ) – u t D t ( τ ) – u x D t ( ξ ),
ς 1 = D x ( η ) – u t D x ( τ ) – u x D x ( ξ ),
(9)
ς 2 = D x ( ς 1 ) – u xx D x ( ξ ) – u tx D x ( τ ).
á‰ÂÒ¸ Dt Ë Dx Ó·ÓÁ̇˜‡˛Ú ÔÓÎÌ˚ ÔÓËÁ‚Ó‰Ì˚ ÔÓ
t Ë x:
∂
∂
∂
∂
D t = ----- + u t ------ + u tt ------- + u tx -------- ,
∂u
∂u
∂u
∂t
t
x
á‡ÏÂÚËÏ, ˜ÚÓ Û‡‚ÌÂÌË (10) ‰ÓÎÊÌÓ ‚˚ÔÓÎÌflÚ¸Òfl ÚÓʉÂÒÚ‚ÂÌÌÓ ÔÓ ÓÚÌÓ¯ÂÌ˲ ÍÓ ‚ÒÂÏ ‚ıÓ‰fl˘ËÏ ‚ ÌÂ„Ó ÔÂÂÏÂÌÌ˚Ï t, x, u, ux, uxx, utx; ÔÓÒΉÌË ‰ÓÎÊÌ˚ ‡ÒÒχÚË‚‡Ú¸Òfl Í‡Í 6 ÌÂÁ‡‚ËÒËÏ˚ı
ÔÂÂÏÂÌÌ˚ı. Ç ÂÁÛθڇÚ ÓÔ‰ÂÎfl˛˘Â Û‡‚ÌÂÌË ‡ÒÔ‡‰‡ÂÚÒfl ̇ ÒËÒÚÂÏÛ ÌÂÒÍÓθÍËı Û‡‚ÌÂÌËÈ, Ë ÔÓÎÛ˜‡ÂÚÒfl ÔÂÂÓÔ‰ÂÎÂÌ̇fl ÒËÒÚÂχ (Ó̇
ÒÓ‰ÂÊËÚ ·Óθ¯Â Û‡‚ÌÂÌËÈ, ˜ÂÏ 3 ÌÂËÁ‚ÂÒÚÌ˚ı
ÙÛÌ͈ËË τ, ξ, η). èÓ˝ÚÓÏÛ Ô‡ÍÚ˘ÂÒÍË ‚Ò„‰‡ ÓÔ‰ÂÎfl˛˘Â Û‡‚ÌÂÌË ۉ‡ÂÚÒfl ΄ÍÓ ¯ËÚ¸.
ÑÎfl ÛÔÓ˘ÂÌËfl ‡Ò˜ÂÚÓ‚ ‚ÓÒÔÓθÁÛÂÏÒfl ÒÎÂ‰Û˛˘ÂÈ ÎÂÏÏÓÈ.
t = f (t, a),
èÓ‰ÒÚ‡Ìӂ͇ (4) Ë (8) ‚ Û‡‚ÌÂÌË (3) ‰‡ÂÚ
u t – F ( t , x, u, u x, u x x ) ≈ u t – F ( t, x, u, u x, u xx ) +
∂F
∂F
∂F
∂F
∂F
+ a  ς 0 – -------- ς 1 – ---------- ς 2 – ------τ – ------ξ – ------η .

∂u xx
∂t
∂x
∂u 
∂u x
èÓ˝ÚÓÏÛ, ‚ ÒËÎÛ Û‡‚ÌÂÌËfl (1), Û‡‚ÌÂÌË (3) ÔË‚Ó‰ËÚ Í ÛÒÎӂ˲
(10)
„‰Â ut Á‡ÏÂÌÂÌÓ Ì‡ F(t, x, u, ux, uxx) ‚ ‚˚‡ÊÂÌËflı
‰Îfl ς0, ς1, ς2.
ì‡‚ÌÂÌË (10) ÓÔ‰ÂÎflÂÚ ‚Ò ËÌÙËÌËÚÂÁËχθÌ˚ ÒËÏÏÂÚËË Û‡‚ÌÂÌËfl (1) Ë ÔÓ˝ÚÓÏÛ Ì‡Á˚‚‡ÂÚÒfl ÓÔ‰ÂÎfl˛˘ËÏ Û‡‚ÌÂÌËÂÏ. Ö„Ó Û‰Ó·ÌÓ
Á‡ÔËÒ‡Ú¸ ‚ ÍÓÏÔ‡ÍÚÌÓÈ ÙÓÏÂ
X [ u t – F ( t, x, u, u x, u xx ) ] = 0.
k
X1, …, Xr, ÚÓ [Xi, Xj] = c ij Xk, „‰Â c ij – ˜ËÒ·, ̇Á˚‚‡ÂÏ˚ ÒÚÛÍÚÛÌ˚ÏË ÍÓÌÒÚ‡ÌÚ‡ÏË ‡Î„·˚ ãË Lr.
ãÂÏχ. Ç ÒÎÛ˜‡Â Û‡‚ÌÂÌËfl ‚ˉ‡ (1) ÔÂÓ·‡ÁÓ‚‡ÌË ÒËÏÏÂÚËË (2) ËÏÂÂÚ ‚ˉ
∂
∂
∂
∂
D x = ------ + u x ------ + u tx ------- + u xx -------- .
∂u
∂u t
∂u x
∂x
∂F
∂F
∂F
∂F
∂F
ς 0 – -------- ς 1 – ---------- ς 2 – ------τ – ------ξ – ------η = 0,
∂u xx
∂t
∂x
∂u
∂u x
ÌËfl fl‚ÎflÂÚÒfl ‡Î„·ÓÈ ãË. Ç ˜‡ÒÚÌÓÒÚË, ÂÒÎË L
ÂÒÚ¸ ÍÓ̘ÌÓÏÂÌÓ ÔÓÒÚ‡ÌÒÚ‚Ó Lr Ò ·‡ÁËÒÓÏ
k
u x ≈ u x + aς 1 ( t, x, u, u t, u x ),
3
(11)
á‰ÂÒ¸ X ÓÁ̇˜‡ÂÚ ÔÓ‰ÓÎÊÂÌË ÓÔÂ‡ÚÓ‡ (5) ̇
ÔÓËÁ‚Ó‰Ì˚ ÔÂ‚Ó„Ó Ë ‚ÚÓÓ„Ó ÔÓfl‰Í‡:
∂
∂
∇
∂
∂
∂
X = τ ----- + ξ ------ + η ------ + ς 0 ------- + ς 1 -------- + ς 2 ---------- .
∂t
∂x
∂u
∂u t
∂u x
∂u xx
éÔ‰ÂÎfl˛˘Â Û‡‚ÌÂÌË (10) (ËÎË Â„Ó ˝Í‚Ë‚‡ÎÂÌÚ (11)) fl‚ÎflÂÚÒfl Ó‰ÌÓÓ‰Ì˚Ï ÎËÌÂÈÌ˚Ï
Û‡‚ÌÂÌËÂÏ ‚ ˜‡ÒÚÌ˚ı ÔÓËÁ‚Ó‰Ì˚ı ‚ÚÓÓ„Ó ÔÓfl‰Í‡ ‰Îfl ÌÂËÁ‚ÂÒÚÌ˚ı ÙÛÌ͈ËÈ τ, ξ, η. èÓ˝ÚÓÏÛ
ÏÌÓÊÂÒÚ‚Ó L ‚ÒÂı ¯ÂÌËÈ ÓÔ‰ÂÎfl˛˘Â„Ó Û‡‚ÌÂÌËfl Ó·‡ÁÛÂÚ ‚ÂÍÚÓÌÓ ÔÓÒÚ‡ÌÒÚ‚Ó. ÅÓÎÂÂ
ÚÓ„Ó, ÓÌÓ Á‡ÏÍÌÛÚÓ ÓÚÌÓÒËÚÂθÌÓ ‚ÁflÚËfl ÍÓÏÏÛÚ‡ÚÓ‡, ÚÓ ÂÒÚ¸ ÔÓÒÚ‡ÌÒÚ‚Ó L ‚ÏÂÒÚÂ Ò ÓÔÂ‡ÚÓ‡ÏË X1, X2 ∈ L ÒÓ‰ÂÊËÚ Ú‡ÍÊ Ëı ÍÓÏÏÛÚ‡ÚÓ
[X1, X2] = X1X2 – X2X1. ùÚÓ Ò‚ÓÈÒÚ‚Ó ÓÁ̇˜‡ÂÚ, ˜ÚÓ
ÏÌÓÊÂÒÚ‚Ó ‚ÒÂı ¯ÂÌËÈ ÓÔ‰ÂÎfl˛˘Â„Ó Û‡‚ÌÂÄäìëíàóÖëäàâ ÜìêçÄã ÚÓÏ 50 ‹ 4 2004
x = g(t, x, u, a),
u = h(t, x, u, a). (12)
ùÚÓ ÓÁ̇˜‡ÂÚ, ˜ÚÓ ËÌÙËÌËÚÂÁËχθÌ˚ ÒËÏÏÂÚËË ÏÓÊÌÓ ËÒ͇ڸ ‚ ‚ˉÂ
∂
∂
∂
X = τ ( t ) ----- + ξ ( t, x, u ) ------ + η ( t, x, u ) ------ .
∂t
∂x
∂u
(13)
ÑÓ͇Á‡ÚÂθÒÚ‚Ó. Ç˚‰ÂÎËÏ ‚ ÓÔ‰ÂÎfl˛˘ÂÏ
Û‡‚ÌÂÌËË (10) ˜ÎÂÌ˚, ÒÓ‰Âʇ˘Ë ÔÂÂÏÂÌÌÛ˛
utx. ä‡Í ÒΉÛÂÚ ËÁ ÙÓÏÛÎ ÔÓ‰ÓÎÊÂÌËfl (9), utx ÒÓ‰ÂÊËÚÒfl ÚÓθÍÓ ‚ ς2, ‡ ËÏÂÌÌÓ, ‚ ÔÓÒΉÌÂÏ Â„Ó
Ò·„‡ÂÏÓÏ utxDx(τ). í‡Í Í‡Í ÓÔ‰ÂÎfl˛˘Â Û‡‚ÌÂÌË (10) ‚˚ÔÓÎÌflÂÚÒfl ÚÓʉÂÒÚ‚ÂÌÌÓ ÔÓ ‚ÒÂÏ ÔÂÂÏÂÌÌ˚Ï t, x, u, ux, uxx, utx, ÏÓÊÌÓ Á‡Íβ˜ËÚ¸, ˜ÚÓ
Dx(τ) ≡ τx + uxτu = 0, ÓÚÍÛ‰‡ τx = τu = 0. í‡ÍËÏ Ó·‡ÁÓÏ, τ = τ(t) Ë „ÂÌÂ‡ÚÓ (5) ÔËÌËχÂÚ ÙÓÏÛ (13),
˜ÚÓ Ë ‰Ó͇Á˚‚‡ÂÚ ÚÂÓÂÏÛ. àÚ‡Í, ÙÓÏÛÎ˚ ÔÓ‰ÓÎÊÂÌËfl (9) Á‡Ô˯ÛÚÒfl ÒÎÂ‰Û˛˘ËÏ Ó·‡ÁÓÏ:
ς 0 = D t ( η ) – u x D t ( ξ ) – τ' ( t )u t ,
ς 1 = D x ( η ) – u x D x ( ξ ),
ς 2 = D x ( ς 1 ) – u xx D x ( ξ ) ≡
(14)
≡ D x ( η ) – u x D x ( ξ ) – 2u xx D x ( ξ ).
2
2
2.2. íÓ˜Ì˚ ¯ÂÌËfl, ÔÓÎÛ˜‡ÂÏ˚Â
Ò ÔÓÏÓ˘¸˛ „ÛÔÔ ÒËÏÏÂÚËË
åÂÚÓ‰˚ „ÛÔÔÓ‚Ó„Ó ‡Ì‡ÎËÁ‡ ÓÚÍ˚‚‡˛Ú ‰‚‡
ÓÒÌÓ‚Ì˚ı ÔÛÚË ÍÓÌÒÚÛËÓ‚‡ÌËfl ÚÓ˜Ì˚ı ¯ÂÌËÈ: „ÛÔÔÓ‚˚ ÔÂÓ·‡ÁÓ‚‡ÌËfl ËÁ‚ÂÒÚÌ˚ı ¯ÂÌËÈ Ë Ì‡ıÓʉÂÌË ËÌ‚‡ˇÌÚÌ˚ı ¯ÂÌËÈ.
ÉÛÔÔÓ‚˚ ÔÂÓ·‡ÁÓ‚‡ÌËfl ËÁ‚ÂÒÚÌ˚ı ¯ÂÌËÈ. ùÚÓÚ ÔÛÚ¸ ÓÒÌÓ‚‡Ì ̇ ÚÓÏ Ù‡ÍÚÂ, ˜ÚÓ „ÛÔÔ‡
ÒËÏÏÂÚËË ÔÂÓ·‡ÁÛÂÚ Î˛·Ó ¯ÂÌË ËÒÒΉÛÂÏÓ„Ó Û‡‚ÌÂÌËfl ‚ ¯ÂÌË ÚÓ„Ó Ê Û‡‚ÌÂÌËfl.
à·‡„ËÏÓ‚, êÛ‰ÂÌÍÓ*
4
àÏÂÌÌÓ, ÔÛÒÚ¸ (2) ÂÒÚ¸ „ÛÔÔ‡ ÔÂÓ·‡ÁÓ‚‡ÌËÈ
ÒËÏÏÂÚËË Û‡‚ÌÂÌËfl (1) Ë ÔÛÒÚ¸ ÙÛÌ͈Ëfl
u = Φ ( t, x )
éÔ‰ÂÎfl˛˘Â Û‡‚ÌÂÌË (10) ËÏÂÂÚ ‚ˉ
ÂÒÚ¸ ¯ÂÌË Û‡‚ÌÂÌËfl (1). èÓÒÍÓθÍÛ (2) ÂÒÚ¸
ÔÂÓ·‡ÁÓ‚‡ÌË ÒËÏÏÂÚËË, ¯ÂÌË (12) ÏÓÊÂÚ
·˚Ú¸ Á‡ÔËÒ‡ÌÓ ‚ ÌÓ‚˚ı ÔÂÂÏÂÌÌ˚ı
u = Φ ( t , x ).
á‡ÏÂÌflfl Á‰ÂÒ¸ u , t , x Ò ÔÓÏÓ˘¸˛ (2), ÔÓÎÛ˜ËÏ
h ( t, x, u, a ) = Φ ( f ( t, x, u, a ), g ( t, x, u, a ) ).
(17)
¯ËÚ¸ Û‡‚ÌÂÌË (17) ÓÚÌÓÒËÚÂθÌÓ u Ë ÔÓ‰ÒÚ‡‚ËÚ¸ ÔÓÎÛ˜ÂÌÌÓ ‚˚‡ÊÂÌË ‚ Û‡‚ÌÂÌË (1). ä‡Í
ÔÓ͇Á‡Î ¢ ãË, ‚ ÂÁÛθڇÚ ÔÓÎÛ˜‡ÂÚÒfl Ó·˚ÍÌÓ‚ÂÌÌÓ ‰ËÙÙÂÂ̈ˇθÌÓ Û‡‚ÌÂÌË ‰Îfl ÌÂËÁ‚ÂÒÚÌÓÈ ÙÛÌ͈ËË φ(λ) Ó‰ÌÓÈ ÔÂÂÏÂÌÌÓÈ. ùÚ‡
Ôӈ‰Û‡ ÛÏÂ̸¯‡ÂÚ ˜ËÒÎÓ ÌÂÁ‡‚ËÒËÏ˚ı ÔÂÂÏÂÌÌ˚ı ̇ ‰ËÌˈÛ. чθÌÂȯ ÛÔÓ˘ÂÌË ÏÓÊÂÚ ·˚Ú¸ ‰ÓÒÚË„ÌÛÚÓ ‡ÒÒÏÓÚÂÌËÂÏ ËÌ‚‡ˇÌÚÌ˚ı ¯ÂÌËÈ, ÔÓÓʉ‡ÂÏ˚ı ‰‚ÛÏfl ËÌÙËÌËÚÂÁËχθÌ˚ÏË ÒËÏÏÂÚËflÏË.
3. ì‡‚ÌÂÌËÂ Å˛„ÂÒ‡ Í‡Í ÔËÏÂ
åÂÚÓ‰˚, ÓÔËÒ‡ÌÌ˚ ‚ Ô‰˚‰Û˘ÂÏ ‡Á‰ÂÎÂ,
ËÎβÒÚËÓ‚‡Ì˚ ÌËÊ ̇ ÔËÏÂ Û‡‚ÌÂÌËfl
Å˛„ÂÒ‡ [11]
u t = uu x + u xx .
(18)
(19)
„‰Â ς0, ς1, ς2 ‰‡˛ÚÒfl ÙÓÏÛ·ÏË (14). Ç˚‰ÂÎËÏ Ë
ÔË‡‚ÌflÂÏ ÌÛβ ˜ÎÂÌ˚, ÒÓ‰Âʇ˘Ë uxx. àÏÂfl ‚
‚ˉÛ, ˜ÚÓ ut ‰ÓÎÊÌÓ ·˚Ú¸ Á‡ÏÂÌÂÌÓ Ì‡ uux + uxx, Ë
ÔÓ‰ÒÚ‡‚Îflfl ‚ ς2 ‚˚‡ÊÂÌËfl
D x ( ξ ) = D x ( ξ x + ξu u x ) =
2
= ξ u u xx + ξ uu u x + 2ξ xu u x + ξ xx ,
2
D x ( η ) = D x ( η x + ηu u x ) =
2
(20)
= η u u xx + η uu u x + 2η xu u x + η xx ,
2
ÔˉÂÏ Í ÒÎÂ‰Û˛˘ÂÏÛ Û‡‚ÌÂÌ˲:
2ξ u u x + 2ξ x – τ' ( t ) = 0.
éÌÓ ‰‡ÂÚ ‰‚‡ Û‡‚ÌÂÌËfl, ‡ ËÏÂÌÌÓ, ξu = 0 Ë 2ξx –
τ'(t) = 0. ëΉӂ‡ÚÂθÌÓ, ξ Á‡‚ËÒËÚ ÚÓθÍÓ ÓÚ t, x Ë
ËÏÂÂÚ ‚ˉ
1
ξ = --- τ' ( t )x + p ( t ).
2
(21)
2
àÁ (21) ÒΉÛÂÚ D x (ξ) = 0. íÂÔÂ¸ ÓÔ‰ÂÎfl˛˘Â Û‡‚ÌÂÌË (19) Ò‚Ó‰ËÚÒfl Í ÒÎÂ‰Û˘ÂÈ ÙÓÏÂ:
1
1
2
u x η uu + --- τ' ( t )u + --- τ'' ( t )x + p' ( t ) + 2η xu + η u x +
2
2
(16)
á‡ÚÂÏ ÒΉÛÂÚ ‚˚‡ÁËÚ¸ Ó‰ËÌ ËÁ ËÌ‚‡ˇÌÚÓ‚ ͇Í
ÙÛÌÍˆË˛ ‰Û„Ó„Ó
µ = φ ( λ ),
ς 0 – ς 2 – uς 1 – ηu x = 0,
(15)
ê¯˂ ˝ÚÓ Û‡‚ÌÂÌË ÓÚÌÓÒËÚÂθÌÓ u, ÔÓÎÛ˜ËÏ
Ó‰ÌÓÔ‡‡ÏÂÚ˘ÂÒÍÓ (Ò Ô‡‡ÏÂÚÓÏ a) ÒÂÏÂÈÒÚ‚Ó ÌÓ‚˚ı ¯ÂÌËÈ Û‡‚ÌÂÌËfl (1). ëΉӂ‡ÚÂθÌÓ,
β·Ó ËÁ‚ÂÒÚÌÓ ¯ÂÌË ÏÓÊÂÚ ÒÎÛÊËÚ¸ ËÒÚÓ˜ÌËÍÓÏ ÏÌÓ„ÓÔ‡‡ÏÂÚ˘ÂÒÍÓ„Ó Í·ÒÒ‡ ÌÓ‚˚ı ¯ÂÌËÈ ÔË ÛÒÎÓ‚ËË, ˜ÚÓ ‡ÒÒχÚË‚‡ÂÏÓ ‰ËÙÂÂ̈ˇθÌÓ Û‡‚ÌÂÌË ‰ÓÔÛÒ͇ÂÚ ÏÌÓ„ÓÔ‡‡ÏÂÚ˘ÂÒÍÛ˛ „ÛÔÔÛ ÒËÏÏÂÚËË. Ç ÒÎÂ‰Û˛˘ÂÏ
‡Á‰ÂΠÓÔËÒ‡Ì̇fl Ôӈ‰Û‡ ËÎβÒÚËÓ‚‡Ì‡ ̇
ÔËÏÂ Û‡‚ÌÂÌËfl Å˛„ÂÒ‡.
àÌ‚‡ˇÌÚÌ˚ ¯ÂÌËfl. ÖÒÎË „ÛÔÔÓ‚Ó ÔÂÓ·‡ÁÓ‚‡ÌË ÔÂ‚ӉËÚ ¯ÂÌË ‚ Ò·fl, ÔÓÎÛ˜‡ÂÏ
Ú‡Í Ì‡Á˚‚‡ÂÏÓ ËÌ‚‡ˇÌÚÌÓ ¯ÂÌËÂ. èË ËÁ‚ÂÒÚÌÓÈ ËÌÙËÌËÚÂÁËχθÌÓÈ ÒËÏÏÂÚËË (5) Û‡‚ÌÂÌËfl (1), ËÌ‚‡ˇÌÚÌÓ ¯ÂÌË ÔÓÎÛ˜‡ÂÚÒfl ÒÎÂ‰Û˛˘ËÏ Ó·‡ÁÓÏ. Ç˚˜ËÒÎfl˛ÚÒfl ‰‚‡ ÌÂÁ‡‚ËÒËÏ˚ı
ËÌ‚‡ˇÌÚ‡ J1 = λ(t, x) Ë J2 = µ(t, x, u) ÔÛÚÂÏ ¯ÂÌËfl Û‡‚ÌÂÌËfl X(J) = 0 (ÒÏ. (6)) ËÎË Â„Ó ı‡‡ÍÚÂËÒÚ˘ÂÒÍÓÈ ÒËÒÚÂÏ˚
du
dt
dx
--------------------- = --------------------- = ---------------------- .
η ( t , x, u )
τ ( t , x, u )
ξ ( t , x, u )
3.1. Ç˚˜ËÒÎÂÌË ËÌÙËÌËÚÂÁËχθÌ˚ı
ÒËÏÏÂÚËÈ
+ uη x + η xx – η t = 0
Ë ‡ÒÔ‡‰‡ÂÚÒfl ̇ ÚË Û‡‚ÌÂÌËfl
η uu = 0,
1
1
--- τ' ( t )u + --- τ'' ( t )x + p' ( t ) + 2η xu + η = 0,
2
2
uη x + η xx – η t = 0.
(22)
èÂ‚Ó Û‡‚ÌÂÌË (22) ‰‡ÂÚ η = σ(t, x)u + µ(t, x),
ÔË ˝ÚÓÏ ‚ÚÓÓ Û‡‚ÌÂÌË (22) ÒÚ‡ÌÓ‚ËÚÒfl Ú‡ÍËÏ:
 1--- τ' ( t ) + σ u + 1--- τ'' ( t )x + p' ( t ) + 2σ + µ = 0,
x
2

2
ÓÚÍÛ‰‡
1
σ = – --- τ' ( t ),
2
1
µ = – --- τ'' ( t )x – p' ( t ).
2
í‡ÍËÏ Ó·‡ÁÓÏ, Ï˚ ËÏÂÂÏ
1
1
η = – --- τ' ( t )u – --- τ'' ( t )x – p' ( t ).
2
2
(23)
ÄäìëíàóÖëäàâ ÜìêçÄã ÚÓÏ 50 ‹ 4 2004
èË̈ËÔ ‡ÔËÓÌÓ„Ó ËÒÔÓθÁÓ‚‡ÌËfl ÒËÏÏÂÚËÈ
ç‡ÍÓ̈, ÔÓ‰ÒÚ‡Ìӂ͇ (23) ‚ ÚÂڸ Û‡‚ÌÂÌËÂ
(22) ‰‡ÂÚ
1
--- τ''' ( t )x + p'' ( t ) = 0,
2
τ ( t ) = C 1 t + 2C 2 t + C 3 ,
2
τ(t) = C 1 t + 2C 2 t + C 3 ,
ξ = C 1 tx + C 2 x + C 4 t + C 5 ,
η = – ( C 1 t + C 2 )u – C 1 x – C 4 .
éÌÓ ÒÓ‰ÂÊËÚ ÔflÚ¸ ÔÓËÁ‚ÓθÌ˚ı ÍÓÌÒÚ‡ÌÚ Ci.
ùÚÓ ÓÁ̇˜‡ÂÚ, ˜ÚÓ ËÌÙËÌËÚÂÁËχθÌ˚ ÒËÏÏÂÚËË Û‡‚ÌÂÌËfl Å˛„ÂÒ‡ (18) Ó·‡ÁÛ˛Ú ÔflÚËÏÂÌÛ˛ ‡Î„·Û ãË, “̇ÚflÌÛÚÛ˛” ̇ ÒÎÂ‰Û˛˘Ë ÎËÌÂÈÌÓ ÌÂÁ‡‚ËÒËÏ˚ ÓÔÂ‡ÚÓ˚:
∂
∂
∂
X 2 = ------ , X 3 = t ------ – ------ ,
∂x ∂u
∂x
∂
∂
∂
X 4 = 2t ----- + x ------ – u ------ ,
∂t
∂x
∂u
∂
∂
2∂
X 5 = t ----- + tx ------ – ( x + tu ) ------ .
∂t
∂x
∂u
∂
X 1 = ----- ,
∂t
dx
------ = ξ ( t , x ),
da
(24)
du
------ = η ( t , x, u ), (25)
da
ÍÓÚÓ‡fl ۉӷ̇ ‰Îfl ÔÓÒΉӂ‡ÚÂθÌÓ„Ó ËÌÚ„ËÓ‚‡ÌËfl Ò ËÒÔÓθÁÓ‚‡ÌËÂÏ ÒÓÓÚ‚ÂÚÒÚ‚Û˛˘Ëı ̇˜‡Î¸Ì˚ı ÛÒÎÓ‚ËÈ (ÒÏ. (7))
ê‡ÒÒÏÓÚËÏ, ̇ÔËÏÂ, „ÂÌÂ‡ÚÓ X5 ËÁ ̇·Ó‡
(24). ì‡‚ÌÂÌËfl ãË (25) ‰Îfl ˝ÚÓ„Ó ÒÎÛ˜‡fl ËϲÚ
‚ˉ
dt
2
------ = t ,
da
dx
------ = t x,
da
x
x = -------------- .
1 – at
du
------ = – ( x + t u ) .
da
àÌÚ„ËÓ‚‡ÌË ÔÂ‚Ó„Ó Û‡‚ÌÂÌËfl ‰‡ÂÚ
1
t = – --------------- .
a + C1
í·Ûfl ‚ ÒÓÓÚ‚ÂÚÒÚ‚ËË Ò Ì‡˜‡Î¸Ì˚Ï ÛÒÎÓ‚ËÂÏ ‰Îfl
Û‡‚ÌÂÌËÈ ãË, ˜ÚÓ·˚ t = t ÔË a = 0, ̇ȉÂÏ Á̇˜ÂÌË ÔÓÒÚÓflÌÌÓÈ C1 (ÍÓÌÒÚ‡ÌÚ˚ ÔÓ ÓÚÌÓ¯ÂÌ˲ Í
ÄäìëíàóÖëäàâ ÜìêçÄã ÚÓÏ 50 ‹ 4 2004
(27)
ì‡‚ÌÂÌËfl (26) Ë (27) ÓÔ‰ÂÎfl˛Ú ÒÔˆˇθÌÛ˛
„ÛÔÔÛ ÔÓÂÍÚË‚ÌÓ„Ó ÔÂÓ·‡ÁÓ‚‡ÌËfl (ÔÂÓ·‡ÁÓ‚‡ÌË åfi·ËÛÒ‡) ̇ ÔÎÓÒÍÓÒÚË. èÓ‰ÒÚ‡‚Îflfl (26)
Ë (27) ‚ ÚÂڸ Û‡‚ÌÂÌË ãË, ÔÓÎÛ˜ËÏ ÎËÌÂÈÌÓÂ
ÌÂÓ‰ÌÓÓ‰ÌÓ Û‡‚ÌÂÌËÂ
t
du
x
------ + -------------- u + -------------- = 0.
da 1 – at
1 – at
àÌÚ„ËÛfl ˝ÚÓ Û‡‚ÌÂÌËÂ Ò Ì‡˜‡Î¸Ì˚Ï ÛÒÎÓ‚ËÂÏ: u = u ÔË a = 0, ̇ȉÂÏ
u = u ( 1 – at ) – ax.
3.2. ê¯ÂÌËfl, ÔÓÎÛ˜‡ÂÏ˚Â
Ò ÔÓÏÓ˘¸˛ „ÛÔÔ˚ ÒËÏÏÂÚËË
çÂÚÛ‰ÌÓ Ì‡ÈÚË ÔÂÓ·‡ÁÓ‚‡ÌË (12) „ÛÔÔ˚,
‰ÓÔÛÒ͇ÂÏÓ Û‡‚ÌÂÌËÂÏ Å˛„ÂÒ‡, ÔÛÚÂÏ ¯ÂÌËfl Û‡‚ÌÂÌËÈ ãË ‰Îfl ÓÒÌÓ‚Ì˚ı ËÌÙËÌËÚÂÁËχθÌ˚ı ÒËÏÏÂÚËÈ (24). ÑÎfl „ÂÌÂ‡ÚÓÓ‚ (24) Û‡‚ÌÂÌËfl ãË (7) ËÏÂ˛Ú “ÚÂÛ„ÓθÌÛ˛” ÙÓÏÛ
dt
------ = τ ( t ),
da
(26)
èÓ‰ÒÚ‡‚Îflfl (26) ‚Ó ‚ÚÓÓ Û‡‚ÌÂÌË ãË, ̇ȉÂÏ
p ( t ) = C4 t + C5 .
èËÌËχfl ‚Ó ‚ÌËχÌË (21) Ë (23), Ï˚ ‚ ÍÓ̘ÌÓÏ
Ò˜ÂÚ ÔËıÓ‰ËÏ Í ÒÎÂ‰Û˛˘ÂÏÛ Ó·˘ÂÏÛ ¯ÂÌ˲
ÓÔ‰ÂÎfl˛˘Â„Ó Û‡‚ÌÂÌËfl (19):
2
ÔÂÂÏÂÌÌÓÈ a ‚ Û‡‚ÌÂÌËflı ãË): C1 = –1/t. í‡ÍËÏ
Ó·‡ÁÓÏ,
t
t = -------------- .
1 – at
ÓÚÍÛ‰‡ τ'''(t) = 0, p''(t) = 0 Ë ÔÓ˝ÚÓÏÛ
5
(28)
àÒÔÓθÁÛfl ÔÓÂÍÚË‚Ì˚ ÔÂÓ·‡ÁÓ‚‡ÌËfl (26)–
(28) Ë ÔËÏÂÌflfl Û‡‚ÌÂÌË (15) Í Î˛·ÓÏÛ ËÁ‚ÂÒÚÌÓÏÛ ¯ÂÌ˲ u = Φ(t, x) Û‡‚ÌÂÌËfl Å˛„ÂÒ‡,
ÏÓÊÌÓ ÔÓÎÛ˜ËÚ¸ ÒÎÂ‰Û˛˘Â ӉÌÓÔ‡‡ÏÂÚ˘ÂÒÍÓ ÒÂÏÂÈÒÚ‚Ó ÌÓ‚˚ı ¯ÂÌËÈ:
x
1
t
ax
u = -------------- + -------------- Φ  --------------, -------------- .
1 – at 1 – at  1 – at 1 – at
(29)
Ç˚‡ÊÂÌË (29) ËÏÂÂÚ ‚‡ÊÌ˚È ÙËÁ˘ÂÒÍËÈ
ÒÏ˚ÒÎ. éÌÓ ÓÔËÒ˚‚‡ÂÚ ‚Á‡ËÏÓ‰ÂÈÒÚ‚Ë Ò˄̇·
u = Φ(t, x) Ò ÎËÌÂÈÌ˚Ï Û˜‡ÒÚÍÓÏ ÔÓÙËÎfl ÔËÎÓÓ·‡ÁÌÓÈ ‚ÓÎÌ˚. ç‡ÔËÏÂ, ÂÒÎË ‚˚ÒÓÍÓ˜‡ÒÚÓÚÌ˚È
ˆÛ„ Í‚‡ÁË„‡ÏÓÌ˘ÂÒÍÓ„Ó Ò˄̇· ÔÓÏ¢ÂÌ Ì‡
Á‡‰ÌËÈ “ÒÍÎÓÌ ÔËÎ˚”, ÍÛÚËÁ̇ ÍÓÚÓÓ„Ó ‚ ÔÓˆÂÒÒ ‡ÒÔÓÒÚ‡ÌÂÌËfl ‚ÓÎÌ˚ Û·˚‚‡ÂÚ (ÓÚˈ‡ÚÂθÌ˚ Á̇˜ÂÌËfl ÍÓÌÒÚ‡ÌÚ˚ a), ÔÓËÒıÓ‰ËÚ
ÛÏÂ̸¯ÂÌË ‡ÏÔÎËÚÛ‰˚ Ò˄̇· Ë Â„Ó ˜‡ÒÚÓÚ˚.
ç‡ÔÓÚË‚, ̇ ÛÍÛ˜‡˛˘ÂÏÒfl ÔÂ‰ÌÂÏ ÒÍÎÓÌÂ
ÌËÁÍÓ˜‡ÒÚÓÚÌÓÈ ‚ÓÎÌ˚ (ÍÓÌÒÚ‡ÌÚ‡ a ÔÓÎÓÊËÚÂθ̇) Ò˄̇ΠÛÒËÎË‚‡ÂÚÒfl, ‡ ˜‡ÒÚÓÚ‡ Â„Ó ‡ÒÚÂÚ. ùÚË
fl‚ÎÂÌËfl ÓÔËÒ‡Ì˚ ÚÂÓÂÚ˘ÂÒÍË Ë Ì‡·Î˛‰‡ÎËÒ¸ ‚
˝ÍÒÔÂËÏÂÌÚ‡ı (ÒÏ. [12, 13] ).
èËÏÂ 1. åÓÊÌÓ ÔÓÎÛ˜ËÚ¸ ÏÌÓÊÂÒÚ‚Ó ÌÓ‚˚ı
¯ÂÌËÈ, ‚˚·Ë‡fl ‚ ͇˜ÂÒÚ‚Â ËÒıÓ‰ÌÓ„Ó u = Φ(t, x)
β·Ó ËÌ‚‡ˇÌÚÌÓ ¯ÂÌËÂ. ÇÓÁ¸ÏÂÏ, Í ÔËÏÂÛ, ¯ÂÌËÂ, ËÌ‚‡ˇÌÚÌÓ ÓÚÌÓÒËÚÂθÌÓ Ò‰‚Ë„‡
‚‰Óθ ÍÓÓ‰Ë̇Ú˚ x, „ÂÌÂËÛÂÏÓ ÓÔÂ‡ÚÓÓÏ X2
ËÁ (24). Ç ˝ÚÓÏ ÒÎÛ˜‡Â ËÌ‚‡ˇÌÚ‡ÏË ·Û‰ÛÚ λ = t,
µ = u, Ë Û‡‚ÌÂÌË (17) Á‡Ô˯ÂÚÒfl Í‡Í u = φ(t). èÓ‰ÒÚ‡Ìӂ͇ ˝ÚÓ„Ó ‚˚‡ÊÂÌËfl ‚ Û‡‚ÌÂÌËÂ Å˛„ÂÒ‡
ÔË‚Ó‰ËÚ Í Ú˂ˇθÌÓÏÛ ¯ÂÌ˲ ‚ ‚ˉ ÍÓÌÒÚ‡ÌÚ˚: u = k. éÌÓ ÔÂÓ·‡ÁÛÂÚÒfl ÔÓ ÙÓÏÛΠ(29)
à·‡„ËÏÓ‚, êÛ‰ÂÌÍÓ*
6
‚ ÒÎÂ‰Û˛˘Â ӉÌÓÔ‡‡ÏÂÚ˘ÂÒÍÓ ÒÂÏÂÈÒÚ‚Ó ¯ÂÌËÈ:
X5. ï‡‡ÍÚÂËÒÚ˘ÂÒ͇fl ÒËÒÚÂχ (16) Á‡Ô˯ÂÚÒfl
Ú‡Í:
k + ax
u = --------------- ,
1 – at
du
dx
dt
----2- = ------ = – -------------- .
x
+ tu
tx
t
ËÒÔÓθÁÛÂÏ˚ı ‚ ÌÂÎËÌÂÈÌÓÈ ‡ÍÛÒÚËÍ ‰Îfl ÓÔËÒ‡ÌËfl „·‰ÍËı Û˜‡ÒÚÍÓ‚ ÔÓÙËÎfl ÔËÎÓÓ·‡ÁÌ˚ı
‚ÓÎÌ [4].
èËÏÂ 2. é‰ÌËÏ ËÁ ̇˷ÓΠËÌÚÂÂÒÌ˚ı Ò
ÙËÁ˘ÂÒÍÓÈ ÚÓ˜ÍË ÁÂÌËfl ¯ÂÌËÈ fl‚ÎflÂÚÒfl ÒÚ‡ˆËÓ̇ÌÓ ¯ÂÌËÂ
u = Φ ( x ),
ÔÓÎÛ˜ÂÌÌÓ ËÁ ÛÒÎÓ‚Ëfl ËÌ‚‡ˇÌÚÌÓÒÚË ÓÚÌÓÒËÚÂθÌÓ „ÛÔÔ˚ ÔÂÂÌÓÒÓ‚ ÔÓ ‚ÂÏÂÌË, „ÂÌÂËÛÂÏÓÈ ÓÔÂ‡ÚÓÓÏ X1. èÓ‰ÒÚ‡Ìӂ͇ ‚ Û‡‚ÌÂÌËÂ
Å˛„ÂÒ‡ ÔË‚Ó‰ËÚ Í Ó·˚ÍÌÓ‚ÂÌÌÓÏÛ ‰ËÙÙÂÂ̈ˇθÌÓÏÛ Û‡‚ÌÂÌ˲
Φ'' + ΦΦ' = 0.
(30)
àÌÚ„ËÛfl Â„Ó Ó‰ËÌ ‡Á: Φ' + Φ2/2 = C1 Ë ËÌÚ„ËÛfl ÒÌÓ‚‡, ÔÓ·„‡fl ÍÓÌÒÚ‡ÌÚÛ ‡‚ÌÓÈ C1 = 0, C1 =
= ν2 > 0, C1 = –ν2 < 0, ÔÓÎÛ˜ËÏ
2
Φ = ------------- ,
x+C
ν
Φ = ν th  C + --- x ,

2 
ν
Φ = ν tg  C – --- x .

2 
(31)
ꯇfl ÂÂ, ̇ȉÂÏ ‰‚‡ ËÌ‚‡ˇÌÚ‡ λ = x/t, µ = x + tu.
í‡ÍËÏ Ó·‡ÁÓÏ, Ó·˘Â ‚˚‡ÊÂÌË (17) ‰Îfl ËÌ‚‡ˇÌÚÌÓ„Ó ¯ÂÌËfl ÔËÏÂÚ ‚ˉ
x 1
u = – -- + --- Φ ( λ ),
t t
x
λ = --.
t
(33)
èÓ‰ÒÚ‡‚Îflfl ˝ÚÓ ‚˚‡ÊÂÌË ‚ Û‡‚ÌÂÌËÂ Å˛„ÂÒ‡ (18), ÔÓÎÛ˜ËÏ ‰Îfl Φ(λ) ‚ ÚÓ˜ÌÓÒÚË Û‡‚ÌÂÌËÂ
(30). ëΉӂ‡ÚÂθÌÓ, Â„Ó Ó·˘Â ¯ÂÌË ÔÓÎÛ˜‡ÂÚÒfl ËÁ (31) Á‡ÏÂÌÓÈ x ̇ λ. ëÓÓÚ‚ÂÚÒÚ‚Û˛˘Ë ËÌ‚‡ˇÌÚÌ˚ ¯ÂÌËfl ÔÓÎÛ˜‡˛ÚÒfl ÔÓ‰ÒÚ‡ÌÓ‚ÍÓÈ ‚
(33) ÂÁÛθڇڇ ‰Îfl Φ(λ). ç‡ÔËÏÂ, ËÒÔÓθÁÛfl
‰Îfl Φ(λ) ‚ÚÓÛ˛ ÙÓÏÛÎÛ (31) Ë ÔÓ·„‡fl ‚ ÌÂÈ ν =
π, ÔËıÓ‰ËÏ Í ¯ÂÌ˲
1
πx
u = --- – x – π th  C + ------ ,

t
2t 
(34)
‚˚‚‰ÂÌÌÓÏÛ ê.Ç.ïÓıÎÓ‚˚Ï ÔÛÚÂÏ ÙËÁ˘ÂÒÍËı
‡ÒÒÛʉÂÌËÈ (ÒÏ., ̇ÔËÏÂ, [14], „Î. 9, § 4). á‡ÏÂÚËÏ, ˜ÚÓ ÙÓÏÛ· (34) ÔÓÎÛ˜‡ÂÚÒfl Ú‡ÍÊ ËÁ ‚ÚÓÓÈ ÙÓÏÛÎ˚ (32), ÂÒÎË ÔÓÎÓÊËÚ¸ ν = –πa Ë ÛÒÚÂÏËÚ¸ a Í ·ÂÒÍÓ̘ÌÓÒÚË.
á‡ÏÂÚËÏ, ˜ÚÓ ÔÂÓ·‡ÁÓ‚‡ÌË ɇÎËÎÂfl t = t,
x = x + at, u = u – a, „ÂÌÂËÛÂÏÓ ÓÔÂ‡ÚÓÓÏ X3,
ÓÚÓ·‡Ê‡ÂÚ X1 ‚ ÒÛÏÏÛ X1 + cX2. ëÓÓÚ‚ÂÚÒÚ‚ÂÌÌÓ,
ÓÌÓ ÓÚÓ·‡Ê‡ÂÚ ÒÚ‡ˆËÓ̇ÌÓ ¯ÂÌË ‚ ·Â„Û˘Û˛ ‚ÓÎÌÛ u = u(x – ct), ΄ÍÓ ÔÓÎÛ˜‡ÂÏÛ˛ ËÁ ¯ÂÌËÈ (31).
ä‡Í ËÁ‚ÂÒÚÌÓ, ¯ÂÌË ‚ ‚ˉ “„ËÔÂ·Ó΢ÂÒÍÓ„Ó Ú‡Ì„ÂÌÒ‡” ÓÔËÒ˚‚‡ÂÚ Ó‰ËÌÓ˜ÌÛ˛ Û‰‡ÌÛ˛
‚ÓÎÌÛ Ò ÍÓ̘ÌÓÈ ¯ËËÌÓÈ ÙÓÌÚ‡ [4, 14]. èÓÙËÎË ‰Û„Ëı ‰‚Ûı ÒÚ‡ˆËÓ̇Ì˚ı ‚ÓÎÌ (31) ÒÓ‰ÂÊ‡Ú ÓÒÓ·ÂÌÌÓÒÚË; ÔÓ˝ÚÓÏÛ ˝ÚË ¯ÂÌËfl ‚ ÙËÁ˘ÂÒÍËı Á‡‰‡˜‡ı ËÒÔÓθÁÛ˛ÚÒfl ÂÊÂ.
èËÏÂ 3. ÖÒÎË ÔËÏÂÌËÚ¸ ÔÂÓ·‡ÁÓ‚‡ÌËÂ
(29) Í ÒÚ‡ˆËÓ̇Ì˚Ï ¯ÂÌËflÏ (31), ÔÓÎÛ˜‡ÚÒfl
ÒÎÂ‰Û˛˘Ë ÌÓ‚˚ ÌÂÒÚ‡ˆËÓ̇Ì˚ ¯ÂÌËfl:
èËÏÂ 5. ê¯ÂÌËfl, ËÌ‚‡ˇÌÚÌ˚ ÓÚÌÓÒËÚÂθÌÓ „ÛÔÔ˚ ‡ÒÚflÊÂÌËÈ, „ÂÌÂËÛÂÏÓÈ ÓÔÂ‡ÚÓÓÏ X4, ‚ ÙËÁ˘ÂÒÍÓÈ ÎËÚÂ‡ÚÛ ˜‡ÒÚÓ Ì‡Á˚‚‡˛Ú ‡‚ÚÓÏÓ‰ÂθÌ˚ÏË ¯ÂÌËflÏË. Ç ˝ÚÓÏ ÒÎÛ˜‡Â
ı‡‡ÍÚÂËÒÚ˘ÂÒ͇fl ÒËÒÚÂχ
2
ax
u = -------------- + -------------------------------- ,
1 – at x + C ( 1 – at )
Ç ÂÁÛθڇÚ ÔÓÎÛ˜‡ÂÚÒfl Û‡‚ÌÂÌË [4] ‰Îfl ̇ıÓʉÂÌËfl ‡‚ÚÓÏÓ‰ÂθÌ˚ı ¯ÂÌËÈ Û‡‚ÌÂÌËfl
Å˛„ÂÒ‡:
1
νx
u = -------------- ax + ν th  C + ---------------------- ,

1 – at
2 ( 1 – at )
(32)
1
νx
u = -------------- ax + ν tg  C – ---------------------- .

1 – at
2 ( 1 – at )
èËÏÂ 4. ç‡È‰ÂÏ ËÌ‚‡ˇÌÚÌ˚ ¯ÂÌËfl ‰Îfl
ÔÓÂÍÚË‚ÌÓÈ „ÛÔÔ˚, „ÂÌÂËÛÂÏÓÈ ÓÔÂ‡ÚÓÓÏ
dt
dx
du
----- = ------ = – -----2t
x
u
‰‡ÂÚ ÒÎÂ‰Û˛˘Ë ËÌ‚‡ˇÌÚ˚: λ = x/ t , µ = t u.
ëΉӂ‡ÚÂθÌÓ, ËÌ‚‡ˇÌÚÌÓ ¯ÂÌË ÌÛÊÌÓ ËÒ͇ڸ ‚ ‚ˉÂ
1
u = -----Φ ( λ ),
t
x
λ = -----.
t
1
Φ'' + ΦΦ' + --- ( λΦ' + Φ ) = 0.
2
(35)
àÌÚ„ËÛfl Â„Ó Ó‰ËÌ ‡Á, ÔÓÎÛ˜ËÏ
1 2
Φ' + --- ( Φ + λΦ ) = C.
2
ÄäìëíàóÖëäàâ ÜìêçÄã ÚÓÏ 50 ‹ 4 2004
èË̈ËÔ ‡ÔËÓÌÓ„Ó ËÒÔÓθÁÓ‚‡ÌËfl ÒËÏÏÂÚËÈ
èË C = 0 ˝ÚÓ Û‡‚ÌÂÌˠ΄ÍÓ ËÌÚ„ËÛÂÚÒfl Ë
‰‡ÂÚ ¯ÂÌËÂ, ËÒ˜ÂÁ‡˛˘Â ̇ ±∞ (ÒÏ. [14], „Î. 9,
§4 ËÎË [10], Ú. 1, ÒÂ͈.11.4):
2
x
exp  – -----

4t
2
u = --------- -------------------------------- ,
x 
πt B + erf  ------- 2 t
z
2
2
erf ( z ) = ------- exp ( – s ) ds,
π
∫
0
„‰Â B – ÔÓËÁ‚Óθ̇fl ÍÓÌÒÚ‡ÌÚ‡ Ë erf – ÙÛÌ͈Ëfl
ӯ˷ÓÍ.
èËÏÂ 6. äÓÌÒÚÛËÓ‚‡ÌË ÚÓ˜Ì˚ı ¯ÂÌËÈ
ÏÓÊÂÚ ·˚Ú¸ ÓÒÌÓ‚‡ÌÓ Ì ÚÓθÍÓ Ì‡ ÓÒÌÓ‚Ì˚ı ËÌÙËÌËÚÂÁËχθÌ˚ı ÒËÏÏÂÚËflı (24), ÌÓ Ú‡ÍÊ ̇
Ëı ÎËÌÂÈÌ˚ı ÍÓÏ·Ë̇ˆËflı. ê‡ÒÒÏÓÚËÏ, Í ÔËÏÂÛ, ÓÔÂ‡ÚÓ
∂
∂
2 ∂
X 1 + X 5 = ( 1 + t ) ----- + tx ------ – ( x + tu ) ------ .
∂t
∂x
∂u
(36)
ï‡‡ÍÚÂËÒÚ˘ÂÒ͇fl ÒËÒÚÂχ ‰‡ÂÚ ÒÎÂ‰Û˛˘Ë ËÌ‚‡ˇÌÚ˚:
x
λ = ----------------- ,
2
1+t
tx
2
µ = ----------------- + u 1 + t .
2
1+t
(37)
í‡ÍËÏ Ó·‡ÁÓÏ, ËÌ‚‡ˇÌÚÌÓ ¯ÂÌË ËÏÂÂÚ ‚ˉ
1
tx
u = – -------------2 + ----------------- Φ ( λ ),
2
1+t
1+t
x
λ = ----------------- .
2
1+t
èÓ‰ÒÚ‡‚Îflfl ˝ÚÓ ‚˚‡ÊÂÌË ‚ Û‡‚ÌÂÌËÂ Å˛„ÂÒ‡, ÔˉÂÏ Í ÒÎÂ‰Û˛˘ÂÏÛ Û‡‚ÌÂÌ˲ ‰Îfl Φ(λ):
Φ'' + ΦΦ' + λ = 0.
(38)
àÌÚ„ËÛfl (37) Ó‰ËÌ ‡Á, ÔÓÎÛ˜ËÏ
1 2
2
Φ' + --- ( Φ + λ ) = C 1 .
2
(39)
d
Φ = 2 ------ ln
dλ
λ
λ
λJ 1/4  ----- + C λY 1/4  ----- ,
 4
 4
2
‚Ë‚‡ÎÂÌÚÌÓÒÚË (ÒÏ. ÓÔ‰ÂÎÂÌË 2 ‚ ‡Á‰ÂΠ4.2.),
ÚÓ „ÛÔÔ‡ ÒËÏÏÂÚËË Ì‡ıÓ‰ËÚÒfl Í‡Í ÔÓ‰ıÓ‰fl˘‡fl
ÔÓ‰„ÛÔÔ‡ „ÛÔÔ˚ ˝Í‚Ë‚‡ÎÂÌÚÌÓÒÚË. Ç ÔÓÚË‚ÌÓÏ
ÒÎÛ˜‡Â ‡ÒÒχÚË‚‡Âχfl ÌÂÎËÌÂÈ̇fl ÏÓ‰Âθ
Ó·Ó·˘‡ÂÚÒfl ÔÛÚÂÏ Â “ÔÓ„ÛÊÂÌËfl” ‚ ·ÓΠ¯ËÓÍÛ˛ ÏÓ‰Âθ ‰ÓÒÚ‡ÚÓ˜ÌÓ “‡ÁÛÏÌ˚Ï” Ó·‡ÁÓÏ,
˜ÚÓ·˚ Ì ÔÓÚÂflÚ¸ ËÒıÓ‰Ì˚È ÙËÁ˘ÂÒÍËÈ ÒÏ˚ÒÎ Ë
‚ ÚÓ Ê ‚ÂÏfl ‰ÓÒÚ˘¸ Ê·ÂÏÓ„Ó ‡Ò¯ËÂÌËfl
„ÛÔÔ˚ ˝Í‚Ë‚‡ÎÂÌÚÌÓÒÚË. èË ˝ÚÓÏ ÌÂÎËÌÂÈÌÓÒÚ¸
ÏÓ‰ÂÎË ÒÛ˘ÂÒÚ‚ÂÌ̇, Ú‡Í Í‡Í Ó̇ Ôˉ‡ÂÚ ‰ÓÒÚ‡ÚÓ˜ÌÛ˛ „Ë·ÍÓÒÚ¸ ‚ ÔÓ‰·Ó ÏÓ‰ÂÎË Ë Ó·ÂÒÔ˜˂‡ÂÚ Ó·˘ÌÓÒÚ¸ ÏÂÚÓ‰‡.
4.1. èÓ„ÛÊÂÌËÂ Ë ÔËÏÂÌÂÌË ËÌ‚‡ˇÌÚÓ‚
ã‡Ô·҇
ê‡ÒÒÏÓÚËÏ Û‡‚ÌÂÌË à̯ÓÛ (ÒÏ., ̇ÔËÏÂ, [15]):
∂v –( γ + 1 ) ∂ v
∂ v
2
--------2- = 0,
--------2- – c  1 + -------

∂ξ 
∂ξ
∂t
2
2
2
„‰Â C – ‚ÚÓ‡fl ÍÓÌÒÚ‡ÌÚ‡.
4. åÂÚÓ‰ ‡ÔËÓÌÓ„Ó
ËÒÔÓθÁÓ‚‡ÌËfl ÒËÏÏÂÚËË
Ç ‡Á‰ÂΠ3 ‰‡Ì˚ ÔËÏÂ˚ ËÒÔÓθÁÓ‚‡ÌËfl
„ÛÔÔ ÒËÏÏÂÚËË. çÓ ‚Ó ÏÌÓ„Ëı Á‡‰‡˜‡ı „ÛÔÔ‡
ÒËÏÏÂÚËË ÓÚÒÛÚÒÚ‚ÛÂÚ ËÎË ·˚‚‡ÂÚ Ì‰ÓÒÚ‡ÚÓ˜ÌÓ
¯ËÓÍÓÈ ‰Îfl ¯ÂÌËfl ‰‡ÌÌÓ„Ó ÏÓ‰ÂθÌÓ„Ó Û‡‚ÌÂÌËfl. è‰·„‡ÂÏ˚È ÏÂÚÓ‰ ̇ˆÂÎÂÌ Ì‡ ÔÓÒÚÓÂÌË ÏÓ‰ÂÎÂÈ Ò ÔÓ‚˚¯ÂÌÌÓÈ ÒËÏÏÂÚËË ·ÂÁ ÔÓÚÂË
ÙËÁ˘ÂÒÍÓ„Ó ÒÓ‰ÂʇÌËfl ËÒıÓ‰ÌÓÈ ÏÓ‰ÂÎË. ëÛÚ¸
ÏÂÚÓ‰‡ ÒÓÒÚÓËÚ ‚ ÒÎÂ‰Û˛˘ÂÏ.
ÖÒÎË ÏÓ‰Âθ ÒÓ‰ÂÊËÚ “ÔÓËÁ‚ÓθÌ˚ ˝ÎÂÏÂÌÚ˚” Ë ‰ÓÔÛÒ͇ÂÚ ‰ÓÒÚ‡ÚÓ˜ÌÓ ¯ËÓÍÛ˛ „ÛÔÔÛ ˝ÍÄäìëíàóÖëäàâ ÜìêçÄã ÚÓÏ 50 ‹ 4 2004
(38)
ÓÔËÒ˚‚‡˛˘Â ‚ ÔÂÂÏÂÌÌ˚ı ㇄‡Ìʇ Ó‰ÌÓÏÂÌÓ ‰‚ËÊÂÌË ÒÊËχÂÏÓ„Ó „‡Á‡. îËÁ˘ÂÒÍËÈ
ÒÏ˚ÒÎ ÔÂÂÏÂÌÌ˚ı: v – ÒÏ¢ÂÌË ˜‡ÒÚˈ Ò‰˚,
c – ÒÍÓÓÒÚ¸ Á‚Û͇, γ – ÔÓ͇Á‡ÚÂθ ‡‰Ë‡·‡Ú˚ ‚
Û‡‚ÌÂÌËË ÒÓÒÚÓflÌËfl.
ì‡‚ÌÂÌË (3) ÏÓÊÂÚ ·˚Ú¸ ÎË̇ËÁÓ‚‡ÌÓ
“ÔÂÓ·‡ÁÓ‚‡ÌËÂÏ „Ó‰Ó„‡Ù‡”:
x = v ξ,
y = v t,
ξ = X ( x, y ),
t = u ( x, y ), (39)
ÚÓ ÂÒÚ¸ ·„‡ÌÊ‚‡ ÍÓÓ‰Ë̇ڇ ξ Ë ‚ÂÏfl t Ò˜ËÚ‡˛ÚÒfl ÙÛÌ͈ËflÏË ÌÓ‚˚ı ÌÂÁ‡‚ËÒËÏ˚ı ÔÂÂÏÂÌÌ˚ı x, y – ÔÂ‚˚ı ÔÓËÁ‚Ó‰Ì˚ı ÓÚ ËÒÍÓÏÓÈ ÙÛÌ͈ËË v.
ëÓÓÚ‚ÂÚÒÚ‚Û˛˘Â ÎË̇ËÁÓ‚‡ÌÌÓ Û‡‚ÌÂÌË ËÏÂÂÚ ‚ˉ
∂ u 2
–( γ + 1 ) ∂ u
-------2- = 0.
--------2 – c ( 1 + x )
∂y
∂x
2
2
äÓ„‰‡ ÔÓÒÚÓflÌ̇fl C1 = 0, ¯ÂÌË ˝ÚÓ„Ó Û‡‚ÌÂÌËfl Û‰‡ÂÚÒfl ‚˚‡ÁËÚ¸ ˜ÂÂÁ ÙÛÌ͈ËË ÅÂÒÒÂÎfl:
7
(40)
èË Ï‡ÎÓÏ ‡ÍÛÒÚ˘ÂÒÍÓÏ ˜ËÒΠå‡ı‡ |vx| = |x|
Û‡‚ÌÂÌË (40) ‡ÔÔÓÍÒËÏËÛ˛Ú ·ÓΠÔÓÒÚ˚Ï
Û‡‚ÌÂÌËÂÏ
∂ u
∂ u 2
--------2 – c [ 1 – ( γ + 1 )x ] -------2- = 0.
∂y
∂x
2
2
(41)
é‰Ì‡ÍÓ ÌË Û‡‚ÌÂÌË (40), ÌË Â„Ó ÛÔÓ˘ÂÌ̇fl
‚ÂÒËfl (41) Ì ¯‡˛ÚÒfl, ÔÓÒÍÓθÍÛ Ì ӷ·‰‡˛Ú
‰ÓÒÚ‡ÚÓ˜ÌÓ ¯ËÓÍÓÈ „ÛÔÔÓÈ ÒËÏÏÂÚËË.
èÓ˝ÚÓÏÛ ‚˚·ÂÂÏ ‡ÔÔÓÍÒËÏËÛ˛˘Â Û‡‚ÌÂÌË Ì ÔÓ ÔÓÒÚÓÚÂ Â„Ó ‚ˉ‡, ‡ ÔÓ ÔË̈ËÔÛ Ì‡Î˘Ëfl ÒËÏÏÂÚËË, ‰ÓÒÚ‡ÚÓ˜ÌÓÈ ‰Îfl Â„Ó ‡Á¯ËÏÓÒÚË. àÏÂÌÌÓ, ‡ÒÒÏÓÚËÏ „ËÔÂ·Ó΢ÂÒÍÓ Û‡‚ÌÂÌËÂ
∂ u 2 2 ∂ u
--------2 – c ψ ( x ) -------2- = 0,
∂y
∂x
2
2
(42)
à·‡„ËÏÓ‚, êÛ‰ÂÌÍÓ*
8
Ó·Ó·˘‡˛˘Â (40) Ë (41) Ë Á‡‰‡‰ËÏÒfl ˆÂθ˛ ̇ÈÚË
Ú‡ÍÛ˛ ÙÛÌÍˆË˛ ψ(x), ÍÓÚÓ‡fl, ‚Ó-ÔÂ‚˚ı, ÒÓ‚Ô‡‰‡Î‡ ·˚ ÔË Ï‡Î˚ı |x| Ò ÒÓÓÚ‚ÂÚÒÚ‚Û˛˘ÂÈ ÙÛÌ͈ËÂÈ ‚ ËÒıÓ‰ÌÓÏ Û‡‚ÌÂÌËË Ë, ‚Ó-‚ÚÓ˚ı, ÓÚÍ˚‚‡Î‡ ‚ÓÁÏÓÊÌÓÒÚ¸ ̇ÈÚË Ó·˘Â ¯ÂÌË Û‡‚ÌÂÌËfl (42). Ç˚·ÂÂÏ ψ(x) Ú‡Í, ˜ÚÓ·˚ Û‡‚ÌÂÌË (42)
‰ÓÔÛÒ͇ÎÓ Ï‡ÍÒËχθÌÓ ¯ËÓÍÛ˛ „ÛÔÔÛ.
ïÓÓ¯Ó ËÁ‚ÂÒÚÌÓ, ˜ÚÓ Û‡‚ÌÂÌË „ËÔÂ·Ó΢ÂÒÍÓ„Ó ÚËÔ‡
∂u
∂u
∂ u
-------------- + A ( α, β ) ------- + B ( α, β ) ------ + P ( α, β )u = 0 (43)
∂α
∂β
∂α∂β
2
‰ÓÔÛÒ͇ÂÚ Ï‡ÍÒËχθÌÓ ¯ËÓÍÛ˛ „ÛÔÔÛ [16, 17]
Ë ÚÂÏ Ò‡Ï˚Ï ÏÓÊÂÚ ·˚Ú¸ ¯ÂÌÓ Ò‚Â‰ÂÌËÂÏ Í
Ó·˚˜ÌÓÏÛ ‚ÓÎÌÓ‚ÓÏÛ Û‡‚ÌÂÌ˲, ÂÒÎË ËÌ‚‡ˇÌÚ˚ ã‡Ô·҇ [18] ‰Îfl Û‡‚ÌÂÌËfl (43)
∂A
h = ------- + AB – P,
∂α
∂B
k = ------ + AB – P
∂β
(44)
Ó·‡˘‡˛ÚÒfl ‚ ÌÛθ. èÓ˝ÚÓÏÛ ‚˚˜ËÒÎËÏ ËÌ‚‡ˇÌÚ˚ ã‡Ô·҇ Û‡‚ÌÂÌËfl (42), ÔÂÂÔËÒ‡‚ Â„Ó ‚ ı‡‡ÍÚÂËÒÚ˘ÂÒÍËı ÔÂÂÏÂÌÌ˚ı
∫
α = c ψ ( x ) d x – y,
∫
β = c ψ ( x ) d x + y.
(45)
Ç ˝ÚËı ÔÂÂÏÂÌÌ˚ı (42) ÔËÌËχÂÚ Í‡ÌÓÌ˘ÂÒÍÛ˛
ÙÓÏÛ (43)
ψ' ( x )  ∂u ∂u
∂ u
- ------- + ------ = 0
-------------- + ------------------∂α∂β 4cψ 2 ( x )  ∂α ∂β
2
(46)
–1
2
2
ë‡‚ÌË‚‡fl Â„Ó Ò Û‡‚ÌÂÌËÂÏ (41), ‚ˉËÏ, ˜ÚÓ ÌÛÊÌÓ ÔÓÎÓÊËÚ¸ ÍÓÌÒÚ‡ÌÚ˚ ‡‚Ì˚ÏË l = 1, s = (γ +
1)/4. í‡ÍËÏ Ó·‡ÁÓÏ, ̇ıÓ‰ËÏ ËÒÍÓÏÓ ‡Á¯ËÏÓ Û‡‚ÌÂÌËÂ
ε
∂ u 2
--------2 – c 1 + --- x
2
∂x
2
∂ u
-------2- = 0,
∂y
–4 2
α+β
z = ------------- = Ψ ( x ) ≡ ψ ( x ) d x. (48)
2c
∫
àÁ Û‡‚ÌÂÌËÈ (47) Ë (48) ̇ıÓ‰ËÏ
γ +1
ε = ------------ ,
2
(50)
ÍÓÚÓÓ ‡ÔÔÓÍÒËÏËÛÂÚ Û‡‚ÌÂÌË (40) Ò ‰ÓÒÚ‡ÚÓ˜ÌÓÈ ÒÚÂÔÂ̸˛ ÚÓ˜ÌÓÒÚË ÔË ε|x| 1.
ÑÎfl ¯ÂÌËfl Û‡‚ÌÂÌËfl (50) ÔÂÂÔ˯ÂÏ Â„Ó ‚
͇ÌÓÌ˘ÂÒÍÓÈ ÙÓÏ (46)
1 ∂u ∂u
∂ u
-------------- + -------------  ------- + ------ = 0.
∂α∂β α + β  ∂α ∂β
2
(51)
ëӄ·ÒÌÓ Ó·˘ÂÈ ÚÂÓËË, Û‡‚ÌÂÌË (51) Ò‚Ó‰ËÚÒfl
Í ÔÓÒÚÂȯÂÏÛ ‚ÓÎÌÓ‚ÓÏÛ Û‡‚ÌÂÌ˲ wαβ = 0 Á‡ÏÂÌÓÈ w = (α + β)u. èÓ˝ÚÓÏÛ Ó·˘Â ¯ÂÌË Û‡‚ÌÂÌËfl (51) ‰‡ÂÚÒfl ÙÓÏÛÎÓÈ
1
u ( α, β ) = ------------- [ Φ 1 ( α ) + Φ 2 ( β ) ]
α+β
(52)
Ò ‰‚ÛÏfl ÔÓËÁ‚ÓθÌ˚ÏË ÙÛÌ͈ËflÏË Φ1, Φ2. íÂÔÂ¸ ÏÓÊÌÓ ‚ÂÌÛÚ¸Òfl ‚ (52) Í ÔÂÂÏÂÌÌ˚Ï x, y.
îÓÏÛÎ˚ (45) ‰‡˛Ú
2c
β = – ---------------------------- – y ,
ε ( 1 + εx/2 )
4c
α + β = – ---------------------------- .
ε ( 1 + εx/2 )
(47)
P = 0,
„‰Â x ‚˚‡Ê‡ÂÚÒfl ˜ÂÂÁ α Ë β Òӄ·ÒÌÓ (45), ‡
ËÏÂÌÌÓ
x = Ψ ( z ),
∂ u 2
–4 ∂ u
--------2 – c [ l + sx ] -------2- = 0.
∂y
∂x
2c
α = – ---------------------------- + y ,
ε ( 1 + εx/2 )
Ò ÍÓ˝ÙÙˈËÂÌÚ‡ÏË
ψ' ( x )
-,
A = B = ------------------2
4cψ ( x )
àÚ‡Í, Û‡‚ÌÂÌË (42) Ò Ï‡ÍÒËχθÌÓ ¯ËÓÍÓÈ
„ÛÔÔÓÈ ÒËÏÂÚËË ËÏÂÂÚ ‚ˉ
èÓ‰ÒÚ‡Ìӂ͇ ˝ÚËı ‚˚‡ÊÂÌËÈ ‚ Û‡‚ÌÂÌË (52) Ò
Á‡ÏÂÌÓÈ ÌÂÒÛ˘ÂÒÚ‚ÂÌÌÓ„Ó Á͇̇ ‚ ÔÓËÁ‚ÓθÌ˚ı
ÙÛÌ͈Ëflı ‰‡ÂÚ ÒÎÂ‰Û˛˘Â ӷ˘Â ¯ÂÌË Û‡‚ÌÂÌËfl (50):
ε ( 1 + εx/2 )
u ( x, y ) = ---------------------------- ×
4c
(53)
∂A z
∂B
1 ∂A
∂A
ψ'
ψ''
------- = ------ ----α = ------------------ ------ = -------------– -------------= ------.
2
3
2
4
∂x z x
∂β
2cψ ( x ) ∂x
∂α
8c ψ 4c ψ
2c
2c
× Φ 1  ---------------------------- + y + Φ 2  ---------------------------- – y .
 ε ( 1 + εx/2 )

 ε ( 1 + εx/2 ) 
èÓθÁÛflÒ¸ ˝ÚËÏ ÒÓÓÚÌÓ¯ÂÌËÂÏ, ‚˚˜ËÒÎflÂÏ ËÌ‚‡ˇÌÚ˚ ã‡Ô·҇ (44) ‰Îfl Û‡‚ÌÂÌËfl (46):
ëÛÏÏËÛfl, ÏÓÊÌÓ ÍÓÌÒÚ‡ÚËÓ‚‡Ú¸, ˜ÚÓ ¯ÂÌËÂ
(53) Û‰‡ÎÓÒ¸ ̇ÈÚË ÔÛÚÂÏ “ÔÓ„ÛÊÂÌËfl” ËÌÚÂÂÒÛ˛˘ÂÈ Ì‡Ò ÏÓ‰ÂÎË (38) ‚ ·ÓΠӷ˘Û˛ ÏÓ‰Âθ
2
1 
3 2
h = k = -------------ψψ'' – --- ψ'  .
2 4
2 
8c ψ
(49)
ëΉӂ‡ÚÂθÌÓ, ÛÒÎÓ‚Ëfl h = k = 0 Ò‚Ó‰flÚÒfl Í Ó‰ÌÓÏÛ Û‡‚ÌÂÌ˲, ¯ÂÌË ÍÓÚÓÓ„Ó
ψ ( x ) = ( l + sx ) ,
–2
s, l = const.
∂ v
2 2 ∂v ∂ v
--------2- – c ψ  ------- --------2- = 0.
 ∂ξ  ∂ξ
∂t
2
2
(54)
LJÊÌÓ ÔÓ‰˜ÂÍÌÛÚ¸, Ӊ̇ÍÓ, ˜ÚÓ ‚ÓÁÏÓÊÌÓÒÚ¸ ÔÓÒÚÓÂÌËfl ¯ÂÌËfl (53) ÒÛ˘ÂÒÚ‚ÂÌÌÓ Ò‚flÁ‡Ì‡ Ò ÚÂÏ
ÒÎÛ˜‡ÈÌ˚Ï Ó·ÒÚÓflÚÂθÒÚ‚ÓÏ, ˜ÚÓ Û‡‚ÌÂÌË (38) Ë
Â„Ó Ó·Ó·˘ÂÌË (50) ÎË̇ËÁÛ˛ÚÒfl ÔÂÓ·‡ÁÓ‚‡ÄäìëíàóÖëäàâ ÜìêçÄã ÚÓÏ 50 ‹ 4 2004
èË̈ËÔ ‡ÔËÓÌÓ„Ó ËÒÔÓθÁÓ‚‡ÌËfl ÒËÏÏÂÚËÈ
ÌËÂÏ (39). èÓ˝ÚÓÏÛ, ıÓÚfl ˝ÚÓÚ ÔËÏÂ ̇„Îfl‰ÌÓ
ËÎβÒÚËÛÂÚ Ë‰Â˛ ÔÓ„ÛÊÂÌËfl Ë ÔÓ͇Á˚‚‡ÂÚ,
Í‡Í Ó·Ó·˘ÂÌË ÏÓ‰ÂÎË ÏÓÊÂÚ ÔË‚ÂÒÚË Í Â ‡Á¯ËÏÓÒÚË, ÌÓ ÓÌ Â˘Â Ì ‰‡ÂÚ Ô‡ÍÚ˘ÂÒÍÓ„Ó ÏÂÚÓ‰‡, ÔÓÁ‚ÓÎfl˛˘Â„Ó ÓÚ·Ë‡Ú¸ ËÁ Ó·Ó·˘ÂÌÌÓÈ ÏÓ‰ÂÎË Ì‡Ë·ÓΠÒËÏÏÂÚ˘Ì˚ Û‡‚ÌÂÌËfl. í‡ÍÓÈ
ÏÂÚÓ‰ Ó·ÒÛʉ‡ÂÚÒfl ‚ ÒÎÂ‰Û˛˘ÂÏ ‡Á‰ÂΠ̇ ÍÓÌÍÂÚÌÓÏ ÔËÏÂÂ.
4.2. åÂÚÓ‰, ÓÒÌÓ‚‡ÌÌ˚È Ì‡ ÚÂÓÂÏÂ Ó ÔÓÂ͈Ëflı
íÂÓÂχ Ó ÔÓÂ͈Ëflı ·˚· ‰Ó͇Á‡Ì‡ ç.ï. à·‡„ËÏÓ‚˚Ï ‚ 1987 „Ó‰Û [19] Ë Á‡ÚÂÏ ËÒÔÓθÁÓ‚‡Ì‡
‚ Á‡‰‡˜‡ı „ÛÔÔÓ‚ÓÈ Í·ÒÒËÙË͇ˆËË [20], ÒÚ‡‚ ÓÒÌÓ‚ÓÈ ÏÂÚÓ‰‡ “Ô‰‚‡ËÚÂθÌÓÈ „ÛÔÔÓ‚ÓÈ Í·ÒÒËÙË͇ˆËË” (ÒÏ.Ú‡ÍÊ [21–23]).
èÂÂȉÂÏ ÚÂÔÂ¸ Í ÓÒÌÓ‚Ì˚Ï ÔËÏÂ‡Ï, ËÎβÒÚËÛ˛˘ËÏ ‚ÓÁÏÓÊÌÓÒÚË Ô‰·„‡ÂÏÓ„Ó
ÔÓ‰ıÓ‰‡. ùÚË ÔËÏÂ˚ ËÌÚÂÂÒÌ˚ Ë Ò‡ÏË ÔÓ Ò·Â,
ÔÓÒÍÓθÍÛ ‡ÒÒχÚË‚‡˛ÚÒfl ÌÓ‚˚ ÌÂÎËÌÂÈÌ˚Â
Û‡‚ÌÂÌËfl. àı ÙËÁ˘ÂÒÍÓ ÒÓ‰ÂʇÌË ӷÒÛʉ‡ÂÚÒfl ‚ ÒÎÂ‰Û˛˘ÂÏ ‡Á‰ÂÎÂ.
燘ÌÂÏ Ò ÌÂÎËÌÂÈÌÓ„Ó Û‡‚ÌÂÌËfl
∂ ∂u
∂u
----- ------ – u ------ = – βu,
∂t ∂x
∂t
β = const ≠ 0.
(55)
éÌÓ ‰ÓÔÛÒ͇ÂÚ ÚÂıÏÂÌÛ˛ ‡Î„·Û ãË Ò ·‡ÁËÒÓÏ
∂
X 1 = ----- ,
∂t
∂
X 2 = ------ ,
∂x
∂
∂
∂
X 3 = t ----- – x ------ + 2u ------ . (56)
∂t
∂x
∂u
ëÓÓÚ‚ÂÚÒÚ‚ÂÌÌÓ, Û‡‚ÌÂÌË (55) Ì ·Ó„‡ÚÓ ËÌ‚‡ˇÌÚÌÓ-„ÛÔÔÓ‚˚ÏË ¯ÂÌËflÏË; Ëı Í·ÒÒ Ó„‡Ì˘ÂÌ ¯ÂÌËflÏË ÚËÔ‡ ·Â„Û˘Ëı ‚ÓÎÌ, ÔÓÒÚÓÂÌÌ˚ÏË Ò ÔÓÏÓ˘¸˛ „ÂÌÂ‡ÚÓÓ‚ ÔÂÂÌÓÒ‡ X1, X2, Ë
‡‚ÚÓÏÓ‰ÂθÌ˚ÏË ¯ÂÌËflÏË, ÔÓÒÚÓÂÌÌ˚ÏË Ò
ÔÓÏÓ˘¸˛ „ÂÌÂ‡ÚÓ‡ ‡ÒÚflÊÂÌËfl X3.
èÓ˝ÚÓÏÛ ËÒÔÓθÁÛÂÏ Ë‰Â˛ ÔÓ„ÛÊÂÌËfl, ‡ÒÒχÚË‚‡fl ‰‚‡ ÚËÔ‡ ÏÓ‰ÂÎÂÈ, Ó·Ó·˘‡˛˘Ëı Û‡‚ÌÂÌË (55). èÂ‚‡fl ËÁ ÌËı ËÏÂÂÚ ‚ˉ
∂ ∂u
∂u
----- ------ – P ( u ) ------ = F ( x, u ),
∂t ∂x
∂t
Q, F) ÔÂÂıÓ‰ËÚ ‚ Û‡‚ÌÂÌË ÚÓ„Ó Ê ÒÂÏÂÈÒÚ‚‡, ÚÓ
ÂÒÚ¸
∂ ∂u
∂u
----- ------ – Q ( x, u ) ------ = F ( x, u ),
∂t
∂ t ∂x
„‰Â, ‚ÓÓ·˘Â „Ó‚Ófl, ÙÛÌ͈ËË Q , F Ì ÒÓ‚Ô‡‰‡˛Ú
Ò Q, F. ÉÂÌÂ‡ÚÓ˚ „ÛÔÔ˚ ˝Í‚Ë‚‡ÎÂÌÚÌÓÒÚË ËÏÂ˛Ú ‚ˉ
∂
1∂
2 ∂
1 ∂
2 ∂
Y = ξ ----- + ξ ------ + η ------ + µ ------- + ξ ------,
∂t
∂x
∂u
∂Q
∂F
ξ = ξ ( t, x, u ),
1
1
η = η ( t, x, u ),
µ = µ ( t, x, u, Q, F ),
1
i = 1, 2.
1
éÌË Ó·‡ÁÛ˛Ú ‡Î„·Û ãË, ÍÓÚÓ‡fl ̇Á˚‚‡ÂÚÒfl
‡Î„·ÓÈ ˝Í‚Ë‚‡ÎÂÌÚÌÓÒÚË Ë Ó·ÓÁ̇˜‡ÂÚÒfl Í‡Í Lε.
Ç ÓÔÂ‡ÚÓ (59) Ë Â ÍÓÓ‰Ë̇ڇı ÙÛÌ͈ËË Q, F
‡ÒÒχÚË‚‡˛ÚÒfl Í‡Í ÌÓ‚˚ ÔÂÂÏÂÌÌ˚ ̇fl‰Û Ò
ÙËÁ˘ÂÒÍËÏË ÔÂÂÏÂÌÌ˚ÏË t, x, u.
ä‡Í ÔÓ͇Á‡Î ã.Ç.é‚ÒflÌÌËÍÓ‚ [8], ‡Î„·Û ˝Í‚Ë‚‡ÎÂÌÚÌÓÒÚË ÏÓÊÌÓ Ì‡ÈÚË Ò ÔÓÏÓ˘¸˛ ËÌÙËÌËÚÂÁËχθÌÓÈ ÚÂıÌËÍË ãË (ÒÏ. ‡Á‰ÂÎ 2.1), ÓÔ‰ÂÎË‚ „ÛÔÔÛ ˝Í‚Ë‚‡ÎÂÌÚÌÓÒÚË Í‡Í „ÛÔÔÛ, ‰ÓÔÛÒ͇ÂÏÛ˛ ÒÎÂ‰Û˛˘ÂÈ ‡Ò¯ËÂÌÌÓÈ ÒËÒÚÂÏÓÈ,
‡‚ÌÓÒËθÌÓÈ ÒÂÏÂÈÒÚ‚Û Û‡‚ÌÂÌËÈ ‚ˉ‡ (58):
2
u tx – Qu tt – Q u u t – F = 0,
Q t = 0,
F t = 0. (60)
çÂÒÏÓÚfl ̇ ÔË̈ËÔˇθÌÓ ÒıÓ‰ÒÚ‚Ó Ò Í·ÒÒ˘ÂÒÍÓÈ ÚÂÓËÂÈ ãË, ËϲÚÒfl Á̇˜ËÚÂθÌ˚ ÚÂıÌ˘ÂÒÍË ‡Á΢Ëfl ÏÂÊ‰Û ‚˚˜ËÒÎÂÌËÂÏ ËÌÙËÌËÚÂÁËχθÌ˚ı ÒËÏÏÂÚËÈ Ë „ÂÌÂ‡ÚÓÓ‚ (59) „ÛÔÔ˚ ˝Í‚Ë‚‡ÎÂÌÚÌÓÒÚË. èÓ‰Ó·Ì˚ ‚˚˜ËÒÎÂÌËfl
ÏÓÊÌÓ Ì‡ÈÚË ‚ [20] Ë [21].
àÒÔÓθÁÛfl ˝ÚÓÚ ÔÓ‰ıÓ‰, ÏÓÊÌÓ ‚˚˜ËÒÎËÚ¸, ˜ÚÓ
‰Îfl Û‡‚ÌÂÌËfl (58) „ÂÌÂ‡ÚÓ˚ (59) „ÛÔÔ˚ ˝Í‚Ë‚‡ÎÂÌÚÌÓÒÚË ËÏÂ˛Ú ÒÎÂ‰Û˛˘Ë ÍÓÓ‰Ë̇Ú˚:
ξ = C 1 t + ϕ ( x ),
(57)
1
ξ = ψ ( x ),
2
η = ( C 1 + C 2 )u + λ ( x ),
µ = – ϕ' ( x ) + [ C 1 – ψ' ( x ) ]Q,
1
(58)
Ç˚˜ËÒÎÂÌËfl ÔÓ͇Á˚‚‡˛Ú, ˜ÚÓ ‚ÚÓÓ ӷӷ˘ÂÌËÂ
̇˷ÓΠۉ‡˜ÌÓ Ë fl‚ÎflÂÚÒfl ËÒÚÓ˜ÌËÍÓÏ „ÛÔÔ˚
ÒËÏÏÂÚËË, ÍÓÚÓ‡fl Á̇˜ËÚÂθÌÓ ·Ó„‡˜Â, ˜ÂÏ Û
Û‡‚ÌÂÌËÈ (55) Ë (57). èÓ˝ÚÓÏÛ Ï˚ ÔÓËÎβÒÚËÛÂÏ ÓÒÌÓ‚Ì˚ ÔÓÎÓÊÂÌËfl ÏÂÚÓ‰‡ ‡ÔËÓÌ˚ı
ÒËÏÏÂÚËÈ ËÏÂÌÌÓ Ì‡ ÏÓ‰ÂÎË (58).
éÔ‰ÂÎÂÌË 2. ÉÛÔÔ‡ ÔÂÓ·‡ÁÓ‚‡ÌËÈ (2) ̇Á˚‚‡ÂÚÒfl „ÛÔÔÓÈ ˝Í‚Ë‚‡ÎÂÌÚÌÓÒÚË, ÂÒÎË Í‡Ê‰Ó Û‡‚ÌÂÌË ‰‡ÌÌÓ„Ó ÒÂÏÂÈÒÚ‚‡ (‚ ‡ÒÒχÚË‚‡ÂÏÓÏ ÒÎÛ˜‡Â ÒÂÏÂÈÒÚ‚‡ (58) Ò Î˛·˚ÏË ÙÛÌ͈ËflÏË
ÄäìëíàóÖëäàâ ÜìêçÄã ÚÓÏ 50 ‹ 4 2004
(59)
„‰Â
‡ ‚ ͇˜ÂÒÚ‚Â ‚ÚÓÓÈ ‚ÓÁ¸ÏÂÏ
∂ ∂u
∂u
----- ------ – Q ( x, u ) ------ = F ( x, u ).
∂t ∂x
∂t
9
(61)
µ = [ C 2 – ψ' ( x ) ]F,
2
„‰Â ϕ, ψ, λ – ÔÓËÁ‚ÓθÌ˚ ÙÛÌ͈ËË ÓÚ x. ùÚÓ ÓÁ̇˜‡ÂÚ, ˜ÚÓ Û‡‚ÌÂÌË (58) ËÏÂÂÚ ·ÂÒÍÓ̘ÌÓÏÂÌÛ˛ ‡Î„·Û Lε Ò ·‡ÁËÒÓÏ
∂
∂
Y 1 = ϕ ( x ) ----- – ϕ' ( x ) -------,
∂t
∂Q
∂
∂
∂
Y 2 = ψ ( x ) ------ – ψ' ( x )Q ------- – ψ' ( x )F ------,
∂x
∂Q
∂F
∂
∂
∂
∂
Y 3 = λ ( x ) ------ , Y 4 = t ----- + u ------ + Q -------,
∂u
∂t
∂u
∂Q
(62)
à·‡„ËÏÓ‚, êÛ‰ÂÌÍÓ*
10
∂
∂
Y 5 = u ------ + F ------.
∂u
∂F
ÒÚÌ˚ı ÔÓËÁ‚Ó‰Ì˚ı ÔÂ‚Ó„Ó ÔÓfl‰Í‡ ‰Îfl ÙÛÌ͈ËÈ Q(x, u), F(x, u):
á‡Ï˜‡ÌË 1. Ä̇Îӄ˘ÌÓ ÏÓÊÌÓ ÔÓ͇Á‡Ú¸,
˜ÚÓ Û‡‚ÌÂÌË (57) ËÏÂÂÚ ÒÂÏËÏÂÌÛ˛ ‡Î„·Û Lε,
̇ÚflÌÛÚÛ˛ ̇ ÓÔÂ‡ÚÓ˚
∂
Y 1 = ----- ,
∂t
∂
Y 2 = ------ ,
∂x
∂
∂
∂
Y 4 = t ----- + u ------ + P ------,
∂t
∂u
∂P
∂
∂
Y 6 = x ----- – ------,
∂t ∂P
∂
Y 3 = ------ ,
∂u
∂
∂
Y 5 = u ------ + F ------,
∂u
∂F
(63)
á‡Ï˜‡ÌË 2. ùÚ‡ ÚÂÓÂχ ÙÓÏÛÎËÛÂÚÒfl ‡Ì‡Îӄ˘ÌÓ ‰Îfl Û‡‚ÌÂÌËfl (57) Ò Á‡ÏÂÌÓÈ ÓÔÂ‡ÚÓÓ‚
(62) ̇ ÓÔÂ‡ÚÓ˚ (63), ‡ Û‡‚ÌÂÌËÈ (68) ̇
P = P ( u ),
чθÌÂȯË ‚˚˜ËÒÎÂÌËfl ÓÒÌÓ‚‡Ì˚ ̇ ÚÂÓÂÏÂ
Ó ÔÓÂ͈Ëflı, ÛÔÓÏflÌÛÚÓÈ ‚ ̇˜‡Î ‡Á‰Â· 4.2.
é·ÓÁ̇˜ËÏ ˜ÂÂÁ X, Z ÔÓÂ͈ËË „ÂÌÂ‡ÚÓ‡ (59)
„ÛÔÔ˚ ˝Í‚Ë‚‡ÎÂÌÚÌÓÒÚË Ì‡ ÙËÁ˘ÂÒÍË ÔÂÂÏÂÌÌ˚ t, x, u Ë ÔÂÂÏÂÌÌ˚ x, u, Q, F ÔÓËÁ‚ÓθÌ˚ı
˝ÎÂÏÂÌÚÓ‚, ÒÓÓÚ‚ÂÚÒÚ‚ÂÌÌÓ:
(64)
∂
2 ∂
1 ∂
2 ∂
Z = pr ( x, u, Q, F )(Y ) ≡ ξ ------ + η ------ + µ ------- + µ ------. (65)
∂x
∂u
∂Q
∂F
èÓ‰ÒÚ‡‚Îflfl (61) ‚ (64) Ë (65), ÔÓÎÛ˜ËÏ ÔÓÂ͈ËË,
ÓÔ‰ÂÎÂÌÌ˚ ÍÓÂÍÚÌÓ ‚ ÚÓÏ ÒÏ˚ÒÎÂ, ˜ÚÓ ÍÓÓ‰Ë̇Ú˚ X Á‡‚ËÒflÚ ÚÓθÍÓ ÓÚ t, x, u, ‡ ÍÓÓ‰Ë̇Ú˚ Z
– ÓÚ ÔÂÂÏÂÌÌ˚ı x, u, Q, F:
∂
∂
X = [ C 1 t + ϕ ( x ) ] ----- + ψ ( x ) ------ +
∂x
∂t
∂
+ [ ( C 1 + C 2 )u + λ ( x ) ] ------ ,
∂u
(70)
∂F
∂F
ψ( x) ------ + [(C 1 + C 2 )u + λ( x)] ------ + [ ψ'( x) – C 2 ]F = 0.
∂x
∂u
∂
∂
∂
Y 7 = x ------ – P ------ – F ------.
∂x
∂P
∂F
∂
1∂
2 ∂
X = pr ( t, x, u ) ( Y ) ≡ ξ ----- + ξ ------ + η ------ ,
∂t
∂x
∂u
∂Q
∂Q
ψ ( x ) ------- + [ ( C 1 + C 2 )u + λ ( x ) ] ------- +
∂x
∂u
+ [ ψ' ( x ) – C 1 ]Q + ϕ' ( x ) = 0,
(66)
F = F ( x, u ).
èËÏÂ 1. ÇÓÁ¸ÏÂÏ ‚ (61)
C 1 = 1,
ϕ ( x ) = 0,
C 2 = 0,
ψ ( x ) = k – x,
λ ( x ) = 0,
ÚÓ ÂÒÚ¸ ‚˚·ÂÂÏ Y' ∈ Lε ˜‡ÒÚÌÓ„Ó ‚ˉ‡
∂
∂
∂
∂
Y' = t ----- + ( x – k ) ------ + u ------ – F ------.
∂t
∂x
∂u
∂F
(71)
íÓ„‰‡ ¯ÂÌËÂÏ ÒËÒÚÂÏ˚ Û‡‚ÌÂÌËÈ (70) ÔÓÎÛ˜‡ÂÏ ÙÛÌ͈ËË
u
Q = Φ  ----------- ,
 x – k
1
u
F = ----------- Γ  ----------- ,

x – k x – k
(72)
„‰Â Φ, Γ – ÔÓËÁ‚ÓθÌ˚ ÙÛÌ͈ËË Ó‰ÌÓ„Ó ‡„ÛÏÂÌÚ‡. íÂÓÂχ Ó ÔÓÂ͈Ëflı ‰‡ÂÚ, ˜ÚÓ Û‡‚ÌÂÌËÂ
(58) ‚ˉ‡
∂ ∂u
u ∂u
1
u
----- ------ – Φ  ----------- ------ = ----------- Γ  -----------
 x – k ∂t
∂t ∂x
x – k  x – k
(73)
∂
∂
Z = ψ ( x ) ------ + [ ( C 1 + C 2 )u + λ ( x ) ] ------ +
∂u
∂x
(67)
∂
∂
+ [ – ϕ' ( x ) + ( C 1 – ψ' ( x )Q ) ] ------- + ( C 1 – ψ' ( x ) )F ------.
∂Q
∂F
‰ÓÔÛÒ͇ÂÚ, ̇fl‰Û Ò Ó˜Â‚Ë‰Ì˚Ï „ÂÌÂ‡ÚÓÓÏ ÔÂÂÌÓÒ‡ ÔÓ ‚ÂÏÂÌË X0 = ∂/∂t, Ó‰ËÌ ‰ÓÔÓÎÌËÚÂθÌ˚È
ÓÔÂ‡ÚÓ
èËÏÂÌËÚÂθÌÓ Í Û‡‚ÌÂÌ˲ (58) ÚÂÓÂχ Ó ÔÓÂ͈Ëflı Á‚Û˜ËÚ Ú‡Í.
íÂÓÂχ Ó ÔÓÂ͈Ëflı. éÔÂ‡ÚÓ X, ÓÔ‰ÂÎÂÌÌ˚È ÙÓÏÛÎÓÈ (66), fl‚ÎflÂÚÒfl ËÌËÙËÌËÚÂÁËχθÌÓÈ ÒËÏÏÂÚËÂÈ Û‡‚ÌÂÌËfl (58) Ò ÙÛÌ͈ËflÏË
àÁ Û‡‚ÌÂÌËfl X'(J) = 0 ̇ıÓ‰ËÏ ËÌ‚‡ˇÌÚ˚
Q = Q ( x, u ),
F = F ( x, u )
(68)
ÚÓ„‰‡ Ë ÚÓθÍÓ ÚÓ„‰‡, ÍÓ„‰‡ ÒËÒÚÂχ Û‡Ï‚ÌÂÌËÈ
(68) ËÌ‚‡ˇÌÚ̇ ÓÚÌÓÒËÚÂθÌÓ „ÛÔÔ˚ Ò „ÂÌÂ‡ÚÓÓÏ Z, ÓÔ‰ÂÎÂÌÌ˚Ï ÙÓÏÛÎÓÈ (67), ÚÓ ÂÒÚ¸
ÍÓ„‰‡
Z [ Q – Q ( x, u ) ] = 0,
Z [ F – F ( x, u ) ] = 0,
(69)
„‰Â ÔÂÂÏÂÌÌ˚ Q, F ̇‰Ó Á‡ÏÂÌËÚ¸ ̇ ÙÛÌ͈ËË
Q(x, u), F(x, u) ‚ ÒÓÓÚ‚ÂÚÒÚ‚ËË Ò Û‡‚ÌÂÌËflÏË (68).
Ç‚Ë‰Û (67), Û‡‚ÌÂÌËfl (69) ÏÓÊÌÓ ÔÂÂÔËÒ‡Ú¸ ‚ ‚ˉ ÒÎÂ‰Û˛˘ÂÈ ÒËÒÚÂÏ˚ ÎËÌÂÈÌ˚ı Û‡‚ÌÂÌËÈ ‚ ˜‡-
∂
∂
∂
X' = t ----- + ( x – k ) ------ + u ------ .
∂t
∂x
∂u
t
λ = ----------- ,
x–k
(74)
u
µ = ----------- .
x–k
àÌ‚‡ˇÌÚÌÓ ¯ÂÌË u = (x – k)G(λ) Û‡‚ÌÂÌËfl
(73) ‰ÓÎÊÌÓ Û‰Ó‚ÎÂÚ‚ÓflÚ¸ Ó·˚ÍÌÓ‚ÂÌÌÓÏÛ ‰ËÙÙÂÂ̈ˇθÌÓÏÛ Û‡‚ÌÂÌ˲
λG'' + ( Φ ( G )G' )' + Γ ( G ) = 0.
àÌÚÂÂÒ Ô‰ÒÚ‡‚ÎflÂÚ ˜‡ÒÚÌ˚È ÒÎÛ˜‡È Û‡‚ÌÂÌËfl
(73) Ò ÎËÌÂÈÌ˚ÏË ÙÛÌ͈ËflÏË Φ, Γ, ÒÓÓÚ‚ÂÚÒÚ‚Û˛˘ËÈ ÔÓfl‚ÎÂÌ˲ ‚ ΂ÓÈ ˜‡ÒÚË Û‡‚ÌÂÌËfl (73) Í‚‡‰‡Ú˘ÌÓ-ÌÂÎËÌÂÈÌÓ„Ó ˜ÎÂ̇:
∂ ∂u
u ∂u
u
----- ------ – α ----------- ------ = – β ------------------2 .
∂t ∂x
k – x ∂t
(k – x)
(75)
ÄäìëíàóÖëäàâ ÜìêçÄã ÚÓÏ 50 ‹ 4 2004
èË̈ËÔ ‡ÔËÓÌÓ„Ó ËÒÔÓθÁÓ‚‡ÌËfl ÒËÏÏÂÚËÈ
ùÚÓÚ ÒÎÛ˜‡È ÓÚ‰ÂθÌÓ ‡ÒÒÏÓÚÂÌ ‚ ‡Á‰ÂΠ5.
èËÏÂ 2. è˂‰ÂÏ ÚÂÔÂ¸ ÔËÏÂ Û‡‚ÌÂÌËfl, Ëϲ˘Â„Ó ‰‚ ‰ÓÔÓÎÌËÚÂθÌ˚ ÒËÏÏÂÚËË.
ÑÎfl ˝ÚÓ„Ó Ì‡‰Ó ‚ÁflÚ¸ ‰‚‡ „ÂÌÂ‡ÚÓ‡ ˝Í‚Ë‚‡ÎÂÌÚÌÓÒÚË, Ó·‡ÁÛ˛˘Ëı ‰‚ÛÏÂÌÛ˛ ‡Î„·Û. Ç Í‡˜ÂÒÚ‚Â ÔÂ‚Ó„Ó ËÁ ÌËı ‚˚·ÂÂÏ ÒÌÓ‚‡ (71), ‡ ‚ÚÓÓÈ „ÂÌÂ‡ÚÓ Y'' ̇ȉÂÏ, ÔÓÎÓÊË‚ ‚ (61)
C 1 = 0,
C 2 = 1,
ϕ ( x ) = ψ ( x ) = λ ( x ) = x – k,
ÚÓ ÂÒÚ¸ ‚˚·ÂÂÏ
∂
∂
∂
∂
Y'' = ( x – k )  ----- + ------ + ( u + x – k ) ------ – ( 1 + Q ) -------.
 ∂t ∂x
∂u
∂Q
éÔÂ‡ÚÓ˚ Y', Y'' ÍÓÏÏÛÚËÛ˛Ú Ë, Á̇˜ËÚ, Ó·‡ÁÛ˛Ú ‡·ÂÎÂ‚Û ‡Î„·Û ãË. èËÏÂÌË‚ Í ˝ÚËÏ ‰‚ÛÏ
ÓÔÂ‡ÚÓ‡Ï ÚÂÓÂÏÛ Ó ÔÓÂ͈Ëflı, ÚÓ ÂÒÚ¸ ̇
Ô‡ÍÚËÍ ¯˂ Û‡‚ÌÂÌËfl (70) Ò ÍÓÓ‰Ë̇ڇÏË
ÓÔÂ‡ÚÓ‡ Y'' Ë Ò ÙÛÌ͈ËflÏË Q, F ‚ˉ‡ (70), ÔÓÎÛ˜ËÏ:
u
Q = – 1 + A exp  – ----------- ,
 x – k
B
u
F = ----------- exp  ----------- ,
 x – k
x–k
A, B = const.
èÓÒÚÓflÌ̇fl L ̇ıÓ‰ËÚÒfl ÔÓ‰ÒÚ‡ÌÓ‚ÍÓÈ ‚ Û‡‚ÌÂÌË (76).
èËÏÂ 3. íÂÔÂ¸ Ó·‡ÚËÏÒfl Í Û‡‚ÌÂÌ˲ (57)
Ë ÔËÏÂÌËÏ ÚÂÓÂÏÛ Ó ÔÂ͈Ëflı Í ÍÓÏ·Ë̇ˆËË Y =
Y3 + Y5 + Y6 + Y7 ·‡ÁËÒÌ˚ı ÓÔÂ‡ÚÓÓ‚ (63), ÚÓ ÂÒÚ¸
Í ÒÎÂ‰Û˛˘ÂÏÛ „ÂÌÂ‡ÚÓÛ ˝Í‚Ë‚‡ÎÂÌÚÌÓÒÚË:
∂
∂
∂
∂
Y = x ----- + x ------ + ( 1 + u ) ------ – ( 1 + P ) ------.
∂t
∂x
∂u
∂P
èÓ ÙÓÏÛ·Ï, ‡Ì‡Îӄ˘Ì˚Ï (64) Ë (65), ÔÓÎÛ˜‡ÂÏ
Â„Ó ÔÓÂ͈ËË
∂
∂
∂
X = x ----- + x ------ + ( 1 + u ) ------ .
∂t
∂x
∂u
(78)
∂
∂
∂
Z = x ------ + ( 1 + u ) ------ – ( 1 + P ) ------.
∂x
∂u
∂P
(79)
ê¯˂ Û‡‚ÌÂÌËfl Z[P(u) – P] = 0, Z[F(x, u) – F] = 0
(Ò. (69) Ë ÒÏ. á‡Ï˜‡ÌË 2), ÍÓÚÓ˚ ËÏÂ˛Ú ÍÓÌÍÂÚÌ˚È ‚ˉ
dP
( 1 + u ) ------- + ( 1 + P ) = 0,
du
∂F
∂F
x ------ + ( 1 + u ) ------ = 0,
∂x
∂u
̇ıÓ‰ËÏ
x
F = Γ  ------------ ,
 1 + u
èÓ ÚÂÓÂÏÂ Ó ÔÓÂ͈Ëflı, Û‡‚ÌÂÌËÂ
∂ ∂u
∂u
u
----- ------ +  1 – A exp  – -----------  ------ =
 x – k  ∂t
∂t ∂x 
B
u
= ----------- exp  -----------
 x – k
x–k
∂
∂
∂
X' = t ----- + ( x – k ) ------ + u ------ ,
∂x
∂u
∂t
∂
∂
∂
X'' = ( x – k )  ----- + ------ + ( u + x – k ) ------ .
 ∂t ∂x
∂u
K
P = ------------ – 1,
1+u
K = const.
ëΉӂ‡ÚÂθÌÓ, Û‡‚ÌÂÌËÂ
(76)
‰ÓÔÛÒ͇ÂÚ ‰‚‡ ‰ÓÔÓÎÌËÚÂθÌ˚ı ÓÔÂ‡ÚÓ‡:
(77)
Ç ˜‡ÒÚÌÓÒÚË, ÏÓÊÌÓ Ì‡ÈÚË ¯ÂÌËÂ, ËÌ‚‡ˇÌÚÌÓ ÓÚÌÓÒËÚÂθÌÓ ‰‚ÛıÔ‡‡ÏÂÚ˘ÂÒÍÓÈ „ÛÔÔ˚
ÒËÏÏÂÚËË Ò ‰‚ÛÏfl „ÂÌÂ‡ÚÓ‡ÏË (77). Ä ËÏÂÌÌÓ,
̇‰Ó ¯ËÚ¸ ÒËÒÚÂÏÛ Û‡‚ÌÂÌËÈ X'(J) = 0, X''(J) = 0.
àÁ ÔÂ‚Ó„Ó Û‡‚ÌÂÌËfl ÔÓÎÛ˜‡ÂÏ ‰‚‡ ËÌ‚‡ˇÌÚ‡
x–k
λ = ----------- ,
t
11
u
µ = --- ,
t
∂ ∂u
x
K ∂u
----- ------ +  1 – ------------ ------ = Γ  ------------



∂t ∂x
1 + u
1 + u ∂t
(80)
‚ˉ‡ (57) ‰ÓÔÛÒ͇ÂÚ, ̇fl‰Û Ò X0 = ∂/∂t, ‰ÓÔÓÎÌËÚÂθÌ˚È ÓÔÂ‡ÚÓ X, ÓÔ‰ÂÎÂÌÌ˚È ‚ (78). ÇÓÒÔÓθÁÛÂÏÒfl ËÏ, ˜ÚÓ·˚ ÔÓÒÚÓËÚ¸ ËÌ‚‡ˇÌÚÌÓÂ
¯ÂÌËÂ. ç‡È‰fl ËÌ‚‡ˇÌÚ˚ λ = t – x, µ = (u + 1)/x
ÓÔÂ‡ÚÓ‡ X, ËÏÂÂÏ ÒÎÂ‰Û˛˘ËÈ ‚ˉ ËÌ‚‡ˇÌÚÌÓ„Ó ¯ÂÌËfl:
u = xΦ ( λ ),
λ = t – x.
èÓ‰ÒÚ‡ÌÓ‚ÍÓÈ ‚ (80) ̇ıÓ‰ËÚÒfl Ó·˚ÍÌÓ‚ÂÌÌÓÂ
‰ËÙÙÂÂ̈ˇθÌÓ Û‡‚ÌÂÌËÂ
K ( Φ'/Φ )' + Φ' = Γ ( Φ ).
(81)
á‡ÏÂÚËÏ, ˜ÚÓ ÔË Ï‡Î˚ı u Û‡‚ÌÂÌË (80) fl‚ÎflÂÚÒfl ıÓÓ¯ÂÈ ‡ÔÔÓÍÒËχˆËÂÈ, ӷ·‰‡˛˘ÂÈ ‰ÓÔÓÎÌËÚÂθÌÓÈ ÒËÏÏÂÚËÂÈ, ‰Îfl ÏÓ‰ÂÎÂÈ, ÓÔËÒ˚‚‡ÂÏ˚ı Û‡‚ÌÂÌËflÏË ÚËÔ‡ (55).
‡ ‚ÚÓÓ Û‡‚ÌÂÌË ‰‡ÂÚ
λ
µ = Lλ + λ ln  ------------ ,
 1 – λ
L = const.
èÓ‰ÒÚ‡‚Ë‚ Ò˛‰‡ Á̇˜ÂÌËfl µ, λ ÔÓÎÛ˜‡ÂÏ Ú‡ÍÓÈ ‚ˉ
ËÌ‚‡ˇÌÚÌÓ„Ó ¯ÂÌËfl:
x–k
u = ( x – k ) L + ln  ------------------- .
 t – x + k
ÄäìëíàóÖëäàâ ÜìêçÄã ÚÓÏ 50 ‹ 4 2004
5. Ç˚‚Ó‰ Û‡‚ÌÂÌËÈ
Ë Ó·ÒÛʉÂÌË ÙËÁËÍË ÔÓˆÂÒÒÓ‚
åÓ‰Âθ (55) ‚ÓÁÌË͇ÂÚ ‚ ÌÂÒÍÓθÍËı Á‡‰‡˜‡ı.
ê‡ÒÒÏÓÚËÏ ‚̇˜‡Î ÍÓη‡ÌËfl ÒÊËχÂÏÓ„Ó „‡Á‡
‚ÌÛÚË ˆËÎË̉‡ Ò ÔÓÔÂ˜Ì˚Ï Ò˜ÂÌËÂÏ S, ÍÓÚÓ˚È Á‡Í˚Ú ÔÓ‰‚ËÊÌ˚Ï ÔÓ¯ÌÂÏ Ï‡ÒÒ˚ m. ÑÌÓ
ˆËÎË̉‡ (x = 0) ÌÂÔÓ‰‚ËÊÌÓ, ÍÓÓ‰Ë̇ڇ x ÓÚÒ˜ËÚ˚‚‡ÂÚÒfl ‚‰Óθ Ó·‡ÁÛ˛˘ÂÈ ‚‚Âı. èÓ¯Â̸ ÏÓ-
à·‡„ËÏÓ‚, êÛ‰ÂÌÍÓ*
12
Ç Û‡‚ÌÂÌËË (84) ËÒÔÓθÁÓ‚‡Ì˚ ·ÂÁ‡ÁÏÂÌ˚Â
Ó·ÓÁ̇˜ÂÌËfl
u
1'
p
-,
u = ------2
c ρ
1
10
2'
π
–π
ωt
T = --------1 ,
π
1m
β = --- ------g ,
πm
„‰Â t1 – “ωÎÂÌÌÓ” ‚ÂÏfl, ÓÔËÒ˚‚‡˛˘Â ÛÒÚ‡ÌÓ‚ÎÂÌË ÍÓη‡ÌËÈ ‚ pÂÁÓ̇ÚÓÂ, mg = ρSH – χÒÒ‡ „‡Á‡ ‚ ˆËÎË̉Â. é˜Â‚ˉÌÓ, Á‡ÏÂÌÓÈ ÔÂÂÏÂÌÌ˚ı Ë ÔÂÂÓ·ÓÁ̇˜ÂÌËÂÏ ÍÓÌÒÚ‡ÌÚ
2
t
0
–10
ÇÎËflÌË ÌËÁÍÓ˜‡ÒÚÓÚÌÓÈ ‰ËÒÔÂÒËË Ì‡ ÔÓˆÂÒÒ ËÒ͇ÊÂÌËfl ‚ÓÎÌ˚. èÓÙËÎË ÔÓÒÚÓÂÌ˚ ‰Îfl ‡ÒÒÚÓflÌËÈ x
= 0.2 (ÍË‚˚ 1 Ë 1') Ë x = 0.5 (ÍË‚˚ 2 Ë 2'). ëÔÎÓ¯Ì˚ ÍË‚˚ ÓÔËÒ˚‚‡˛ÚÒfl ‡‚ÚÓÏÓ‰ÂθÌ˚Ï ¯ÂÌËÂÏ
(84) ‰Îfl β = 3. òÚËıÓ‚˚ ÍË‚˚ ÔÓÒÚÓÂÌ˚ Ò Û˜ÂÚÓÏ ÚÓθÍÓ ÌÂÎËÌÂÈÌÓÒÚË (β = 0, ‰ËÒÔÂÒËË ÌÂÚ) Ë Ô˂‰ÂÌ˚ ‰Îfl Ò‡‚ÌÂÌËfl Ò ÔÓÙËÎflÏË, ËÁÓ·‡ÊÂÌÌ˚ÏË
ÒÔÎÓ¯Ì˚ÏË ÍË‚˚ÏË.
ÊÂÚ ÒÓ‚Â¯‡Ú¸ ÍÓη‡ÌËfl, ÒÏ¢‡flÒ¸ ̇ ‡ÒÒÚÓflÌË ζ ÓÚÌÓÒËÚÂθÌÓ Ò‰ÌÂ„Ó ÔÓÎÓÊÂÌËfl x = H.
ëËÒÚÂχ, ÓÔËÒ˚‚‡˛˘‡fl ‰‚ËÊÂÌË ÔÓ¯Ìfl Ò Û˜ÂÚÓÏ ÌÂÎËÌÂÈÌ˚ı ‚ÓÎÌÓ‚˚ı ‰‚ËÊÂÌËÈ „‡Á‡, ËÏÂÂÚ
‚ˉ
∂ζ
ρc ------ = p ( t – ) – p ( t + ),
∂t
ξ = ωt + π,
m∂ ζ
---- -------2- = p ( t + ) + p ( t – ). (82)
S ∂t
2
á‰ÂÒ¸ ρ Ë c – ÔÎÓÚÌÓÒÚ¸ Ë ÒÍÓÓÒÚ¸ Á‚Û͇ ‚ „‡ÁÂ,
p(t) – ÙÓχ β·ÓÈ ËÁ ‰‚Ûı ·Â„Û˘Ëı ̇‚ÒÚ˜Û
‰Û„ ‰Û„Û ‚ÓÎÌ ‡ÍÛÒÚ˘ÂÒÍÓ„Ó ‰‡‚ÎÂÌËfl. Ä„ÛÏÂÌÚ˚ ÒÓ‰ÂÊ‡Ú Ò‰‚Ë„ ‚Ó ‚ÂÏÂÌË, ÓÔ‰ÂÎflÂÏ˚È
‰ÎËÌÓÈ H ÂÁÓ̇ÚÓ‡ Ë ÌÂÎËÌÂÈÌ˚ÏË Ò‚ÓÈÒÚ‚‡ÏË
„‡Á‡:
t = ξ – ∆T ,
T = x,
u = πεU
Û‡‚ÌÂÌË (83) Û‰‡ÂÚÒfl ÔÂÓ·‡ÁÓ‚‡Ú¸ Í ‚Ë‰Û (55).
èË·ÎËÊÂÌÌ˚ ¯ÂÌËfl Û‡‚ÌÂÌËfl (83), Óڂ˜‡˛˘Ë ÙËÁ˘ÂÒÍË ËÌÚÂÂÒÌ˚Ï ÔÓÒÚ‡ÌÓ‚Í‡Ï Á‡‰‡˜Ë, ÔÓÎÛ˜ÂÌ˚ ‚ ‡·ÓÚ [25]. é‰Ì‡ÍÓ Ë fl‰ ÚÓ˜Ì˚ı ¯ÂÌËÈ ËÏÂÂÚ ‚‡ÊÌ˚È ÙËÁ˘ÂÒÍËÈ ÒÏ˚ÒÎ.
ê‡ÒÒÏÓÚËÏ, ̇ÔËÏÂ, ÚÓ˜ÌÓ ¯ÂÌËÂ, ÔÓÎÛ˜‡ÂÏÓÂ Ò ÔÓÏÓ˘¸˛ „ÂÌÂ‡ÚÓ‡ ‡ÒÚflÊÂÌËfl X3 (56).
ëÓÓÚ‚ÂÚÒÚ‚Û˛˘Ë ËÌ‚‡ˇÌÚ˚ Ë ‡‚ÚÓÏÓ‰Âθ̇fl
ÙÓχ ËÏÂ˛Ú ‚ˉ
λ = xt,
µ = ux ,
2
1
u ( x, t ) = -----2 Φ ( λ ).
x
èÓ‰ÒÚ‡Ìӂ͇ ÔÓÒΉÌÂ„Ó ‚˚‡ÊÂÌËfl ‚ (55) ÔË‚Ó‰ËÚ Í Ó·˚ÍÌÓ‚ÂÌÌÓÏÛ ‰ËÙÙÂÂ̈ˇθÌÓÏÛ Û‡‚ÌÂÌ˲
ΦΦ'' + Φ' – λΦ'' + Φ' – βΦ = 0.
2
Ö„Ó ˜‡ÒÚÌ˚Ï ¯ÂÌËÂÏ, ‚ Ô‰ÂΠβ
0 ÔÂÂıÓ‰fl˘ËÏ ‚ ¯ÂÌË u = –x/t ‰Îfl ËχÌÓ‚˚ı ‚ÓÎÌ, ·Û‰ÂÚ
β 2
Φ ( λ ) = – λ + --- λ ,
6
β 2 t
u ( x, t ) = --- t – --.
6
x
(84)
èÓ ‡Ì‡ÎÓ„ËË Ò ËÁ‚ÂÒÚÌÓÈ Ôӈ‰ÛÓÈ ÔÓÒÚÓÂÌËfl
ÔËÎÓÓ·‡ÁÌÓ„Ó Ò˄̇· [4], ‚ ÍÓÚÓÓÈ ÔÂËӉ˘ÂÒÍË ÔÓ‰ÓÎÊÂÌÌÓ ¯ÂÌË u = –x/t ËÒÔÓθÁÛÂÚÒfl
‰Îfl ÓÔËÒ‡ÌËfl „·‰ÍËı ÎËÌÂÈÌ˚ı Û˜‡ÒÚÍÓ‚ ÔÓÓÙËÎfl (ÒÏ. ËÒ. 1, ¯ÚËıÓ‚˚ ÍË‚˚Â), Á‰ÂÒ¸ Ú‡ÍÊ ÌÂÓ·ıÓ‰ËÏÓ ÔÓ‰ÓÎÊËÚ¸ ¯ÂÌË (84) Ò ÔÂËÓ‰ÓÏ 2π. 쉇Ì˚ ÙÓÌÚ˚ ÎÓ͇ÎËÁÛ˛ÚÒfl ‚ ÚӘ͇ı
3 2 π
3
t n = π ( 2n + 1 ) + ------ –  ------ – ----- ,
 βx
βx
3
2
H
p 
- .
t ± = t ± ----  1 – ε ------2 
c
c ρ
n = 0, ± 1, ± 2, …,
àÒÔÓθÁÛfl ÒÔÓÒÓ· ÔÂÓ·‡ÁÓ‚‡ÌËfl ÙÛÌ͈ËÓ̇θÌ˚ı Û‡‚ÌÂÌËÈ ÚËÔ‡ (82) Í ‰ËÙÙÂÂ̈ˇθÌ˚Ï
˝‚ÓβˆËÓÌÌ˚Ï Û‡‚ÌÂÌËflÏ, ÓÔËÒ‡ÌÌ˚È ‚ ‡·ÓÚÂ
[24], ‚ ÓÍÂÒÚÌÓÒÚË ÓÒÌÓ‚ÌÓ„Ó ‡ÍÛÒÚ˘ÂÒÍÓ„Ó ÂÁÓ̇ÌÒ‡ ωH/c = π + ∆ („‰Â ∆ – χ·fl ‡ÒÒÚÓÈ͇ ˜‡ÒÚÓÚ˚), Û‰‡ÂÚÒfl ÔÓÎÛ˜ËÚ¸ [25]
∂ ∂U
∂U
∂U
------ ------- + ∆ ------- – πεU ------- = – βU.
∂ξ ∂T
∂ξ
∂ξ
(83)
ÍÓÓ‰Ë̇Ú˚ ÍÓÚÓ˚ı ‚˚˜ËÒÎfl˛ÚÒfl ËÁ ÛÒÎÓ‚Ëfl ‡‚ÂÌÒÚ‚‡ ÌÛβ Ò‰ÌÂ„Ó Á‡ ÔÂËÓ‰ Á̇˜ÂÌËfl ÙÛÌ͈ËË u(x, t).
ä‡Í ÔÓ͇Á‡ÌÓ Ì‡ ËÒ. 1, Û˜ÂÚ ÌËÁÍÓ˜‡ÒÚÓÚÌÓÈ
‰ËÒÔÂÒËË (β ≠ 0) ÔË‚Ó‰ËÚ Í ÌÂÒËÏÏÂÚ˘ÌÓÏÛ
ËÒ͇ÊÂÌ˲ ÙÓÏ˚ ‚ÓÎÌ˚. ÑÎËÚÂθÌÓÒÚ¸ Ù‡Á˚
ÒʇÚËfl Ó͇Á˚‚‡ÂÚÒfl ·ÓΠÍÓÓÚÍÓÈ, ‡ ‡ÁÂÊÂÌËfl – Û‚Â΢ÂÌÌÓÈ. äË‚˚ ̇ ËÒ. 1 ÔÓÒÚÓÂÌ˚
‰Îfl β = 3 Ë ·ÂÁ‡ÁÏÂÌ˚ı ‡ÒÒÚÓflÌËÈ x = 0.2 (ÍË‚˚ 1 Ë 1'), x = 0.5 (ÍË‚˚ 2 Ë 2'). ê‡ÁÛÏÂÂÚÒfl,
ÄäìëíàóÖëäàâ ÜìêçÄã ÚÓÏ 50 ‹ 4 2004
èË̈ËÔ ‡ÔËÓÌÓ„Ó ËÒÔÓθÁÓ‚‡ÌËfl ÒËÏÏÂÚËÈ
ÔÓÒÚÓÂÌË ÒÔ‡‚‰ÎË‚Ó ‚ÔÎÓÚ¸ ‰Ó ‡ÒÒÚÓflÌËÈ
x ≤ 3 3 /πβ.
éÔËÒ˚‚‡ÂÏ˚È ËÒ. 1 ı‡‡ÍÚÂ ËÒ͇ÊÂÌËÈ ÔÓÙËÎfl ‚ÓÎÌ˚ ‡Ì‡Îӄ˘ÂÌ Ì‡·Î˛‰‡ÂÏÓÏÛ ‚ ˝ÍÒÔÂËÏÂÌÚ‡ı Ò ‰ËÙ‡„ËÛ˛˘ËÏË Ô͇ۘÏË ·Óθ¯ÓÈ
ËÌÚÂÌÒË‚ÌÓÒÚË. í‡ÍÓ Ê Ôӂ‰ÂÌË Ô‰Ò͇Á˚‚‡ÂÚ ÚÂÓËfl (ÒÏ. [26, 14]). àÏÂÌÌÓ ÔÓ˝ÚÓÏÛ ‚ÚÓ‡fl
ÙËÁ˘ÂÒ͇fl Á‡‰‡˜‡, Ò‚flÁ‡Ì̇fl Ò ÏÓ‰Âθ˛ (55), ÓÚÌÓÒËÚÒfl Í ÚÂÓËË ÌÂÎËÌÂÈÌ˚ı ‡ÍÛÒÚ˘ÂÒÍËı ÔÛ˜ÍÓ‚ [26]. ê‡ÒÒÏÓÚËÏ Û‡‚ÌÂÌË ïÓıÎÓ‚‡-ᇷÓÎÓÚÒÍÓÈ ‚ ÙÓÏ (ÒÏ. [14], ÒÚ. 346):
∂ ∂p
ε ∂p 1 ∂p  ∂Ψ 1  ∂Ψ 2
- p ------ – --- ------ -------- + --- ------------ ------ – ------+
∂τ ∂x c 3 ρ ∂τ c ∂τ  ∂x 2  ∂r  
∂p ∂Ψ p
+ ------ -------- + --- ∆ ⊥ Ψ
∂r ∂r 2
(85)
c
= --- ∆ ⊥ p.
2
ì‡‚ÌÂÌË (85) ÓÔËÒ˚‚‡ÂÚ ÍÛ„Î˚ ‚ ÔÓÔÂ˜ÌÓÏ
Ò˜ÂÌËË ÔÛ˜ÍË, ‰Îfl ÍÓÚÓ˚ı
1∂
∂
∆ ⊥ = -------2 + --- -----,
∂r r ∂r
2
∂ ∂p q' ( x ) ∂p
ε ∂p
- p ------ + Φ ( x ) p =
----- ------ – ------------ ------ – ------∂τ ∂x
c ∂τ c 3 ρ ∂τ
c 2
= – -----2 Φ ( x ) p.
a
r
∂Ψ 1  ∂Ψ
-------- + --- -------- = ---- f ( x ) + q' ( x ),


2
∂x 2 ∂r
1
t = c τ + --- q ( x ) ,
c
(86)
„‰Â f, q' – ËÁ‚ÂÒÚÌ˚ ÙÛÌ͈ËË, ÓÔËÒ˚‚‡˛˘Ë ËÁÏÂÌÂÌË ÔÓ͇Á‡ÚÂÎfl ÔÂÎÓÏÎÂÌËfl ‚ Ò‰Â. éÔ‰ÂÎflfl ˝ÈÍÓ̇ΠËÁ (86),
2
r
Ψ = ---- Φ ( x ) + q ( x ),
2
Φ' + Φ = f ,
2
Ô˂‰ÂÏ (85) Í ÒÎÂ‰Û˛˘ÂÏÛ ‚ˉÛ:
ε ∂p
∂ ∂p f ∂p 1 ∂p ν f
- p ------ + Φp =
----- ------ + ---- ν ------ – --- ------  ----- ------2 + q' – ------3


∂τ ∂x Φ ∂ν c ∂τ 2 Φ
c ρ ∂τ
(87)
2
c 2  ∂ p 1 ∂p
= --- Φ --------2 + --- ------ ,
2  ∂ν ν ∂ν
2
ν = rΦ(x). Ç ÔË·ÎËÊÂÌËË ÌÂÎËÌÂÈÌÓÈ „ÂÓÏÂÚ˘ÂÒÍÓÈ ‡ÍÛÒÚËÍË Ô‡‚Û˛ ˜‡ÒÚ¸ (85) ËÎË (87) ÔÓ·„‡˛Ú ‡‚ÌÓÈ ÌÛβ. åÓÊÌÓ ÛÚÓ˜ÌËÚ¸ ˝ÚÓ ÔË·ÎËÊÂÌËÂ Ë Û˜ÂÒÚ¸ ‰ËÙ‡ÍˆËÓÌÌ˚ ÔÓÔ‡‚ÍË, ÂÒÎË
‚ÓÒÔÓθÁÓ‚‡Ú¸Òfl ‰Îfl Ô‡‚ÓÈ ˜‡ÒÚË (87) ÒÎÂ‰Û˛˘ÂÈ ÏÓ‰Âθ˛:
p 1 ∂p
 ∂-------+ --- ----- ∂ν 2 ν ∂ν
2
– -----2 p,
a
(88)
ÄäìëíàóÖëäàâ ÜìêçÄã ÚÓÏ 50 ‹ 4 2004
(89)
Ç ˜‡ÒÚÌÓÒÚË, ‰Îfl ÒÙÓÍÛÒËÓ‚‡ÌÌÓÈ ‚ÓÎÌ˚, ÔÓ·„‡fl Φ = 1/(x – k), q' = 0 (k – ÙÓÍÛÒÌÓ ‡ÒÒÚÓflÌËÂ),
҂‰ÂÏ Û‡‚ÌÂÌË (89) Í ‚Ë‰Û (75), „‰Â ÚÂÔÂ¸ u =
= p(x – k), a = ε/c3ρ, β = c/a2.
á‡ÏÂÌÓÈ ÔÂÂÏÂÌÌ˚ı
ε
- exp ( Φ ( x ) d x )
u = ------3
c ρ
∫
Û‡‚ÌÂÌË (89) Ò‚Ó‰ËÚÒfl Í ·ÓΠÔÓÒÚÓÈ ÙÓÏÂ
∂ ∂u
∂u
Φ ( x)
-u,
----- ------ – Q ( x )u ------ = – ------------2
∂t ∂x
∂t
a
2
(90)
„‰Â
∫
Q ( x ) = exp ( – Φ ( x ) d x ).
2
2
2
„‰Â a – ËÒıӉ̇fl ¯ËË̇ Ô͇ۘ. á‡ÏÂÚËÏ, ˜ÚÓ ‰Îfl
ÔÓÔÂ˜ÌÓÈ ÒÚÛÍÚÛ˚, ÓÔËÒ˚‚‡ÂÏÓÈ ÙÛÌ͈ËÂÈ
ÅÂÒÒÂÎfl J0( 2 ν/a), Ô‰ÒÚ‡‚ÎÂÌË (88) Ó͇Á˚‚‡ÂÚÒfl ÚÓ˜Ì˚Ï. àÁÏÂÌflfl ÚÂÔÂ¸ Ô‡‚Û˛ ˜‡ÒÚ¸ (87) ‚
ÒÓÓÚ‚ÂÚÒÚ‚ËË Ò (88) Ë ÛÒÚÂÏÎflfl ν Í ÌÛβ, ÔÓÎÛ˜ËÏ Û‡‚ÌÂÌËÂ, ÓÔËÒ˚‚˛˘Â ÔÓΠ‡ÍÛÒÚ˘ÂÒÍËı
ÔÛ˜ÍÓ‚ ‚·ÎËÁË ÓÒË:
x 1
τ = t – -- – --- Ψ ( x, r ),
c c
ε = (γ + 1)/2 – ÌÂÎËÌÂÈÌ˚È Ô‡‡ÏÂÚ, Ψ – ˝ÈÍÓ̇Î.
ÑÎfl Ó‰ÌÓÓ‰ÌÓÈ Ò‰˚ ‡ÒÒÚÓflÌË x, ÔÓȉÂÌÌÓÂ
‚ÓÎÌÓÈ, ÓÚÒ˜ËÚ˚‚‡ÂÚÒfl ‚‰Óθ ÌÂÍÓÚÓÓÈ ÔflÏÓÈ;
‚ ÌÂÓ‰ÌÓÓ‰ÌÓÈ Ò‰ x ÏÓÊÂÚ ËÁÏÂflÚ¸Òfl ‚‰Óθ
ÎÛ˜‡, fl‚Îfl˛˘Â„ÓÒfl ÓÒ¸˛ Ô͇ۘ [27] . é„‡Ì˘˂‡flÒ¸ ‡ÒÒÏÓÚÂÌËÂÏ ÔÓÎfl ‚·ÎËÁË ÓÒË, ÔÓÎÓÊËÏ ‚
(85)
13
Ç Ó·˘ÂÏ ÒÎÛ˜‡Â, ÍÓ„‰‡ ÓÚ ÍÓÓ‰Ë̇Ú˚ x ÏÓ„ÛÚ Á‡‚ËÒÂÚ¸ ‚Ò ı‡‡ÍÚÂËÒÚËÍË Ò‰˚, ‚Íβ˜‡fl Ô‡‡ÏÂÚ ÌÂÎËÌÂÈÌÓÒÚË, Ó·Ó·˘ÂÌÌÓ Û‡‚ÌÂÌË ÏÓÊÌÓ Á‡ÔËÒ‡Ú¸ ‚ ‚ˉÂ
∂ ∂u
∂u
----- ------ – Q ( x, u ) ------ = F ( x, u ).
∂t ∂x
∂t
(91)
Ç Û‡‚ÌÂÌËË (91) Û˜ÚÂ̇ Ú‡ÍÊ ‚ÓÁÏÓÊÌÓÒÚ¸ ÔÓfl‚ÎÂÌËfl ·ÓΠÒÎÓÊÌÓ„Ó ÌÂÎËÌÂÈÌÓ„Ó ÓÚÍÎË͇
Ò‰˚, Ì ÓÔËÒ˚‚‡ÂÏÓ„Ó Ó·˚˜ÌÓÈ Í‚‡‰‡Ú˘ÌÓÈ
ÌÂÎËÌÂÈÌÓÒÚ¸˛.
ÖÒÎË ‚ Û‡‚ÌÂÌËË (91) ÔÓÎÓÊËÚ¸
α
Q = – ----------- ,
k–x
β
F = – ------------------2 u,
(k – x)
ÓÌÓ ÔËÏÂÚ ‚ˉ (75) Ë ·Û‰ÂÚ ÓÔËÒ˚‚‡Ú¸ ÙÓÍÛÒËÓ‚ÍÛ Ô͇ۘ ‚ Ó‰ÌÓÓ‰ÌÓÈ Ò‰Â. àÌ‚‡ˇÌÚÌÓÂ
¯ÂÌËÂ, Óڂ˜‡˛˘Â ÓÔÂ‡ÚÓÛ (74), ÂÒÚ¸
u = ( k – x )G ( λ ),
t
λ = ----------- .
k–x
îÛÌ͈Ëfl G(λ) Û‰Ó‚ÎÂÚ‚ÓflÂÚ Û‡‚ÌÂÌ˲
λG'' – α ( GG' ) + βG = 0.
(92)
à·‡„ËÏÓ‚, êÛ‰ÂÌÍÓ*
14
íÓ˜ÌÓ ¯ÂÌË Û‡‚ÌÂÌË (92) ÔË α = 0 Á‡ÔËÒ˚‚‡ÂÚÒfl ˜ÂÂÁ ÙÛÌ͈ËË ÅÂÒÒÂÎfl
G =
λ C 1 J 1 ( 2 βλ ) + C 2 Y 1 ( 2 βλ ) ,
‡ ÔË β = 0 ÓÌÓ ËÏÂÂÚ ‚ˉ
λ = – αD 1 + D 2 ( G + D 1 ) – α ( G + D 1 ) ln G + D 1 .
ê‡ÁÛÏÂÂÚÒfl, Í ÏÓ‰ÂÎflÏ (55), (57), (58) ÔË‚Ó‰flÚ
Ì ÚÓθÍÓ Ó·ÒÛʉ‡‚¯ËÂÒfl ‚˚¯Â Ú ‰‚ Á‡‰‡˜Ë.
é˜Â‚ˉÌÓ, ˜ÚÓ Î˛·ÓÈ ÎËÌÂÈÌÓÈ ‡ÒÔ‰ÂÎÂÌÌÓÈ
ÒËÒÚÂÏÂ Ò ÌËÁÍÓ˜‡ÒÚÓÚÌÓÈ ‰ËÒÔÂÒËÂÈ, ÓÔËÒ˚‚‡ÂÏÓÈ Á‡ÍÓÌÓÏ
ω β
k ( ω ) = ---- – ---c ω
ëèàëéä ãàíÖêÄíìêõ
ÏÓÊÌÓ ÒÓÔÓÒÚ‡‚ËÚ¸ ‰ËÙÙÂÂ̈ˇθÌÓ Û‡‚ÌÂÌËÂ
∂ u
----------- = – βu.
∂t∂x
2
(93)
ÖÒÎË ÌÂÎËÌÂÈÌÓÒÚ¸ Ò··‡fl, ÒÓÓÚ‚ÂÚÒÚ‚Û˛˘ËÈ
˜ÎÂÌ ‰Ó·‡‚ÎflÂÚÒfl ‚ ˝‚ÓβˆËÓÌÌÓ Û‡‚ÌÂÌË ‡‰‰ËÚË‚ÌÓ, Ë (93) ÔÂÂıÓ‰ËÚ, ̇ÔËÏÂ, ‚ (55).
á‡ÏÂÚËÏ, ˜ÚÓ “Ó·˘‡fl” ÙÓχ (91) ‚Ò Ê ‰ÓÒÚ‡ÚÓ˜ÌÓ ÍÓÌÍÂÚ̇. Ö ҂ÓÈÒÚ‚‡ ÒËÏÏÂÚËË, Ә‚ˉÌÓ, ·ÓΠÚÓ˜ÌÓ ÓÚ‡Ê‡˛Ú ÙËÁËÍÛ ËÒÒΉÛÂÏ˚ı ÔÓˆÂÒÒÓ‚, ˜ÂÏ, ̇ÔËÏÂ, ÒËÏÏÂÚËË ÛÔÓ˘ÂÌÌÓÈ ÏÓ‰ÂÎË (55) ËÎË ÒÓ‚Â¯ÂÌÌÓ Ó·˘ÂÈ
ÏÓ‰ÂÎË
∂u ∂ u
∂ u
----------- = G  x, t, u, ------, -------2- ,

∂t ∂t 
∂t∂x
2
ÏÓÊÌÓ “‰ÓÒÚÓËÚ¸” ÏÓ‰Âθ ‰Ó ·ÓΠÒËÏÏÂÚ˘ÌÓÈ ÔÛÚÂÏ Â ÛÒÎÓÊÌÂÌËfl. ÖÒÎË ÒÎÓÊ̇fl ÏÓ‰Âθ
‰ÓÔÛÒ͇ÂÚ ÔÓ‰ıÓ‰fl˘Â ÚÓ˜ÌÓ ¯ÂÌËÂ, ÌÂÓ·ıÓ‰ËÏ˚ ÛÔÓ˘ÂÌËfl ÏÓÊÌÓ ÔÓËÁ‚ÂÒÚË Ì‡ Á‡‚Â¯‡˛˘ÂÏ ˝Ú‡Ô ‡Ò˜ÂÚ‡, ÚÓ ÂÒÚ¸ ‚ ÓÍÓ̘‡ÚÂθÌ˚ı
ÙÓÏÛ·ı.
ꇷÓÚ‡ ‚˚ÔÓÎÌÂ̇ ‚ ‡Ï͇ı ÑÓ„Ó‚Ó‡ Ó ÒÓÚÛ‰Ì˘ÂÒÚ‚Â ÏÂÊ‰Û Í‡Ù‰ÓÈ ‡ÍÛÒÚËÍË ÙËÁ˘ÂÒÍÓ„Ó Ù‡ÍÛθÚÂÚ‡ åÉì Ë ñÂÌÚÓÏ ALGA BTH.
é̇ ˜‡ÒÚ˘ÌÓ ÔÓ‰‰Âʇ̇ „‡ÌÚ‡ÏË ÇÂ‰Û˘Ëı ̇ۘÌ˚ı ¯ÍÓÎ, êÄç Ë êîîà.
2
ÍÓÚÓÛ˛ ÏÓÊÌÓ ·˚ÎÓ ·˚ ËÁÛ˜‡Ú¸ Ô‰ÎÓÊÂÌÌ˚Ï
ÏÂÚÓ‰ÓÏ, Ì ÒÎ˯ÍÓÏ ÒËθÌÓ Ó„‡Ì˘˂‡fl Í·ÒÒ
‡ÒÒχÚË‚‡ÂÏ˚ı Á‡‰‡˜.
6. á‡Íβ˜ÂÌËÂ
Ç ÓÒÌÓ‚Â ‰‡ÌÌÓÈ ‡·ÓÚ˚ ÎÂÊËÚ Ô‡‡‰ÓÍ҇θÌÓÂ, ̇ ÔÂ‚˚È ‚Á„Îfl‰, ÛÚ‚ÂʉÂÌËÂ Ó ˆÂÎÂÒÓÓ·‡ÁÌÓÒÚË ‡Ì‡ÎËÁ‡ ËÌÚÂÂÒÛ˛˘Ëı Ì‡Ò ÌÂÎËÌÂÈÌ˚ı
Á‡‰‡˜ ÔÛÚÂÏ Ëı “ÔÓ„ÛÊÂÌËfl” ‚ Í·ÒÒ ·ÓΠӷ˘Ëı
Ë, ÒΉӂ‡ÚÂθÌÓ, ·ÓΠÒÎÓÊÌ˚ı ÏÓ‰ÂÎÂÈ. éÔ˚Ú
ÚÂÓËË ÌÂÎËÌÂÈÌ˚ı ÍÓη‡ÌËÈ Ë ‚ÓÎÌ, ÓÒÌÓ‚‡ÌÌ˚È Ì‡ ÙËÁ˘ÂÒÍË Ó·ÓÒÌÓ‚‡ÌÌÓÏ ÛÔÓ˘ÂÌËË ÏÓ‰ÂÎÂÈ, ͇Á‡ÎÓÒ¸ ·˚, ÔÓÚË‚Ó˜ËÚ ÓÔËÒ‡ÌÌÓÏÛ
‚˚¯Â ÔÓ‰ıÓ‰Û, ‚ ÓÒÌÓ‚Â ÍÓÚÓÓ„Ó – ÔË̈ËÔ ‡ÔËÓÌÓ„Ó ËÒÔÓθÁÓ‚‡ÌËfl ÒËÏÏÂÚËÈ. ÇÏÂÒÚÂ Ò ÚÂÏ,
Á‡ ‚̯ÌËÏË ‡Á΢ËflÏË ÒÍ˚ÚÓ ‚ÌÛÚÂÌÌÂÂ
‰ËÌÒÚ‚Ó ‰‚Ûı ÔÓ‰ıÓ‰Ó‚. é˜Â‚ˉÌÓ, ˜ÚÓ ·ÓÎÂÂ
ÒËÏÏÂÚ˘̇fl ÏÓ‰Âθ ‰ÓÎÊ̇ ÒÓ‰Âʇڸ ·Óθ¯Â
ÚÓ˜Ì˚ı ¯ÂÌËÈ. ä‡Í ÏÓÊÌÓ ‰Ó·ËÚ¸Òfl ·Óθ¯ÂÈ
ÒËÏÏÂÚËË? ë Ó‰ÌÓÈ ÒÚÓÓÌ˚, ÏÓÊÌÓ ‰‚Ë„‡Ú¸Òfl
ÔÓ Ú‡‰ËˆËÓÌÌÓÏÛ ÔÛÚË ÛÔÓ˘ÂÌËfl, “ÓÚÒÂ͇fl”
˝ÎÂÏÂÌÚ˚ ÏÓ‰ÂÎË, ̇Û¯‡˛˘Ë Â ÒËÏÏÂÚ˲
(ÔÂÌ·„‡fl ÌÂÍÓÚÓ˚ÏË ˜ÎÂ̇ÏË Û‡‚ÌÂÌËfl
ËÎË Í‡Í-ÚÓ ‚ˉÓËÁÏÂÌflfl Ëı). ë ‰Û„ÓÈ ÒÚÓÓÌ˚,
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