Numerical determination of the heat transfer coefficient at the boundary between two media

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Abstract

Subject of research: mathematical model of heat transfer.

Purpose of research: to develop an algorithm for the numerical solution of the inverse problem of determining the heat transfer coefficient at the boundary of two media.

Research methods: the finite element method is used in the work, the algorithm is based on a special iterative scheme.

Object of research: the process of heat transfer at the interface of two media with imperfect contact.

Research findings: the work describes an algorithm that allows calculating the heat transfer coefficient at the boundary of two media when the contact is not ideal. The algorithm is based on the finite element method and a special iterative scheme, in which the solution is sought in the form of a finite segment of a series. A number of experiments are presented, the results are analyzed and conclusions are made on the use of the algorithm.

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Рассматривается уравнение

Mu= u t Lu= u t div c x,t u +b x,t u+a x,t u=f, b x,t = b 1 x,t ,, b n x,t T ,u= u x 1 ,, x x n T ,n=2,3 MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaauaabeqaceaaae aaqaaaaaaaaaWdbiaad2eacaWG1bGaeyypa0JaamyDa8aadaWgaaWc baWdbiaadshaa8aabeaak8qacqGHsislcaWGmbGaamyDaiabg2da9i aadwhapaWaaSbaaSqaa8qacaWG0baapaqabaGcpeGaeyOeI0Iaamiz aiaadMgacaWG2bWaaeWaa8aabaWdbiaadogadaqadaWdaeaapeGaam iEaiaacYcacaWG0baacaGLOaGaayzkaaGaey4bIeTaamyDaaGaayjk aiaawMcaaiabgUcaRiaadkgadaqadaWdaeaapeGaamiEaiaacYcaca WG0baacaGLOaGaayzkaaGaey4bIeTaamyDaiabgUcaRiaadggadaqa daWdaeaapeGaamiEaiaacYcacaWG0baacaGLOaGaayzkaaGaamyDai abg2da9iaadAgacaGGSaaapaqaa8qacaWGIbWaaeWaa8aabaWdbiaa dIhacaGGSaGaamiDaaGaayjkaiaawMcaaiabg2da9maabmaapaqaa8 qacaWGIbWdamaaBaaaleaapeGaaGymaaWdaeqaaOWdbmaabmaapaqa a8qacaWG4bGaaiilaiaadshaaiaawIcacaGLPaaacaGGSaGaeyOjGW RaaiilaiaadkgapaWaaSbaaSqaa8qacaWGUbaapaqabaGcpeWaaeWa a8aabaWdbiaadIhacaGGSaGaamiDaaGaayjkaiaawMcaaaGaayjkai aawMcaa8aadaahaaWcbeqaa8qacaWGubaaaOGaaiilaiabgEGirlaa dwhacqGH9aqpdaqadaWdaeaapeWaaSaaa8aabaWdbiabgkGi2kaadw haa8aabaWdbiabgkGi2kaadIhapaWaaSbaaSqaa8qacaaIXaaapaqa baaaaOWdbiaacYcacqGHMacVcaGGSaWaaSaaa8aabaWdbiabgkGi2k aadIhaa8aabaWdbiabgkGi2kaadIhapaWaaSbaaSqaa8qacaWGUbaa paqabaaaaaGcpeGaayjkaiaawMcaa8aadaahaaWcbeqaa8qacaWGub aaaOGaaiilaiaad6gacqGH9aqpcaaIYaGaaiilaiaaiodaaaaaaa@9584@  (1)

в области Q= 0,T ×G MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaaeyuaiabg2da9maabmaapaqaa8qacaaIWaGaaiilaiaabsfaaiaa wIcacaGLPaaacqGHxdaTcaqGhbaaaa@3EB1@ . Считаем, что пространственная область имеет вид G=Ω× 0,Z MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaae4raiabg2da9iaabM6acqGHxdaTdaqadaWdaeaapeGaaGimaiaa cYcacaqGAbaacaGLOaGaayzkaaaaaa@3F12@  в случае n=3 MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamOBaiabg2da9iaaiodaaaa@38C3@  и G MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaam4raaaa@36D9@   MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSn0BKvguHDwzZbqef00uGuvsGC0B0H wAJbWexLMBbXgBcf2CPn2qVrwzqf2zLnharyavP1wzZbItLDhis9wB H5garmWu51MyVXgaryWqVvNCPvMCG4uz3bqee0evGueE0jxyaibaie Ydh9qrpeeu0dXdh9vqqj=hEeeu0xXdbba9arpi0=irpK0dbba91qpK 0=vr0RYxir=dbbc9q8aq0=yqpe0xbba9suk9fr=xfr=xfrpiWZqaai aaciWacmaadaGabiaaeaGaauaaaOqaaGGaaKqzafaeaaaaaaaaa8qa caWFtacaaa@42F3@  прямоугольник в случае n=2 MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamOBaiabg2da9iaaikdaaaa@38C2@ , т. е. Ω= 0,X MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaaeyQdiabg2da9maabmaapaqaa8qacaaIWaGaaiilaiaadIfaaiaa wIcacaGLPaaaaaa@3C31@ . Считаем, что область G MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaam4raaaa@36D9@  разделена на две части G ± MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaam4ra8aadaahaaWcbeqaa8qacqGHXcqSaaaaaa@3913@ , G + =Ω× l,Z , G =Ω× 0,l ,0<l<Z MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaam4ra8aadaahaaWcbeqaa8qacqGHRaWkaaGccqGH9aqpcaqGPoGa ey41aq7aaeWaa8aabaWdbiaadYgacaGGSaGaamOwaaGaayjkaiaawM caaiaacYcacaWGhbWdamaaCaaaleqabaWdbiabgkHiTaaakiabg2da 9iaabM6acqGHxdaTdaqadaWdaeaapeGaaGimaiaacYcacaWGSbaaca GLOaGaayzkaaGaaiilaiaaicdacqGH8aapcaWGSbGaeyipaWJaamOw aaaa@50D5@ . На плоскости x 3 = l 0 MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamiEa8aadaWgaaWcbaWdbiaaiodaa8aabeaak8qacqGH9aqpcaWG SbWdamaaBaaaleaapeGaaGimaaWdaeqaaaaa@3B46@  (прямой x 2 =l MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamiEa8aadaWgaaWcbaWdbiaaikdaa8aabeaak8qacqGH9aqpcaWG Sbaaaa@3A31@  в двумерном случае), т. е. на множестве Γ 0 = x , l 0 , x Ω MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaeu4KdC0damaaBaaaleaapeGaaGimaaWdaeqaaOWdbiabg2da9maa cmaapaqaa8qadaqadaWdaeaapeGabmiEa8aagaqba8qacaGGSaGaam iBa8aadaWgaaWcbaWdbiaaicdaa8aabeaaaOWdbiaawIcacaGLPaaa caGGSaGabmiEa8aagaqba8qacqGHiiIZcqqHPoWvaiaawUhacaGL9b aaaaa@4682@  заданы условия сопряжения типа неидеального контакта

c n + u x n + =β u + u +g, c n + u x n + = c n u x n , x n = l 0   MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaam4ya8aadaqhaaWcbaWdbiaad6gaa8aabaWdbiabgUcaRaaakiaa dwhapaWaa0baaSqaa8qacaWG4bWdamaaBaaameaapeGaamOBaaWdae qaaaWcbaWdbiabgUcaRaaakiabg2da9iabek7aInaabmaapaqaa8qa caWG1bWdamaaCaaaleqabaWdbiabgUcaRaaakiabgkHiTiaadwhapa WaaWbaaSqabeaapeGaeyOeI0caaaGccaGLOaGaayzkaaGaey4kaSIa am4zaiaacYcacaWGJbWdamaaDaaaleaapeGaamOBaaWdaeaapeGaey 4kaScaaOGaamyDa8aadaqhaaWcbaWdbiaadIhapaWaaSbaaWqaa8qa caWGUbaapaqabaaaleaapeGaey4kaScaaOGaeyypa0Jaam4ya8aada qhaaWcbaWdbiaad6gaa8aabaWdbiabgkHiTaaakiaadwhapaWaa0ba aSqaa8qacaWG4bWdamaaBaaameaapeGaamOBaaWdaeqaaaWcbaWdbi abgkHiTaaakiaacYcacaWG4bWdamaaBaaaleaapeGaamOBaaWdaeqa aOWdbiabg2da9iaadYgapaWaaSbaaSqaa8qacaaIWaaapaqabaGcpe GaaeiOaaaa@620C@ , (2)

где cnuxn±t,x0=limxG±,xx0Γ0u±=limxG±,xx0Γ0ut,x. Далее иногда используем обозначение u ± = u G± MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamyDa8aadaahaaWcbeqaa8qacqGHXcqSaaGccqGH9aqpdaabcaWd aeaapeGaamyDaaGaayjcSdWdamaaBaaaleaapeGaam4raiabgglaXc Wdaeqaaaaa@4014@  и записываем функцию u MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamyDaaaa@3707@  в виде вектора u= u + , u MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamyDaiabg2da9maabmaapaqaa8qacaWG1bWdamaaCaaaleqabaWd biabgUcaRaaakiaacYcacaWG1bWdamaaCaaaleqabaWdbiabgkHiTa aaaOGaayjkaiaawMcaaaaa@3ED4@ . К условиям сопряжения мы добавляем условия переопределения вида

u + t, y i = ψ i t i=1,2,, r 1 ,  u t, y i = ψ i t i= r 1 +1,,r , MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamyDa8aadaahaaWcbeqaa8qacqGHRaWkaaGcdaqadaWdaeaapeGa amiDaiaacYcacaWG5bWdamaaBaaaleaapeGaamyAaaWdaeqaaaGcpe GaayjkaiaawMcaaiabg2da9iabeI8a59aadaWgaaWcbaWdbiaadMga a8aabeaak8qadaqadaWdaeaapeGaamiDaaGaayjkaiaawMcaamaabm aapaqaa8qacaWGPbGaeyypa0JaaGymaiaacYcacaaIYaGaaiilaiab gAci8kaacYcacaWGYbWdamaaBaaaleaapeGaaGymaaWdaeqaaaGcpe GaayjkaiaawMcaaiaacYcacaqGGcGaamyDa8aadaahaaWcbeqaa8qa cqGHsislaaGcdaqadaWdaeaapeGaamiDaiaacYcacaWG5bWdamaaBa aaleaapeGaamyAaaWdaeqaaaGcpeGaayjkaiaawMcaaiabg2da9iab eI8a59aadaWgaaWcbaWdbiaadMgaa8aabeaak8qadaqadaWdaeaape GaamiDaaGaayjkaiaawMcaamaabmaapaqaa8qacaWGPbGaeyypa0Ja amOCa8aadaWgaaWcbaWdbiaaigdaa8aabeaak8qacqGHRaWkcaaIXa GaaiilaiabgAci8kaacYcacaWGYbaacaGLOaGaayzkaaGaaiilaaaa @6C2A@  (3)

где y i G ± ¯ i=1,2,,r MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamyEa8aadaWgaaWcbaWdbiaadMgaa8aabeaak8qacqGHiiIZpaWa a0aaaeaapeGaam4ra8aadaahaaWcbeqaa8qacqGHXcqSaaaaaOWaae Waa8aabaWdbiaadMgacqGH9aqpcaaIXaGaaiilaiaaikdacaGGSaGa eyOjGWRaaiilaiaadkhaaiaawIcacaGLPaaaaaa@46D9@ , т. е. возможен случай y i Γ 0 .Ha S= 0,T ×G MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamyEa8aadaWgaaWcbaWdbiaadMgaa8aabeaak8qacqGHiiIZcqqH toWrpaWaaSbaaSqaa8qacaaIWaaapaqabaGcpeGaaiOlaiaabIeaca qGHbGaaeiOaiaadofacqGH9aqpdaqadaWdaeaapeGaaGimaiaacYca caWGubaacaGLOaGaayzkaaGaey41aqRaeyOaIyRaam4raaaa@4A1D@  задаем какие-либо краевые условия: Дирихле, Робина или смешанные условия. Например, варианты:

c 3 u x 1 t, x ,Z = g 1 t, x , c 3 u x 3 t, x ,0 = g 0 t, x ,  MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaam4ya8aadaWgaaWcbaWdbiaaiodaa8aabeaak8qacaWG1bWdamaa BaaaleaapeGaamiEa8aadaWgaaadbaWdbiaaigdaa8aabeaaaSqaba GcpeWaaeWaa8aabaWdbiaadshacaGGSaGabmiEa8aagaqba8qacaGG SaGaamOwaaGaayjkaiaawMcaaiabg2da9iaadEgapaWaaSbaaSqaa8 qacaaIXaaapaqabaGcpeWaaeWaa8aabaWdbiaadshacaGGSaGabmiE a8aagaqbaaWdbiaawIcacaGLPaaacaGGSaGaam4ya8aadaWgaaWcba Wdbiaaiodaa8aabeaak8qacaWG1bWdamaaBaaaleaapeGaamiEa8aa daWgaaadbaWdbiaaiodaa8aabeaaaSqabaGcpeWaaeWaa8aabaWdbi aadshacaGGSaGabmiEa8aagaqba8qacaGGSaGaaGimaaGaayjkaiaa wMcaaiabg2da9iaadEgapaWaaSbaaSqaa8qacaaIWaaapaqabaGcpe WaaeWaa8aabaWdbiaadshacaGGSaGabmiEa8aagaqbaaWdbiaawIca caGLPaaacaGGSaGaaiiOaaaa@5EDD@

u 0,T ×Ω =0, u t=0 = u 0 x , MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape WaaqGaa8aabaWdbiaadwhaaiaawIa7a8aadaWgaaWcbaWdbmaabmaa paqaa8qacaaIWaGaaiilaiaadsfaaiaawIcacaGLPaaacqGHxdaTcq GHciITcaqGPoaapaqabaGcpeGaeyypa0JaaGimaiaacYcadaabcaWd aeaapeGaamyDaaGaayjcSdWdamaaBaaaleaapeGaamiDaiabg2da9i aaicdaa8aabeaak8qacqGH9aqpcaWG1bWdamaaBaaaleaapeGaaGim aaWdaeqaaOWdbmaabmaapaqaa8qacaWG4baacaGLOaGaayzkaaGaai ilaaaa@5096@  (4)

u t, x ,Z =0,u t, x ,0 = 0 u 0,T ×Ω =0, u t=0 = u 0 x . MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamyDamaabmaapaqaa8qacaWG0bGaaiilaiqadIhapaGbauaapeGa aiilaiaadQfaaiaawIcacaGLPaaacqGH9aqpcaaIWaGaaiilaiaadw hadaqadaWdaeaapeGaamiDaiaacYcaceWG4bWdayaafaWdbiaacYca caaIWaaacaGLOaGaayzkaaGaeyypa0ZaaqGaa8aabaWdbiaaicdaca GGGcGaamyDaaGaayjcSdWdamaaBaaaleaapeWaaeWaa8aabaWdbiaa icdacaGGSaGaamivaaGaayjkaiaawMcaaiabgEna0kabgkGi2kaabM 6aa8aabeaak8qacqGH9aqpcaaIWaGaaiilamaaeiaapaqaa8qacaWG 1baacaGLiWoapaWaaSbaaSqaa8qacaWG0bGaeyypa0JaaGimaaWdae qaaOWdbiabg2da9iaadwhapaWaaSbaaSqaa8qacaaIWaaapaqabaGc peWaaeWaa8aabaWdbiaadIhaaiaawIcacaGLPaaacaGGUaaaaa@63CB@  (5)

Условия могут быть как однородными, так и неоднородными. Задача состоит в нахождении решения уравнения (1), удовлетворяющего условиям (2) MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSn0BKvguHDwzZbqef00uGuvsGC0B0H wAJbWexLMBbXgBcf2CPn2qVrwzqf2zLnharyavP1wzZbItLDhis9wB H5garmWu51MyVXgaryWqVvNCPvMCG4uz3bqee0evGueE0jxyaibaie Ydh9qrpeeu0dXdh9vqqj=hEeeu0xXdbba9arpi0=irpK0dbba91qpK 0=vr0RYxir=dbbc9q8aq0=yqpe0xbba9suk9fr=xfr=xfrpiWZqaai aaciWacmaadaGabiaaeaGaauaaaOqaaGGaaKqzafaeaaaaaaaaa8qa caWFtacaaa@42F3@ (4) и неизвестной функции β MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaaeOSdaaa@3745@ вида β=j=1rβjtΦit,x', где функции Φ MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaaeOPdaaa@3739@  заданы, а функции β j MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaaeOSd8aadaWgaaWcbaWdbiaabQgaa8aabeaaaaa@388C@  считаются неизвестными. Условия сопряжения (2) совпадают с известными в теории тепломассопереноса условиями на границе двух сред, когда контакт не является идеальным. В этом случае   MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSn0BKvguHDwzZbqef00uGuvsGC0B0H wAJbWexLMBbXgBcf2CPn2qVrwzqf2zLnharyavP1wzZbItLDhis9wB H5garmWu51MyVXgaryWqVvNCPvMCG4uz3bqee0evGueE0jxyaibaie Ydh9qrpeeu0dXdh9vqqj=hEeeu0xXdbba9arpi0=irpK0dbba91qpK 0=vr0RYxir=dbbc9q8aq0=yqpe0xbba9suk9fr=xfr=xfrpiWZqaai aaciWacmaadaGabiaaeaGaauaaaOqaaGGaaKqzafaeaaaaaaaaa8qa caWFtacaaa@42F3@  коэффициент теплообмена.

Обратные задачи нахождения неизвестных граничных режимов, в частности задачи конвективного теплообмена, являются классическими. Они возникают в самых различных задачах математической физики: управление процессами теплообмена и проектирование тепловой защиты, диагностика и идентификация теплопередачи в сверхзвуковых гетерогенных потоках, идентификация и моделирование теплопереноса в теплозащитных материалах и покрытиях, моделирование свойств и тепловых режимов многоразовой тепловой защиты аэрокосмических аппаратов, исследование композитных материалов и т. п. (см. [1], [5]).

В настоящее время имеется большое количество работ, посвященных численному решению задач типа (1) MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSn0BKvguHDwzZbqef00uGuvsGC0B0H wAJbWexLMBbXgBcf2CPn2qVrwzqf2zLnharyavP1wzZbItLDhis9wB H5garmWu51MyVXgaryWqVvNCPvMCG4uz3bqee0evGueE0jxyaibaie Ydh9qrpeeu0dXdh9vqqj=hEeeu0xXdbba9arpi0=irpK0dbba91qpK 0=vr0RYxir=dbbc9q8aq0=yqpe0xbba9suk9fr=xfr=xfrpiWZqaai aaciWacmaadaGabiaaeaGaauaaaOqaaGGaaKqzafaeaaaaaaaaa8qa caWFtacaaa@42F3@ (5) в различных постановках, возникающих в приложениях; как правило, ищутся коэффициенты β MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaaeOSdaaa@3745@ , зависящие от времени или, наоборот, от пространственных переменных, точки β j MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaaeOSd8aadaWgaaWcbaWdbiaabQgaa8aabeaaaaa@388C@  в (4) чаще всего являются внутренними точками областей G + MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaam4ra8aadaahaaWcbeqaa8qacqGHRaWkaaaaaa@3807@ , G MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaam4ra8aadaahaaWcbeqaa8qacqGHsislaaaaaa@3812@ . Отметим, например, работы [4], [7], [8], [10] MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSn0BKvguHDwzZbqef00uGuvsGC0B0H wAJbWexLMBbXgBcf2CPn2qVrwzqf2zLnharyavP1wzZbItLDhis9wB H5garmWu51MyVXgaryWqVvNCPvMCG4uz3bqee0evGueE0jxyaibaie Ydh9qrpeeu0dXdh9vqqj=hEeeu0xXdbba9arpi0=irpK0dbba91qpK 0=vr0RYxir=dbbc9q8aq0=yqpe0xbba9suk9fr=xfr=xfrpiWZqaai aaciWacmaadaGabiaaeaGaauaaaOqaaGGaaKqzafaeaaaaaaaaa8qa caWFtacaaa@42F3@ [14]. В качестве метода почти во всех работах используется сведение обратной задачи к некоторой задаче управления и минимизация соответствующего квадратичного функционала ([4], [8], [10], [11], [13], [14]). Опишем некоторые рассмотренные задачи. В работе [3] рассматриваются задачи определения коэффициента теплообмена на границе раздела сред. Полученные результаты и методы позволяют подойти к построению численных методов, но в работе получены теоретические результаты. В случае одной пространственной переменной зависящий от температуры коэффициент теплообмена по точечным условиям переопределения численно определяется в статье [8]. Двумерная обратная задача определения коэффициентов теплообмена (зависящих специальным образом от дополнительных параметров, которые и подлежат определению) по набору значений решений в заданных точках численно решается в работе [10]. В работах [7], [12] рассматриваются и численно решаются обратные задачи определения коэффициента теплообмена, зависящего от двух пространственных переменных с помощью метода Монте-Карло. В качестве условий переопределения берется значение решения на части границы области. Одновременное определение коэффициента, входящего в параболическое уравнение, и коэффициента теплообмена осуществляется в работе [13]. В качестве условий переопределения используются значения замеров температур в точках на границе раздела слоев (как и в условии (4). Точечные условия переопределения также используются в [4] и [11], в последней была рассмотрена одномерная обратная задача одновременного определения теплового потока на одной из боковых поверхностей цилиндра и термического контактного сопротивления на границе раздела сред. Численное определение коэффициента теплообмена по данным замеров на доступной части внешней границы рассматриваемой области осуществляется в работе [14]. Задачи численного определения точечных источников в обратных задачах тепломассопереноса рассмотрены в работе [6], где источники задаются в виде суммы дельта-функций Дирака с коэффициентами, зависящими от времени и характеризующими мощность соответствующего источника.

РЕЗУЛЬТАТЫ И ОБСУЖДЕНИЕ

В ходе работы будем основываться на результатах, полученных в работах [2] и [9], в которых получены и доказаны теоремы о существовании и единственности решения.

Рассмотрим случай n=2 MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamOBaiabg2da9iaaikdaaaa@38C2@ , G= 0,X × 0,Z MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaam4raiabg2da9maabmaapaqaa8qacaaIWaGaaiilaiaadIfaaiaa wIcacaGLPaaacqGHxdaTdaqadaWdaeaapeGaaGimaiaacYcacaWGAb aacaGLOaGaayzkaaaaaa@41D6@ . Положим Γ=G MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaeu4KdCKaeyypa0JaeyOaIyRaam4raaaa@3AAD@ , Γ 0 = x 1 , l 0 :  x 1 0,X MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaeu4KdC0damaaBaaaleaapeGaaGimaaWdaeqaaOWdbiabg2da9maa cmaapaqaa8qadaqadaWdaeaapeGaamiEa8aadaWgaaWcbaWdbiaaig daa8aabeaak8qacaGGSaGaamiBa8aadaWgaaWcbaWdbiaaicdaa8aa beaaaOWdbiaawIcacaGLPaaacaGG6aGaaeiOaiaadIhapaWaaSbaaS qaa8qacaaIXaaapaqabaGcpeGaeyicI48aaeWaa8aabaWdbiaaicda caGGSaGaamiwaaGaayjkaiaawMcaaaGaay5Eaiaaw2haaaaa@4C1C@   S= 0,T ×Γ MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaam4uaiabg2da9maabmaapaqaa8qacaaIWaGaaiilaiaadsfaaiaa wIcacaGLPaaacqGHxdaTcqqHtoWraaa@3F55@ , S 0 = 0,T × Γ 0 MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaam4ua8aadaWgaaWcbaWdbiaaicdaa8aabeaak8qacqGH9aqpdaqa daWdaeaapeGaaGimaiaacYcacaWGubaacaGLOaGaayzkaaGaey41aq Raeu4KdC0damaaBaaaleaapeGaaGimaaWdaeqaaaaa@4197@ .

Условия согласования данных имеют вид:

u 0 x 1 ,0 = u 0 x 1 ,Z =0,  u 0 y k = ψ k 0 . MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaaeyDa8aadaWgaaWcbaWdbiaaicdaa8aabeaak8qadaqadaWdaeaa peGaaeiEa8aadaWgaaWcbaWdbiaaigdaa8aabeaak8qacaGGSaGaaG imaaGaayjkaiaawMcaaiabg2da9iaabwhapaWaaSbaaSqaa8qacaaI WaaapaqabaGcpeWaaeWaa8aabaWdbiaabIhapaWaaSbaaSqaa8qaca aIXaaapaqabaGcpeGaaiilaiaabQfaaiaawIcacaGLPaaacqGH9aqp caaIWaGaaiilaiaabckacaqG1bWdamaaBaaaleaapeGaaGimaaWdae qaaOWdbmaabmaapaqaa8qacaqG5bWdamaaBaaaleaapeGaae4AaaWd aeqaaaGcpeGaayjkaiaawMcaaiabg2da9iaabI8apaWaaSbaaSqaa8 qacaqGRbaapaqabaGcpeWaaeWaa8aabaWdbiaaicdaaiaawIcacaGL PaaacaGGUaaaaa@5683@  (6)

Опишем метод в случае n=2 MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamOBaiabg2da9iaaikdaaaa@38C2@ . Для численного решения используем метод конечных элементов. Далее для простоты рассматриваем условия (3) с условиями согласования (6).

Ищем функцию β MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaeqOSdigaaa@37AE@  в виде β=j=1rβjtΦix1, где функции βj подлежат определению, а функции Φj,g0 известны. Считаем, что точки y i MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamyEa8aadaWgaaWcbaWdbiaadMgaa8aabeaaaaa@3853@  с i r 1 MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamyAaiabgsMiJkaadkhapaWaaSbaaSqaa8qacaaIXaaapaqabaaa aa@3ABC@  лежат во множестве G + Γ 0 MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaam4ra8aadaahaaWcbeqaa8qacqGHRaWkaaGccqGHQicYcqqHtoWr paWaaSbaaSqaa8qacaaIWaaapaqabaaaaa@3C2D@ , соответственно точки y i MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamyEa8aadaWgaaWcbaWdbiaadMgaa8aabeaaaaa@3853@  с i r 1 +1 MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamyAaiabgwMiZkaadkhapaWaaSbaaSqaa8qacaaIXaaapaqabaGc peGaey4kaSIaaGymaaaa@3C84@  во множестве G Γ 0 MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaam4ra8aadaahaaWcbeqaa8qacqGHsislaaGccqGHQicYcqqHtoWr paWaaSbaaSqaa8qacaaIWaaapaqabaaaaa@3C38@ .

Опишем метод решения прямой задачи. Задана триангуляция областей G ± MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaam4ra8aadaahaaWcbeqaa8qacqGHXcqSaaaaaa@3913@  и соответствующие базисы метода конечных элементов { φ i } i=1 s MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaai4EaiabeA8aQ9aadaWgaaWcbaWdbiaadMgaa8aabeaak8qacaGG 9bWdamaaDaaaleaapeGaamyAaiabg2da9iaaigdaa8aabaWdbiaado haaaaaaa@3F3E@ , { φ i } i=s+1 N MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaai4EaiabeA8aQ9aadaWgaaWcbaWdbiaadMgaa8aabeaak8qacaGG 9bWdamaaDaaaleaapeGaamyAaiabg2da9iaadohacqGHRaWkcaaIXa aapaqaa8qacaWGobaaaaaa@40F3@ . Узлы сетки обозначим через b i MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape WaaiWaa8aabaWdbiaadkgapaWaaSbaaSqaa8qacaWGPbaapaqabaaa k8qacaGL7bGaayzFaaaaaa@3AA6@ .

Ищем приближенное решение в виде

v=i=1NCitφi.

Для удобства далее считаем, что точки y i MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamyEa8aadaWgaaWcbaWdbiaadMgaa8aabeaaaaa@3853@  ( i r 1 MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamyAaiabgsMiJkaadkhapaWaaSbaaSqaa8qacaaIXaaapaqabaaa aa@3ABC@  ) совпадают с узлами сетки b 1 ,, b r 1 MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamOya8aadaWgaaWcbaWdbiaaigdaa8aabeaak8qacaGGSaGaeyOj GWRaaiilaiaadkgapaWaaSbaaSqaa8qacaWGYbWdamaaBaaameaape GaaGymaaWdaeqaaaWcbeaaaaa@3E5B@ , а точки y i MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamyEa8aadaWgaaWcbaWdbiaadMgaa8aabeaaaaa@3853@  ( r 1 +1<ir MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamOCa8aadaWgaaWcbaWdbiaaigdaa8aabeaak8qacqGHRaWkcaaI XaGaeyipaWJaamyAaiabgsMiJkaadkhaaaa@3E6E@  ) совпадают с узлами сетки b s+1 ,, b s+r r 1 MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamOya8aadaWgaaWcbaWdbiaadohacqGHRaWkcaaIXaaapaqabaGc peGaaiilaiabgAci8kaacYcacaWGIbWdamaaBaaaleaapeGaam4Cai abgUcaRiaadkhacqGHsislcaWGYbWdamaaBaaameaapeGaaGymaaWd aeqaaaWcbeaaaaa@43F3@ . Функции C i MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaam4qa8aadaWgaaWcbaWdbiaadMgaa8aabeaaaaa@381D@  определяем из системы

R 0 C t + R 1 t C = F + f ,  C = ( C 1 , C 2 ,, C N ) T MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamOua8aadaWgaaWcbaWdbiaaicdaa8aabeaak8qaceWGdbWdayaa laWaaSbaaSqaa8qacaWG0baapaqabaGcpeGaey4kaSIaamOua8aada WgaaWcbaWdbiaaigdaa8aabeaak8qadaqadaWdaeaapeGaamiDaaGa ayjkaiaawMcaaiqadoeapaGbaSaapeGaeyypa0JabmOra8aagaWca8 qacqGHRaWkceWGMbWdayaalaWdbiaacYcacaqGGcGabm4qa8aagaWc a8qacqGH9aqpcaGGOaGaam4qa8aadaWgaaWcbaWdbiaaigdaa8aabe aak8qacaGGSaGaam4qa8aadaWgaaWcbaWdbiaaikdaa8aabeaak8qa caGGSaGaeyOjGWRaaiilaiaadoeapaWaaSbaaSqaa8qacaWGobaapa qabaGcpeGaaiyka8aadaahaaWcbeqaa8qacaWGubaaaaaa@54C7@ , (7)

где координаты f имеют вид

fi=G+ft,xφix dx+0Xg1t,x1φix1,Z dx10Xgt,x1φix1,l0 dx1, 

при is MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamyAaiabgsMiJkaadohaaaa@39A8@  и при   i>s MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaaiiOaiaacckacaWGPbGaeyOpa4Jaam4Caaaa@3B43@

f i = G f t,x φ i x  dx 0 X g 0 t, x 1 φ i x 1 ,0  d x 1 + 0 X g t, x 1 φ i x 1 , l 0  d x 1 , MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaauaabeqabeaaae aaqaaaaaaaaaWdbiaadAgapaWaaSbaaSqaa8qacaWGPbaapaqabaGc peGaeyypa0Zaaubeaeqal8aabaWdbiaadEeapaWaaWbaaWqabeaape GaeyOeI0caaaWcbeqdpaqaa8qacqGHRiI8aaGccaaMk8UaaGPcVlaa dAgadaqadaWdaeaapeGaamiDaiaacYcacaWG4baacaGLOaGaayzkaa GaeqOXdO2damaaBaaaleaapeGaamyAaaWdaeqaaOWdbmaabmaapaqa a8qacaWG4baacaGLOaGaayzkaaGaaiiOaiaadsgacaWG4bGaeyOeI0 Yaaubmaeqal8aabaWdbiaaicdaa8aabaWdbiaadIfaa0WdaeaapeGa ey4kIipaaOGaaGPcVlaayQW7caWGNbWdamaaBaaaleaapeGaaGimaa WdaeqaaOWdbmaabmaapaqaa8qacaWG0bGaaiilaiaadIhapaWaaSba aSqaa8qacaaIXaaapaqabaaak8qacaGLOaGaayzkaaGaeqOXdO2dam aaBaaaleaapeGaamyAaaWdaeqaaOWdbmaabmaapaqaa8qacaWG4bWd amaaBaaaleaapeGaaGymaaWdaeqaaOWdbiaacYcacaaIWaaacaGLOa GaayzkaaGaaiiOaiaadsgacaWG4bWdamaaBaaaleaapeGaaGymaaWd aeqaaOWdbiabgUcaRmaavadabeWcpaqaa8qacaaIWaaapaqaa8qaca WGybaan8aabaWdbiabgUIiYdaakiaayQW7caaMk8Uaam4zamaabmaa paqaa8qacaWG0bGaaiilaiaadIhapaWaaSbaaSqaa8qacaaIXaaapa qabaaak8qacaGLOaGaayzkaaGaeqOXdO2damaaBaaaleaapeGaamyA aaWdaeqaaOWdbmaabmaapaqaa8qacaWG4bWdamaaBaaaleaapeGaaG ymaaWdaeqaaOWdbiaacYcacaWGSbWdamaaBaaaleaapeGaaGimaaWd aeqaaaGcpeGaayjkaiaawMcaaiaacckacaWGKbGaamiEa8aadaWgaa WcbaWdbiaaigdaa8aabeaak8qacaGGSaaaaaaa@88BA@   R 0 MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamOua8aadaWgaaWcbaWdbiaaicdaa8aabeaaaaa@37F8@   MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSn0BKvguHDwzZbqef00uGuvsGC0B0H wAJbWexLMBbXgBcf2CPn2qVrwzqf2zLnharyavP1wzZbItLDhis9wB H5garmWu51MyVXgaryWqVvNCPvMCG4uz3bqee0evGueE0jxyaibaie Ydh9qrpeeu0dXdh9vqqj=hEeeu0xXdbba9arpi0=irpK0dbba91qpK 0=vr0RYxir=dbbc9q8aq0=yqpe0xbba9suk9fr=xfr=xfrpiWZqaai aaciWacmaadaGabiaaeaGaauaaaOqaaGGaaKqzafaeaaaaaaaaa8qa caWFtacaaa@42F3@  матрица с элементами r ij = φ i , φ j = G + φ i x φ j x  dx MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamOCa8aadaWgaaWcbaWdbiaadMgacaWGQbaapaqabaGcpeGaeyyp a0ZaaeWaa8aabaWdbiabeA8aQ9aadaWgaaWcbaWdbiaadMgaa8aabe aak8qacaGGSaGaeqOXdO2damaaBaaaleaapeGaamOAaaWdaeqaaaGc peGaayjkaiaawMcaaiabg2da9maavababeWcpaqaa8qacaWGhbWdam aaCaaameqabaWdbiabgUcaRaaaaSqab0WdaeaapeGaey4kIipaaOGa aGPcVlabeA8aQ9aadaWgaaWcbaWdbiaadMgaa8aabeaak8qadaqada WdaeaapeGaamiEaaGaayjkaiaawMcaaiabeA8aQ9aadaWgaaWcbaWd biaadQgaa8aabeaak8qadaqadaWdaeaapeGaamiEaaGaayjkaiaawM caaiaacckacaWGKbGaamiEaaaa@587D@  при i,js MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamyAaiaacYcacaWGQbGaeyizImQaam4Caaaa@3B47@ , r ij = φ i , φ j = G φ i x φ j x  dx MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamOCa8aadaWgaaWcbaWdbiaadMgacaWGQbaapaqabaGcpeGaeyyp a0ZaaeWaa8aabaWdbiabeA8aQ9aadaWgaaWcbaWdbiaadMgaa8aabe aak8qacaGGSaGaeqOXdO2damaaBaaaleaapeGaamOAaaWdaeqaaaGc peGaayjkaiaawMcaaiabg2da9maavababeWcpaqaa8qacaWGhbWdam aaCaaameqabaWdbiabgkHiTaaaaSqab0WdaeaapeGaey4kIipaaOGa aGPcVlabeA8aQ9aadaWgaaWcbaWdbiaadMgaa8aabeaak8qadaqada WdaeaapeGaamiEaaGaayjkaiaawMcaaiabeA8aQ9aadaWgaaWcbaWd biaadQgaa8aabeaak8qadaqadaWdaeaapeGaamiEaaGaayjkaiaawM caaiaacckacaWGKbGaamiEaaaa@5888@ при i,j>s MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamyAaiaacYcacaWGQbGaeyOpa4Jaam4Caaaa@3A9A@ , r ij =0, MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamOCa8aadaWgaaWcbaWdbiaadMgacaWGQbaapaqabaGcpeGaeyyp a0JaaGimaiaacYcaaaa@3BC5@  если is MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaaeyAaiabgsMiJkaabohaaaa@39A4@  и j>s MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamOAaiabg6da+iaadohaaaa@38FC@  или i>s MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamyAaiabg6da+iaadohaaaa@38FB@  и js MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamOAaiabgsMiJkaadohaaaa@39A9@ .

R 1 MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamOua8aadaWgaaWcbaWdbiaaigdaa8aabeaaaaa@37F9@   MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSn0BKvguHDwzZbqef00uGuvsGC0B0H wAJbWexLMBbXgBcf2CPn2qVrwzqf2zLnharyavP1wzZbItLDhis9wB H5garmWu51MyVXgaryWqVvNCPvMCG4uz3bqee0evGueE0jxyaibaie Ydh9qrpeeu0dXdh9vqqj=hEeeu0xXdbba9arpi0=irpK0dbba91qpK 0=vr0RYxir=dbbc9q8aq0=yqpe0xbba9suk9fr=xfr=xfrpiWZqaai aaciWacmaadaGabiaaeaGaauaaaOqaaGGaaKqzafaeaaaaaaaaa8qa caWFtacaaa@42F3@  матрица с элементами:

Rjk=(c1t,xφkx1,φjx1)±+(c2t,xφkx2,φjx2)±+(bt,xφk,φj)±+(at,xφk,φj)±,

  (u,v)±=G±uv dx , (8)

при j,ks MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamOAaiaacYcacaWGRbGaeyizImQaam4Caaaa@3B49@  (в этом случае интегралы берутся по G + MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaam4ra8aadaahaaWcbeqaa8qacqGHRaWkaaaaaa@3807@  ) или k,j>s MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaam4AaiaacYcacaWGQbGaeyOpa4Jaam4Caaaa@3A9C@  (в этом случае интегралы берутся по G MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaam4ra8aadaahaaWcbeqaa8qacqGHsislaaaaaa@3812@  ), считаем, что R kj =0 MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamOua8aadaWgaaWcbaWdbiaadUgacaWGQbaapaqabaGcpeGaeyyp a0JaaGimaaaa@3AF7@ , если ks MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaam4AaiabgsMiJkaadohaaaa@39AA@  и j>s MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamOAaiabg6da+iaadohaaaa@38FC@  или k>s MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaam4Aaiabg6da+iaadohaaaa@38FD@  и js MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamOAaiabgsMiJkaadohaaaa@39A9@ . Имеем, чтo C 0 = C 0 = u 0 b 1 ,, u 0 b N MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gabm4qa8aagaWca8qadaqadaWdaeaapeGaaGimaaGaayjkaiaawMca aiabg2da9iqadoeapaGbaSaadaWgaaWcbaWdbiaaicdaa8aabeaak8 qacqGH9aqpdaqadaWdaeaapeGaamyDa8aadaWgaaWcbaWdbiaaicda a8aabeaak8qadaqadaWdaeaapeGaamOya8aadaWgaaWcbaWdbiaaig daa8aabeaaaOWdbiaawIcacaGLPaaacaGGSaGaeyOjGWRaaiilaiaa dwhapaWaaSbaaSqaa8qacaaIWaaapaqabaGcpeWaaeWaa8aabaWdbi aadkgapaWaaSbaaSqaa8qacaWGobaapaqabaaak8qacaGLOaGaayzk aaaacaGLOaGaayzkaaaaaa@4DF6@ . Координаты вектора F MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GabmOra8aagaWcaaaa@36F9@  имеют вид

F i = 0 X β t, x 1 v + t, x 1 , l 0 v t, x 1 , l 0 φ i x 1 , l 0  d x 1 MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamOra8aadaWgaaWcbaWdbiaadMgaa8aabeaak8qacqGH9aqpcqGH sisldaqfWaqabSWdaeaapeGaaGimaaWdaeaapeGaamiwaaqdpaqaa8 qacqGHRiI8aaGccaaMk8UaeqOSdi2aaeWaa8aabaWdbiaadshacaGG SaGaamiEa8aadaWgaaWcbaWdbiaaigdaa8aabeaaaOWdbiaawIcaca GLPaaadaqadaWdaeaapeGaamODa8aadaahaaWcbeqaa8qacqGHRaWk aaGcdaqadaWdaeaapeGaamiDaiaacYcacaWG4bWdamaaBaaaleaape GaaGymaaWdaeqaaOWdbiaacYcacaWGSbWdamaaBaaaleaapeGaaGim aaWdaeqaaaGcpeGaayjkaiaawMcaaiabgkHiTiaadAhapaWaaWbaaS qabeaapeGaeyOeI0caaOWaaeWaa8aabaWdbiaadshacaGGSaGaamiE a8aadaWgaaWcbaWdbiaaigdaa8aabeaak8qacaGGSaGaamiBa8aada WgaaWcbaWdbiaaicdaa8aabeaaaOWdbiaawIcacaGLPaaaaiaawIca caGLPaaacqaHgpGApaWaaSbaaSqaa8qacaWGPbaapaqabaGcpeWaae Waa8aabaWdbiaadIhapaWaaSbaaSqaa8qacaaIXaaapaqabaGcpeGa aiilaiaadYgapaWaaSbaaSqaa8qacaaIWaaapaqabaaak8qacaGLOa GaayzkaaGaaiiOaiaadsgacaWG4bWdamaaBaaaleaapeGaaGymaaWd aeqaaaaa@6C8B@ при is MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamyAaiabgsMiJkaadohaaaa@39A8@

F i = 0 X β t, x 1 v + t, x 1 , l 0 v t, x 1 , l 0 φ i x 1 , l 0  d x 1 MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamOra8aadaWgaaWcbaWdbiaadMgaa8aabeaak8qacqGH9aqpdaqf WaqabSWdaeaapeGaaGimaaWdaeaapeGaamiwaaqdpaqaa8qacqGHRi I8aaGccaaMk8UaeqOSdi2aaeWaa8aabaWdbiaadshacaGGSaGaamiE a8aadaWgaaWcbaWdbiaaigdaa8aabeaaaOWdbiaawIcacaGLPaaada qadaWdaeaapeGaamODa8aadaahaaWcbeqaa8qacqGHRaWkaaGcdaqa daWdaeaapeGaamiDaiaacYcacaWG4bWdamaaBaaaleaapeGaaGymaa WdaeqaaOWdbiaacYcacaWGSbWdamaaBaaaleaapeGaaGimaaWdaeqa aaGcpeGaayjkaiaawMcaaiabgkHiTiaadAhapaWaaWbaaSqabeaape GaeyOeI0caaOWaaeWaa8aabaWdbiaadshacaGGSaGaamiEa8aadaWg aaWcbaWdbiaaigdaa8aabeaak8qacaGGSaGaamiBa8aadaWgaaWcba Wdbiaaicdaa8aabeaaaOWdbiaawIcacaGLPaaaaiaawIcacaGLPaaa cqaHgpGApaWaaSbaaSqaa8qacaWGPbaapaqabaGcpeWaaeWaa8aaba WdbiaadIhapaWaaSbaaSqaa8qacaaIXaaapaqabaGcpeGaaiilaiaa dYgapaWaaSbaaSqaa8qacaaIWaaapaqabaaak8qacaGLOaGaayzkaa GaaiiOaiaadsgacaWG4bWdamaaBaaaleaapeGaaGymaaWdaeqaaaaa @6B9E@  при i>s MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamyAaiabg6da+iaadohaaaa@38FB@ .

Здесь v ± t,x, l 0 = lim ε0 v t, x 1 , l 0 ±ε MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamODa8aadaahaaWcbeqaa8qacqGHXcqSaaGcdaqadaWdaeaapeGa amiDaiaacYcacaWG4bGaaiilaiaadYgapaWaaSbaaSqaa8qacaaIWa aapaqabaaak8qacaGLOaGaayzkaaGaeyypa0ZdamaaxababaWdbiaa dYgacaWGPbGaamyBaaWcpaqaa8qacqaH1oqzcqGHsgIRcaaIWaaapa qabaGcpeGaaGPcVlaadAhadaqadaWdaeaapeGaamiDaiaacYcacaWG 4bWdamaaBaaaleaapeGaaGymaaWdaeqaaOWdbiaacYcacaWGSbWdam aaBaaaleaapeGaaGimaaWdaeqaaOWdbiabgglaXkabew7aLbGaayjk aiaawMcaaaaa@579D@ . Решение системы ищем методом конечных разностей. Пусть τ=T/M MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaeqiXdqNaeyypa0Jaamivaiaac+cacaWGnbaaaa@3B36@ . MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSn0BKvguHDwzZbqef00uGuvsGC0B0H wAJbWexLMBbXgBcf2CPn2qVrwzqf2zLnharyavP1wzZbItLDhis9wB H5garmWu51MyVXgaryWqVvNCPvMCG4uz3bqee0evGueE0jxyaibaie Ydh9qrpeeu0dXdh9vqqj=hEeeu0xXdbba9arpi0=irpK0dbba91qpK 0=vr0RYxir=dbbc9q8aq0=yqpe0xbba9suk9fr=xfr=xfrpiWZqaai aaciWacmaadaGabiaaeaGaauaaaOqaaGGaaKqzafaeaaaaaaaaa8qa caWFtacaaa@42F3@  шаг по времени. Заменим уравнение (7) системой (9)

где C i k C k τi , MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaam4qa8aadaqhaaWcbaWdbiaadMgaa8aabaWdbiaadUgaaaGccqGH ijYUcaWGdbWdamaaBaaaleaapeGaam4AaaWdaeqaaOWdbmaabmaapa qaa8qacqaHepaDcaWGPbaacaGLOaGaayzkaaGaaiilaaaa@4210@   F i F τi MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiqadAeagaWcam aaBaaaleaacaWGPbaabeaakabaaaaaaaaapeGaeyisIS7daiqadAea gaWca8qadaqadaWdaeaapeGaaeiXdiaabMgaaiaawIcacaGLPaaaaa a@3E99@ , f i = f τi ,  A i = R 1 τi . MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GabmOza8aagaWcamaaBaaaleaapeGaamyAaaWdaeqaaOWdbiabg2da 9iqadAgapaGbaSaapeWaaeWaa8aabaWdbiabes8a0jaadMgaaiaawI cacaGLPaaacaGGSaGaaeiOaiaadgeapaWaaSbaaSqaa8qacaWGPbaa paqabaGcpeGaeyypa0JaamOua8aadaWgaaWcbaWdbiaaigdaa8aabe aak8qadaqadaWdaeaapeGaeqiXdqNaamyAaaGaayjkaiaawMcaaiaa c6caaaa@4AFD@  Пусть Ψ=(ψ1,ψ2,,ψr)T, Ψi=Ψτi.

Положим ψ i k = ψ k τi MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaeqiYdK3damaaDaaaleaapeGaamyAaaWdaeaapeGaam4Aaaaakiab g2da9iabeI8a59aadaWgaaWcbaWdbiaadUgaa8aabeaak8qadaqada WdaeaapeGaeqiXdqNaamyAaaGaayjkaiaawMcaaaaa@42C1@ . β i = ( β i 1 ,, β i r ) T MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GafqOSdi2dayaalaWaaSbaaSqaa8qacaWGPbaapaqabaGcpeGaeyyp a0Jaaiikaiabek7aI9aadaqhaaWcbaWdbiaadMgaa8aabaWdbiaaig daaaGccaGGSaGaeyOjGWRaaiilaiabek7aI9aadaqhaaWcbaWdbiaa dMgaa8aabaWdbiaadkhaaaGccaGGPaWdamaaCaaaleqabaWdbiaads faaaaaaa@474E@ , β i β τi MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GafqOSdi2dayaalaWaaSbaaSqaa8qacaWGPbaapaqabaGcpeGaeyis ISRafqOSdi2dayaalaWdbmaabmaapaqaa8qacqaHepaDcaWGPbaaca GLOaGaayzkaaaaaa@4100@ , β i k β k iτ MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaeqOSdi2damaaDaaaleaapeGaamyAaaWdaeaapeGaam4Aaaaakiab gIKi7kabek7aI9aadaWgaaWcbaWdbiaadUgaa8aabeaak8qadaqada WdaeaapeGaamyAaiabes8a0bGaayjkaiaawMcaaaaa@4312@ .

Запишем координаты вектора Fi+1. Возьмем

F i+1 k = j=1 r β i+1 j ( l=1 r 1 ψ i l 0 X Φ j x 1 φ l x 1 , l 0 φ k x 1 , l 0  dx 1 + MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaaeOra8aadaqhaaWcbaWdbiaabMgacqGHRaWkcaaIXaaapaqaa8qa caqGRbaaaOGaeyypa0JaeyOeI0YaaybCaeqal8aabaWdbiaabQgacq GH9aqpcaaIXaaapaqaa8qacaqGYbaan8aabaWdbiabggHiLdaakiaa yQW7caaMk8UaaeOSd8aadaqhaaWcbaWdbiaabMgacqGHRaWkcaaIXa aapaqaa8qacaqGQbaaaOGaaiikamaawahabeWcpaqaa8qacaqGSbGa eyypa0JaaGymaaWdaeaapeGaaeOCa8aadaWgaaadbaWdbiaaigdaa8 aabeaaa0qaa8qacqGHris5aaGccaaMk8UaaGPcVlaabI8apaWaa0ba aSqaa8qacaqGPbaapaqaa8qacaqGSbaaaOWaaubmaeqal8aabaWdbi aaicdaa8aabaWdbiaabIfaa0WdaeaapeGaey4kIipaaOGaaGPcVlaa yQW7caqGMoWdamaaBaaaleaapeGaaeOAaaWdaeqaaOWdbmaabmaapa qaa8qacaqG4bWdamaaBaaaleaapeGaaGymaaWdaeqaaaGcpeGaayjk aiaawMcaaiaabA8apaWaaSbaaSqaa8qacaqGSbaapaqabaGcpeWaae Waa8aabaWdbiaabIhapaWaaSbaaSqaa8qacaaIXaaapaqabaGcpeGa aiilaiaabYgapaWaaSbaaSqaa8qacaaIWaaapaqabaaak8qacaGLOa GaayzkaaGaaeOXd8aadaWgaaWcbaWdbiaabUgaa8aabeaak8qadaqa daWdaeaapeGaaeiEa8aadaWgaaWcbaWdbiaaigdaa8aabeaak8qaca GGSaGaaeiBa8aadaWgaaWcbaWdbiaaicdaa8aabeaaaOWdbiaawIca caGLPaaacaqGGcGaaeizaiaabIhapaWaaSbaaSqaa8qacaaIXaaapa qabaGcpeGaey4kaScaaa@7F81@

l=r1+1sCil0XΦjx1φlx1,l0φkx1,l0dx1
-l=s+1s+rr1ψils+r10XΦjx1φlx1,l0φkx1,l0dx1-l=s+rr1+1NCil0XΦjx1φlx1,l0φkx1,l0 dx1), k=1,2,,s,

F i+1 k = j=1 r β i+1 j ( l=1 r 1 ψ i l 0 X Φ j x 1 φ l x 1 , l 0 φ k x 1 , l 0  dx 1 + MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaaeOra8aadaqhaaWcbaWdbiaabMgacqGHRaWkcaaIXaaapaqaa8qa caqGRbaaaOGaeyypa0ZaaybCaeqal8aabaWdbiaabQgacqGH9aqpca aIXaaapaqaa8qacaqGYbaan8aabaWdbiabggHiLdaakiaayQW7caaM k8UaaeOSd8aadaqhaaWcbaWdbiaabMgacqGHRaWkcaaIXaaapaqaa8 qacaqGQbaaaOGaaiikamaawahabeWcpaqaa8qacaqGSbGaeyypa0Ja aGymaaWdaeaapeGaaeOCa8aadaWgaaadbaWdbiaaigdaa8aabeaaa0 qaa8qacqGHris5aaGccaaMk8UaaGPcVlaabI8apaWaa0baaSqaa8qa caqGPbaapaqaa8qacaqGSbaaaOWaaubmaeqal8aabaWdbiaaicdaa8 aabaWdbiaabIfaa0WdaeaapeGaey4kIipaaOGaaGPcVlaayQW7caqG MoWdamaaBaaaleaapeGaaeOAaaWdaeqaaOWdbmaabmaapaqaa8qaca qG4bWdamaaBaaaleaapeGaaGymaaWdaeqaaaGcpeGaayjkaiaawMca aiaabA8apaWaaSbaaSqaa8qacaqGSbaapaqabaGcpeWaaeWaa8aaba WdbiaabIhapaWaaSbaaSqaa8qacaaIXaaapaqabaGcpeGaaiilaiaa bYgapaWaaSbaaSqaa8qacaaIWaaapaqabaaak8qacaGLOaGaayzkaa GaaeOXd8aadaWgaaWcbaWdbiaabUgaa8aabeaak8qadaqadaWdaeaa peGaaeiEa8aadaWgaaWcbaWdbiaaigdaa8aabeaak8qacaGGSaGaae iBa8aadaWgaaWcbaWdbiaaicdaa8aabeaaaOWdbiaawIcacaGLPaaa caqGGcGaaeizaiaabIhapaWaaSbaaSqaa8qacaaIXaaapaqabaGcpe Gaey4kaScaaa@7E94@
+l=r1+1sCil0XΦjx1φlx1,l0φkx1,l0 dx1l=s+1s+rr1ψils+r10XΦjx1φlx1,l0φkx1,l0dx1l=s+rr1+1NCil0XΦjx1φlx1,l0φkx1,l0 dx1), k=s+1,s+2,,N.

Опишем ситуацию более подробно. Положим

a kj i+1 = l=1 s C i l 0 X Φ j x 1 φ l x 1 , l 0 φ k x 1 , l 0  dx 1 + l=s+1 N C i l 0 X Φ j x 1 φ l x 1 , l 0 φ k x 1 , l 0  dx 1 MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaaeyya8aadaqhaaWcbaWdbiaabUgacaqGQbaapaqaa8qacaqGPbGa ey4kaSIaaGymaaaakiabg2da9iabgkHiTmaawahabeWcpaqaa8qaca qGSbGaeyypa0JaaGymaaWdaeaapeGaae4Caaqdpaqaa8qacqGHris5 aaGccaaMk8Uaae4qa8aadaqhaaWcbaWdbiaabMgaa8aabaWdbiaabY gaaaGcdaqfWaqabSWdaeaapeGaaGimaaWdaeaapeGaaeiwaaqdpaqa a8qacqGHRiI8aaGccaaMk8UaaeOPd8aadaWgaaWcbaWdbiaabQgaa8 aabeaak8qadaqadaWdaeaapeGaaeiEa8aadaWgaaWcbaWdbiaaigda a8aabeaaaOWdbiaawIcacaGLPaaacaqGgpWdamaaBaaaleaapeGaae iBaaWdaeqaaOWdbmaabmaapaqaa8qacaqG4bWdamaaBaaaleaapeGa aGymaaWdaeqaaOWdbiaacYcacaqGSbWdamaaBaaaleaapeGaaGimaa WdaeqaaaGcpeGaayjkaiaawMcaaiaabA8apaWaaSbaaSqaa8qacaqG RbaapaqabaGcpeWaaeWaa8aabaWdbiaabIhapaWaaSbaaSqaa8qaca aIXaaapaqabaGcpeGaaiilaiaabYgapaWaaSbaaSqaa8qacaaIWaaa paqabaaak8qacaGLOaGaayzkaaGaaeiOaiaabsgacaqG4bWdamaaBa aaleaapeGaaGymaaWdaeqaaOWdbiabgUcaRmaawahabeWcpaqaa8qa caqGSbGaeyypa0Jaae4CaiabgUcaRiaaigdaa8aabaWdbiaab6eaa0 WdaeaapeGaeyyeIuoaaOGaaGPcVlaaboeapaWaa0baaSqaa8qacaqG Pbaapaqaa8qacaqGSbaaaOWaaubmaeqal8aabaWdbiaaicdaa8aaba WdbiaabIfaa0WdaeaapeGaey4kIipaaOGaaGPcVlaabA6apaWaaSba aSqaa8qacaqGQbaapaqabaGcpeWaaeWaa8aabaWdbiaabIhapaWaaS baaSqaa8qacaaIXaaapaqabaaak8qacaGLOaGaayzkaaGaaeOXd8aa daWgaaWcbaWdbiaabYgaa8aabeaak8qadaqadaWdaeaapeGaaeiEa8 aadaWgaaWcbaWdbiaaigdaa8aabeaak8qacaGGSaGaaeiBa8aadaWg aaWcbaWdbiaaicdaa8aabeaaaOWdbiaawIcacaGLPaaacaqGgpWdam aaBaaaleaapeGaae4AaaWdaeqaaOWdbmaabmaapaqaa8qacaqG4bWd amaaBaaaleaapeGaaGymaaWdaeqaaOWdbiaacYcacaqGSbWdamaaBa aaleaapeGaaGimaaWdaeqaaaGcpeGaayjkaiaawMcaaiaabckacaqG KbGaaeiEa8aadaWgaaWcbaWdbiaaigdaa8aabeaaaaa@9C12@

при ks MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaae4AaiabgsMiJkaabohaaaa@39A6@  и

a kj i+1 = l=1 s C i l 0 X Φ j x 1 φ l x 1 , l 0 φ k x 1 , l 0  dx 1 l=s+1 N C i l 0 X Φ j x 1 φ l x 1 , l 0 φ k x 1 , l 0  dx 1 MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaaeyya8aadaqhaaWcbaWdbiaabUgacaqGQbaapaqaa8qacaqGPbGa ey4kaSIaaGymaaaakiabg2da9maawahabeWcpaqaa8qacaqGSbGaey ypa0JaaGymaaWdaeaapeGaae4Caaqdpaqaa8qacqGHris5aaGccaaM k8Uaae4qa8aadaqhaaWcbaWdbiaabMgaa8aabaWdbiaabYgaaaGcda qfWaqabSWdaeaapeGaaGimaaWdaeaapeGaaeiwaaqdpaqaa8qacqGH RiI8aaGccaaMk8UaaeOPd8aadaWgaaWcbaWdbiaabQgaa8aabeaak8 qadaqadaWdaeaapeGaaeiEa8aadaWgaaWcbaWdbiaaigdaa8aabeaa aOWdbiaawIcacaGLPaaacaqGgpWdamaaBaaaleaapeGaaeiBaaWdae qaaOWdbmaabmaapaqaa8qacaqG4bWdamaaBaaaleaapeGaaGymaaWd aeqaaOWdbiaacYcacaqGSbWdamaaBaaaleaapeGaaGimaaWdaeqaaa GcpeGaayjkaiaawMcaaiaabA8apaWaaSbaaSqaa8qacaqGRbaapaqa baGcpeWaaeWaa8aabaWdbiaabIhapaWaaSbaaSqaa8qacaaIXaaapa qabaGcpeGaaiilaiaabYgapaWaaSbaaSqaa8qacaaIWaaapaqabaaa k8qacaGLOaGaayzkaaGaaeiOaiaabsgacaqG4bWdamaaBaaaleaape GaaGymaaWdaeqaaOWdbiabgkHiTmaawahabeWcpaqaa8qacaqGSbGa eyypa0Jaae4CaiabgUcaRiaaigdaa8aabaWdbiaab6eaa0Wdaeaape GaeyyeIuoaaOGaaGPcVlaaboeapaWaa0baaSqaa8qacaqGPbaapaqa a8qacaqGSbaaaOWaaubmaeqal8aabaWdbiaaicdaa8aabaWdbiaabI faa0WdaeaapeGaey4kIipaaOGaaGPcVlaabA6apaWaaSbaaSqaa8qa caqGQbaapaqabaGcpeWaaeWaa8aabaWdbiaabIhapaWaaSbaaSqaa8 qacaaIXaaapaqabaaak8qacaGLOaGaayzkaaGaaeOXd8aadaWgaaWc baWdbiaabYgaa8aabeaak8qadaqadaWdaeaapeGaaeiEa8aadaWgaa WcbaWdbiaaigdaa8aabeaak8qacaGGSaGaaeiBa8aadaWgaaWcbaWd biaaicdaa8aabeaaaOWdbiaawIcacaGLPaaacaqGgpWdamaaBaaale aapeGaae4AaaWdaeqaaOWdbmaabmaapaqaa8qacaqG4bWdamaaBaaa leaapeGaaGymaaWdaeqaaOWdbiaacYcacaqGSbWdamaaBaaaleaape GaaGimaaWdaeqaaaGcpeGaayjkaiaawMcaaiaabckacaqGKbGaaeiE a8aadaWgaaWcbaWdbiaaigdaa8aabeaaaaa@9B30@

при k>s MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaae4Aaiabg6da+iaabohaaaa@38F9@ .

Здесь Cil=ψillr1, Cil=ψils+r1l=s+1,,s+rr1. Тогда

Fi+1k=j=1rβi+1jakji+1 k=1,,r1, s+1,,s+rr1, Fi+1=Ai+1βi+1,

где матрица A i+1 MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaamyqa8aadaahaaWcbeqaa8qacaWGPbGaey4kaSIaaGymaaaaaaa@39AA@  имеет размерность N×r MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaaeOtaiabgEna0kaabkhaaaa@39EA@ . Перепишем равенство (9) в виде

R i+1 C i+1 =τ A i+1 β i+1 +τ f i+1 + R 0 C i ,  R i+1 = R 0 +τ A i+1  i=0,1,,M1. MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamOua8aadaWgaaWcbaWdbiaadMgacqGHRaWkcaaIXaaapaqabaGc peGabm4qa8aagaWcamaaBaaaleaapeGaamyAaiabgUcaRiaaigdaa8 aabeaak8qacqGH9aqpcqaHepaDcaWGbbWdamaaCaaaleqabaWdbiaa dMgacqGHRaWkcaaIXaaaaOGafqOSdi2dayaalaWaaSbaaSqaa8qaca WGPbGaey4kaSIaaGymaaWdaeqaaOWdbiabgUcaRiabes8a0jqadAga paGbaSaadaWgaaWcbaWdbiaadMgacqGHRaWkcaaIXaaapaqabaGcpe Gaey4kaSIaamOua8aadaWgaaWcbaWdbiaaicdaa8aabeaak8qaceWG dbWdayaalaWaaSbaaSqaa8qacaWGPbaapaqabaGcpeGaaiilaiaabc kacaWGsbWdamaaBaaaleaapeGaamyAaiabgUcaRiaaigdaa8aabeaa k8qacqGH9aqpcaWGsbWdamaaBaaaleaapeGaaGimaaWdaeqaaOWdbi abgUcaRiabes8a0jaadgeapaWaaSbaaSqaa8qacaWGPbGaey4kaSIa aGymaaWdaeqaaOWdbiaabckacaWGPbGaeyypa0JaaGimaiaacYcaca aIXaGaaiilaiabgAci8kaacYcacaWGnbGaeyOeI0IaaGymaiaac6ca aaa@6F1C@  (10)

Построим r×N MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaaeOCaiabgEna0kaab6eaaaa@39EA@  матрицу D 0 MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaaeira8aadaWgaaWcbaWdbiaaicdaa8aabeaaaaa@37E8@  такую, что d ii =1 MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaaeiza8aadaWgaaWcbaWdbiaabMgacaqGPbaapaqabaGcpeGaeyyp a0JaaGymaaaa@3B01@  при i=1,, r 1 MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaaeyAaiabg2da9iaaigdacaGGSaGaeyOjGWRaaiilaiaabkhapaWa aSbaaSqaa8qacaaIXaaapaqabaaaaa@3DB2@ , d ii+s r 1 =1 MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaaeiza8aadaWgaaWcbaWdbiaabMgacaqGPbGaey4kaSIaae4Caiab gkHiTiaabkhapaWaaSbaaWqaa8qacaaIXaaapaqabaaaleqaaOWdbi abg2da9iaaigdaaaa@3FCD@  при i= r 1 +1,,r MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaaeyAaiabg2da9iaabkhapaWaaSbaaSqaa8qacaaIXaaapaqabaGc peGaey4kaSIaaGymaiaacYcacqGHMacVcaGGSaGaaeOCaaaa@3FA3@ , а остальные элементы матрицы D 0 MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaaeira8aadaWgaaWcbaWdbiaaicdaa8aabeaaaaa@37E8@  равны нулю. Обращая матрицу R i+1 MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaaeOua8aadaWgaaWcbaWdbiaabMgacqGHRaWkcaaIXaaapaqabaaa aa@39C5@  из (10), получим

Ci+1=τRi+1-1Ai+1βi+1+τRi+1-1fi+1+Ri+1-1R0Ci,     i=0,1,2,…, M-1, (11)

Применив матрицу D 0 MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaaeira8aadaWgaaWcbaWdbiaaicdaa8aabeaaaaa@37E8@  и используя условия переопределения, получим

Ci+1=τD0Ri+11Ai+1βi+1+τD0Ri+11fi+1+D0Ri+11R0Ci,     i= 0,1,2,,M1. (12)

Обозначим B i+1 = D 0 R i+1 1 A i+1 MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamOqa8aadaWgaaWcbaWdbiaadMgacqGHRaWkcaaIXaaapaqabaGc peGaeyypa0Jaamira8aadaWgaaWcbaWdbiaaicdaa8aabeaak8qaca WGsbWdamaaDaaaleaapeGaamyAaiabgUcaRiaaigdaa8aabaWdbiab gkHiTiaaigdaaaGccaWGbbWdamaaCaaaleqabaWdbiaadMgacqGHRa WkcaaIXaaaaaaa@45EC@  (матрица имеет размерность r×r MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaaeOCaiabgEna0kaabkhaaaa@3A0E@  ).
Отсюда, из равенства (12), находим вектор β i+1 MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaaeOSd8aadaWgaaWcbaWdbiaabMgacqGHRaWkcaaIXaaapaqabaaa aa@3A28@ :

Ci+1=Bi+11βi+1τBi+11D0Ri+11fi+1Bi+11D0Ri+11R0Ci, i=0,1,,M1. (13)

Определим начальные данные. Имеем C 0 k = u 0 b k MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaae4qa8aadaqhaaWcbaWdbiaaicdaa8aabaWdbiaabUgaaaGccqGH 9aqpcaqG1bWdamaaBaaaleaapeGaaGimaaWdaeqaaOWdbmaabmaapa qaa8qacaqGIbWdamaaBaaaleaapeGaae4AaaWdaeqaaaGcpeGaayjk aiaawMcaaaaa@400B@ . При i=0 MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaaeyAaiabg2da9iaaicdaaaa@38B9@  правая часть системы (13) известна, тем самым найдем β1, используя равенство (11), найдем вектор C . Далее повторяем рассуждения: на i MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaaeyAaaaa@36F9@  -м шаге известны βi, Ci. Используя равенство (13), найдем βi+1, затем из (11) найдем вектор Ci+1. Матрица B i+1 MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaaeOqa8aadaWgaaWcbaWdbiaabMgacqGHRaWkcaaIXaaapaqabaaa aa@39B5@  может быть сингулярной, поэтому для улучшения сходимости используем регуляризацию и заменяем в формуле (13) матрицу B i+1 1 MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaaeOqa8aadaqhaaWcbaWdbiaabMgacqGHRaWkcaaIXaaapaqaa8qa cqGHsislcaaIXaaaaaaa@3B6E@  матрицей ( B i+1 B i+1 * +ε) 1 B i+1 * MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaaiikaiaabkeapaWaaSbaaSqaa8qacaqGPbGaey4kaSIaaGymaaWd aeqaaOWdbiaabkeapaWaa0baaSqaa8qacaqGPbGaey4kaSIaaGymaa WdaeaapeGaaeOkaaaakiabgUcaRiaabw7acaGGPaWdamaaCaaaleqa baWdbiabgkHiTiaaigdaaaGccaqGcbWdamaaDaaaleaapeGaaeyAai abgUcaRiaaigdaa8aabaWdbiaabQcaaaaaaa@4819@ .

Сходимость алгоритма. Исходя из построения, легко увидеть, что система (8) эквивалентна системе:

G+k=1sCikCi1kτφkxφlxdx+G+m=12cmk=1sCikφkxm+φlxmxdx++G+k=1sCikbφk+aφkφlxdx==G+fφl dx+0Xg1t,x1φlx1,Zdx10Xgt,x1φlx1,l0dx1-0Xβ~iNk=1sCi1kφkx1,l0k=1+sNCi1kφkx1,l0φlx1,l0 dx1 (14)

GK=1+sN(CjkCj1k)τφk(x)φl(x)dx+Gm=12Cmk=1+sNCjkφkxm(x)φlxm(x)dx==Gk=1+sNCjk(bφk+aφk)φl(x)dxoXg0(t,x1)φl(x1,0)dx1++0Xg(t,x1)φl(x1,l0)dx1+Gfφdx+0Xβ~iNk=1sCi1kφkx1,l0k=1+sNCi1kφkx1,l0φlx1,l0dx1,  (15)

где β~iN=k=1rβNikΦkx1(мы добавили индекс N MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamOtaaaa@36E0@  в определении функции βi=k=1rβikΦkx1 ). Кроме того, здесь C i1 l = ψ i1 l l r 1   MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaae4qa8aadaqhaaWcbaWdbiaabMgacqGHsislcaaIXaaapaqaa8qa caqGSbaaaOGaeyypa0JaaeiYd8aadaqhaaWcbaWdbiaabMgacqGHsi slcaaIXaaapaqaa8qacaqGSbaaaOWaaeWaa8aabaWdbiaabYgacqGH KjYOcaqGYbWdamaaBaaaleaapeGaaGymaaWdaeqaaaGcpeGaayjkai aawMcaaiaabckaaaa@48AA@   C i1 l = ψ i1 ls+ r 1 l=s+1,,s+r r 1 .  MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaam4qa8aadaqhaaWcbaWdbiaadMgacqGHsislcaaIXaaapaqaa8qa caWGSbaaaOGaeyypa0JaeqiYdK3damaaDaaaleaapeGaamyAaiabgk HiTiaaigdaa8aabaWdbiaadYgacqGHsislcaWGZbGaey4kaSIaamOC a8aadaWgaaadbaWdbiaaigdaa8aabeaaaaGcpeWaaeWaa8aabaWdbi aadYgacqGH9aqpcaWGZbGaey4kaSIaaGymaiaacYcacqGHMacVcaGG SaGaam4CaiabgUcaRiaadkhacqGHsislcaWGYbWdamaaBaaaleaape GaaGymaaWdaeqaaaGcpeGaayjkaiaawMcaaiaac6cacaGGGcaaaa@5761@  Положим также, что β ˜ N t,x = β ˜ i N MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaaxacabaaeaa aaaaaaa8qacqaHYoGyaSWdaeqabaWdbiaacYTaaaGcpaWaaSbaaSqa a8qacaWGobaapaqabaGcpeWaaeWaa8aabaWdbiaadshacaGGSaGaam iEaaGaayjkaiaawMcaaiabg2da98aadaWfGaqaa8qacqaHYoGyaSWd aeqabaWdbiaacYTaaaGcpaWaa0baaSqaa8qacaWGPbaapaqaa8qaca WGobaaaaaa@45DD@  при xG MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamiEaiabgIGiolaadEeaaaa@395A@ , t i1 τ,iτ MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamiDaiabgIGiopaajibapaqaa8qadaqadaWdaeaapeGaamyAaiab gkHiTiaaigdaaiaawIcacaGLPaaacqaHepaDcaGGSaGaamyAaiabes 8a0bGaay5waiaawMcaaaaa@43E2@ , i=1,2,,M MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamyAaiabg2da9iaaigdacaGGSaGaaGOmaiaacYcacqGHMacVcaGG SaGaamytaaaa@3DE8@ .

Умножим равенства (14), (15) на постоянные v i l MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamODa8aadaqhaaWcbaWdbiaadMgaa8aabaWdbiaadYgaaaaaaa@3952@  и суммируем по l MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamiBaaaa@36FE@  (в соответствующих диапазонах). Получим

G+k=1sCikCi1kτφkxvi+ dx+G+m=12cmk=1sCikφkxmxvixm+x dx+G+k=1sCikbφk+aφkvi+xdx=0Xg1t,x1vi+x1,Zdx10Xgt,x1vi+x1,l0 dx1+G+fvi+ dx0Xβ~iN(k=1sCi1kφkx1,l0-k=1+sNCi1kφkx1,l0)vi+x1,l0 dx1,  (16)

GK=1+sN(CjkCj1k)τφk(x)vj(x)dx++Gm=12cmk=1+sNCikφkxm(x)vix(x)dx++Gk=1+sNCik(bφk+aφk)vi(x)dx==oXg0(t,x1)vi(x1,0)dx1+oXg(t,x1)vi(x1,l0)dx1++Gfvidx+oxβiN(k=1SCi1kφk(x1,l0)    -k=1+sNCi1kφkx1,l0)vix1,l0 dx1, (17)

где v i + = l=1 s v i l φ l MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamODa8aadaqhaaWcbaWdbiaadMgaa8aabaWdbiabgUcaRaaakiab g2da9maawahabeWcpaqaa8qacaWGSbGaeyypa0JaaGymaaWdaeaape Gaam4Caaqdpaqaa8qacqGHris5aaGccaaMk8UaamODa8aadaqhaaWc baWdbiaadMgaa8aabaWdbiaadYgaaaGccqaHgpGApaWaaSbaaSqaa8 qacaWGSbaapaqabaaaaa@4898@ , v i = l=1+s N v i l φ l MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamODa8aadaqhaaWcbaWdbiaadMgaa8aabaWdbiabgkHiTaaakiab g2da9maawahabeWcpaqaa8qacaWGSbGaeyypa0JaaGymaiabgUcaRi aadohaa8aabaWdbiaad6eaa0WdaeaapeGaeyyeIuoaaOGaaGPcVlaa dAhapaWaa0baaSqaa8qacaWGPbaapaqaa8qacaWGSbaaaOGaeqOXdO 2damaaBaaaleaapeGaamiBaaWdaeqaaaaa@4A58@ . Суммируя равенства (16), (17) по i MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamyAaaaa@36FB@  и меняя суммирование в первом слагаемом (используем равенства

i=1 M a i a i1 b i = i=1 r a i b i b i+1 a M b M+1 + a 0 b 1 MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape WaaybCaeqal8aabaWdbiaadMgacqGH9aqpcaaIXaaapaqaa8qacaWG nbaan8aabaWdbiabggHiLdaakiaayQW7daqadaWdaeaapeGaamyya8 aadaWgaaWcbaWdbiaadMgaa8aabeaak8qacqGHsislcaWGHbWdamaa BaaaleaapeGaamyAaiabgkHiTiaaigdaa8aabeaaaOWdbiaawIcaca GLPaaacaWGIbWdamaaBaaaleaapeGaamyAaaWdaeqaaOWdbiabg2da 9maawahabeWcpaqaa8qacaWGPbGaeyypa0JaaGymaaWdaeaapeGaam OCaaqdpaqaa8qacqGHris5aaGccaaMk8Uaamyya8aadaWgaaWcbaWd biaadMgaa8aabeaak8qadaqadaWdaeaapeGaamOya8aadaWgaaWcba WdbiaadMgaa8aabeaak8qacqGHsislcaWGIbWdamaaBaaaleaapeGa amyAaiabgUcaRiaaigdaa8aabeaaaOWdbiaawIcacaGLPaaacqGHsi slcaWGHbWdamaaBaaaleaapeGaamytaaWdaeqaaOWdbiaadkgapaWa aSbaaSqaa8qacaWGnbGaey4kaSIaaGymaaWdaeqaaOWdbiabgUcaRi aadggapaWaaSbaaSqaa8qacaaIWaaapaqabaGcpeGaamOya8aadaWg aaWcbaWdbiaaigdaa8aabeaaaaa@68B9@ где полагаем b M+1 =0 MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamOya8aadaWgaaWcbaWdbiaad2eacqGHRaWkcaaIXaaapaqabaGc peGaeyypa0JaaGimaaaa@3B97@ ), получим

i=1M[G+k=1sCikφkxvi+vi+1+τ dx+G+m=12cmk=1sCikφkxmxvixm+x dx+k=1sCikbφk+aφkvi+x dx]=G+k=1sC0kφkxv1+ dx+0Xg1t,x1vi+x1,Z dx10Xgt,x1vi+x1,l0 dx1i=1M[G+fφl dx0Xβ~iNk=1sCi1kφkx1,l0k=1+sNCi1kφkx1,l0vi+x1,l0 dx1], (18)

0Xg0t,x1vix1,0 dx1+0Xgt,x1vix1,l0 dx1+k=1+sNC0kφkxv1 dx+i=1M[Gfvi dx++0Xβ~iNk=1sCi1kφkx1,l0k=1+sNCi1kφkx1,l0vix1,l0 dx1]. (19)

Положим v ¯ N + t,x = l=1 s v i1 l τit τ + v i l tτ i1 τ φ l x MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GabmODayaaraWdamaaDaaaleaapeGaamOtaaWdaeaapeGaey4kaSca aOWaaeWaa8aabaWdbiaadshacaGGSaGaamiEaaGaayjkaiaawMcaai abg2da9maawahabeWcpaqaa8qacaWGSbGaeyypa0JaaGymaaWdaeaa peGaam4Caaqdpaqaa8qacqGHris5aaGccaaMk8+aaeWaa8aabaWdbi aadAhapaWaa0baaSqaa8qacaWGPbGaeyOeI0IaaGymaaWdaeaapeGa amiBaaaakmaalaaapaqaa8qadaqadaWdaeaapeGaeqiXdqNaamyAai abgkHiTiaadshaaiaawIcacaGLPaaaa8aabaWdbiabes8a0baacqGH RaWkcaWG2bWdamaaDaaaleaapeGaamyAaaWdaeaapeGaamiBaaaakm aalaaapaqaa8qadaqadaWdaeaapeGaamiDaiabgkHiTiabes8a0naa bmaapaqaa8qacaWGPbGaeyOeI0IaaGymaaGaayjkaiaawMcaaaGaay jkaiaawMcaaaWdaeaapeGaeqiXdqhaaaGaayjkaiaawMcaaiabeA8a Q9aadaWgaaWcbaWdbiaadYgaa8aabeaak8qadaqadaWdaeaapeGaam iEaaGaayjkaiaawMcaaaaa@6B1B@  при xG MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamiEaiabgIGiolaadEeaaaa@395A@ , t i1 τ,iτ MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamiDaiabgIGiopaajibapaqaa8qadaqadaWdaeaapeGaamyAaiab gkHiTiaaigdaaiaawIcacaGLPaaacqaHepaDcaGGSaGaamyAaiabes 8a0bGaay5waiaawMcaaaaa@43E2@ , i=1,2,,M.  v ˜ N + t,x = l=1 s v i l φ l x MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamyAaiabg2da9iaaigdacaGGSaGaaGOmaiaacYcacqGHMacVcaGG SaGaamytaiaac6cacaGGGcWdamaaxacabaWdbiaadAhaaSWdaeqaba WdbiaacYTaaaGcpaWaa0baaSqaa8qacaWGobaapaqaa8qacqGHRaWk aaGcdaqadaWdaeaapeGaamiDaiaacYcacaWG4baacaGLOaGaayzkaa Gaeyypa0ZaaybCaeqal8aabaWdbiaadYgacqGH9aqpcaaIXaaapaqa a8qacaWGZbaan8aabaWdbiabggHiLdaakiaayQW7caWG2bWdamaaDa aaleaapeGaamyAaaWdaeaapeGaamiBaaaakiabeA8aQ9aadaWgaaWc baWdbiaadYgaa8aabeaak8qadaqadaWdaeaapeGaamiEaaGaayjkai aawMcaaaaa@5B2E@  при xG MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamiEaiabgIGiolaadEeaaaa@395A@ , t i1 τ,iτ MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamiDaiabgIGiopaajibapaqaa8qadaqadaWdaeaapeGaamyAaiab gkHiTiaaigdaaiaawIcacaGLPaaacqaHepaDcaGGSaGaamyAaiabes 8a0bGaay5waiaawMcaaaaa@43E2@ , i=1,2,,M MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamyAaiabg2da9iaaigdacaGGSaGaaGOmaiaacYcacqGHMacVcaGG SaGaamytaaaa@3DE8@ , v ¯ N t,x = l=1+s N v i1 l τit τ + v i l tτ i1 τ φ l x MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GabmODayaaraWdamaaDaaaleaapeGaamOtaaWdaeaapeGaeyOeI0ca aOWaaeWaa8aabaWdbiaadshacaGGSaGaamiEaaGaayjkaiaawMcaai abg2da9maawahabeWcpaqaa8qacaWGSbGaeyypa0JaaGymaiabgUca Riaadohaa8aabaWdbiaad6eaa0WdaeaapeGaeyyeIuoaaOGaaGPcVp aabmaapaqaa8qacaWG2bWdamaaDaaaleaapeGaamyAaiabgkHiTiaa igdaa8aabaWdbiaadYgaaaGcdaWcaaWdaeaapeWaaeWaa8aabaWdbi abes8a0jaadMgacqGHsislcaWG0baacaGLOaGaayzkaaaapaqaa8qa cqaHepaDaaGaey4kaSIaamODa8aadaqhaaWcbaWdbiaadMgaa8aaba WdbiaadYgaaaGcdaWcaaWdaeaapeWaaeWaa8aabaWdbiaadshacqGH sislcqaHepaDdaqadaWdaeaapeGaamyAaiabgkHiTiaaigdaaiaawI cacaGLPaaaaiaawIcacaGLPaaaa8aabaWdbiabes8a0baaaiaawIca caGLPaaacqaHgpGApaWaaSbaaSqaa8qacaWGSbaapaqabaGcpeWaae Waa8aabaWdbiaadIhaaiaawIcacaGLPaaaaaa@6CDB@ , при xG MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamiEaiabgIGiolaadEeaaaa@395A@ , t i1 τ,iτ MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamiDaiabgIGiopaajibapaqaa8qadaqadaWdaeaapeGaamyAaiab gkHiTiaaigdaaiaawIcacaGLPaaacqaHepaDcaGGSaGaamyAaiabes 8a0bGaay5waiaawMcaaaaa@43E2@ , i=1,2,,M MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamyAaiabg2da9iaaigdacaGGSaGaaGOmaiaacYcacqGHMacVcaGG SaGaamytaaaa@3DE8@ , v ˜ N t,x = l=1+s N v i l φ l x MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaaxacabaaeaa aaaaaaa8qacaWG2baal8aabeqaa8qacaGGClaaaOWdamaaDaaaleaa peGaamOtaaWdaeaapeGaeyOeI0caaOWaaeWaa8aabaWdbiaadshaca GGSaGaamiEaaGaayjkaiaawMcaaiabg2da9maawahabeWcpaqaa8qa caWGSbGaeyypa0JaaGymaiabgUcaRiaadohaa8aabaWdbiaad6eaa0 WdaeaapeGaeyyeIuoaaOGaaGPcVlaadAhapaWaa0baaSqaa8qacaWG Pbaapaqaa8qacaWGSbaaaOGaeqOXdO2damaaBaaaleaapeGaamiBaa WdaeqaaOWdbmaabmaapaqaa8qacaWG4baacaGLOaGaayzkaaaaaa@531E@  при xG MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamiEaiabgIGiolaadEeaaaa@395A@ , t i1 τ,iτ MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamiDaiabgIGiopaajibapaqaa8qadaqadaWdaeaapeGaamyAaiab gkHiTiaaigdaaiaawIcacaGLPaaacqaHepaDcaGGSaGaamyAaiabes 8a0bGaay5waiaawMcaaaaa@43E2@ , i=1,2,,M MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamyAaiabg2da9iaaigdacaGGSaGaaGOmaiaacYcacqGHMacVcaGG SaGaamytaaaa@3DE8@ .

Аналогичным образом определяем функции u ˜ N + MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaaxacabaaeaa aaaaaaa8qacaWG1baal8aabeqaa8qacaGGClaaaOWdamaaDaaaleaa peGaamOtaaWdaeaapeGaey4kaScaaaaa@3AFB@ , u ˜ N MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaaxacabaaeaa aaaaaaa8qacaWG1baal8aabeqaa8qacaGGClaaaOWdamaaDaaaleaa peGaamOtaaWdaeaapeGaeyOeI0caaaaa@3B06@ , например, u ˜ N t,x = l=1+s N C i l φ l x MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaaxacabaaeaa aaaaaaa8qacaWG1baal8aabeqaa8qacaGGClaaaOWdamaaDaaaleaa peGaamOtaaWdaeaapeGaeyOeI0caaOWaaeWaa8aabaWdbiaadshaca GGSaGaamiEaaGaayjkaiaawMcaaiabg2da9maawahabeWcpaqaa8qa caWGSbGaeyypa0JaaGymaiabgUcaRiaadohaa8aabaWdbiaad6eaa0 WdaeaapeGaeyyeIuoaaOGaaGPcVlaadoeapaWaa0baaSqaa8qacaWG Pbaapaqaa8qacaWGSbaaaOGaeqOXdO2damaaBaaaleaapeGaamiBaa WdaeqaaOWdbmaabmaapaqaa8qacaWG4baacaGLOaGaayzkaaaaaa@52EA@  при xG MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamiEaiabgIGiolaadEeaaaa@395A@ , t i1 τ,iτ MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamiDaiabgIGiopaajibapaqaa8qadaqadaWdaeaapeGaamyAaiab gkHiTiaaigdaaiaawIcacaGLPaaacqaHepaDcaGGSaGaamyAaiabes 8a0bGaay5waiaawMcaaaaa@43E2@ , i=1,2,,M MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamyAaiabg2da9iaaigdacaGGSaGaaGOmaiaacYcacqGHMacVcaGG SaGaamytaaaa@3DE8@ . Используя эти определения, можно переписать равенства (19), (18) в виде:

0 T [ G + u ˜ N + v ¯ Nt +  dx+ G + m=1 2 c m u ˜ N x m + v ˜ N x m + t,x  dx+ MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Waaubmaeqal8aabaWdbiaaicdaa8aabaWdbiaadsfaa0WdaeaapeGa ey4kIipaaOGaaGPcVlaayQW7caGGBbWaaubeaeqal8aabaWdbiaadE eapaWaaWbaaWqabeaapeGaey4kaScaaaWcbeqdpaqaa8qacqGHRiI8 aaGccaaMk8UaaGPcV=aadaWfGaqaa8qacaWG1baal8aabeqaa8qaca GGClaaaOWdamaaDaaaleaapeGaamOtaaWdaeaapeGaey4kaScaaOGa bmODayaaraWdamaaDaaaleaapeGaamOtaiaadshaa8aabaWdbiabgU caRaaakiaacckacaWGKbGaamiEaiabgUcaRmaavababeWcpaqaa8qa caWGhbWdamaaCaaameqabaWdbiabgUcaRaaaaSqab0WdaeaapeGaey 4kIipaaOGaaGPcVlaayQW7daGfWbqabSWdaeaapeGaamyBaiabg2da 9iaaigdaa8aabaWdbiaaikdaa0WdaeaapeGaeyyeIuoaaOGaaGPcVl aayQW7caWGJbWdamaaBaaaleaapeGaamyBaaWdaeqaaOWaaCbiaeaa peGaamyDaaWcpaqabeaapeGaaii3caaak8aadaqhaaWcbaWdbiaad6 eacaWG4bWdamaaBaaameaapeGaamyBaaWdaeqaaaWcbaWdbiabgUca Raaak8aadaWfGaqaa8qacaWG2baal8aabeqaa8qacaGGClaaaOWdam aaDaaaleaapeGaamOtaiaadIhapaWaaSbaaWqaa8qacaWGTbaapaqa baaaleaapeGaey4kaScaaOWaaeWaa8aabaWdbiaadshacaGGSaGaam iEaaGaayjkaiaawMcaaiaacckacaWGKbGaamiEaiabgUcaRaaa@7CE9@

G + b u ˜ N + +a u ˜ N + v ˜ N + t,x ) dx] dt= 0 T 0 X g 1 t, x 1 v N + t, x 1 ,Z  d x 1 MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Waaubeaeqal8aabaWdbiaadEeapaWaaWbaaWqabeaapeGaey4kaSca aaWcbeqdpaqaa8qacqGHRiI8aaGccaaMk8UaaGPcVpaabmaapaqaa8 qaceWGIbWdayaalaWdbiabgEGir=aadaWfGaqaa8qacaWG1baal8aa beqaa8qacaGGClaaaOWdamaaDaaaleaapeGaamOtaaWdaeaapeGaey 4kaScaaOGaey4kaSIaamyya8aadaWfGaqaa8qacaWG1baal8aabeqa a8qacaGGClaaaOWdamaaDaaaleaapeGaamOtaaWdaeaapeGaey4kaS caaaGccaGLOaGaayzkaaWdamaaxacabaWdbiaadAhaaSWdaeqabaWd biaacYTaaaGcpaWaa0baaSqaa8qacaWGobaapaqaa8qacqGHRaWkaa GcdaqadaWdaeaapeGaamiDaiaacYcacaWG4baacaGLOaGaayzkaaGa aiykaiaacckacaWGKbGaamiEaiaac2facaGGGcGaamizaiaadshacq GH9aqpdaqfWaqabSWdaeaapeGaaGimaaWdaeaapeGaamivaaqdpaqa a8qacqGHRiI8aaGccaaMk8UaaGPcVpaavadabeWcpaqaa8qacaaIWa aapaqaa8qacaWGybaan8aabaWdbiabgUIiYdaakiaayQW7caaMk8Ua am4za8aadaWgaaWcbaWdbiaaigdaa8aabeaak8qadaqadaWdaeaape GaamiDaiaacYcacaWG4bWdamaaBaaaleaapeGaaGymaaWdaeqaaaGc peGaayjkaiaawMcaaiaadAhapaWaa0baaSqaa8qacaWGobaapaqaa8 qacqGHRaWkaaGcdaqadaWdaeaapeGaamiDaiaacYcacaWG4bWdamaa BaaaleaapeGaaGymaaWdaeqaaOWdbiaacYcacaWGAbaacaGLOaGaay zkaaGaaiiOaiaadsgacaWG4bWdamaaBaaaleaapeGaaGymaaWdaeqa aOWdbiabgkHiTaaa@856B@

0 X g t, x 1 v N + t, x 1 , l 0  d x 1  dt+ G + u 0N + x v ¯ N + τ,x  dx+ 0 T [ G + f v N +  dx MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Waaubmaeqal8aabaWdbiaaicdaa8aabaWdbiaadIfaa0WdaeaapeGa ey4kIipaaOGaaGPcVlaayQW7caWGNbWaaeWaa8aabaWdbiaadshaca GGSaGaamiEa8aadaWgaaWcbaWdbiaaigdaa8aabeaaaOWdbiaawIca caGLPaaacaWG2bWdamaaDaaaleaapeGaamOtaaWdaeaapeGaey4kaS caaOWaaeWaa8aabaWdbiaadshacaGGSaGaamiEa8aadaWgaaWcbaWd biaaigdaa8aabeaak8qacaGGSaGaamiBa8aadaWgaaWcbaWdbiaaic daa8aabeaaaOWdbiaawIcacaGLPaaacaGGGcGaamizaiaadIhapaWa aSbaaSqaa8qacaaIXaaapaqabaGcpeGaaiiOaiaadsgacaWG0bGaey 4kaSYaaubeaeqal8aabaWdbiaadEeapaWaaWbaaWqabeaapeGaey4k aScaaaWcbeqdpaqaa8qacqGHRiI8aaGccaaMk8UaaGPcVlaadwhapa Waa0baaSqaa8qacaaIWaGaamOtaaWdaeaapeGaey4kaScaaOWaaeWa a8aabaWdbiaadIhaaiaawIcacaGLPaaaceWG2bGbaebapaWaa0baaS qaa8qacaWGobaapaqaa8qacqGHRaWkaaGcdaqadaWdaeaapeGaeqiX dqNaaiilaiaadIhaaiaawIcacaGLPaaacaGGGcGaamizaiaadIhacq GHRaWkdaqfWaqabSWdaeaapeGaaGimaaWdaeaapeGaamivaaqdpaqa a8qacqGHRiI8aaGccaaMk8UaaGPcVlaacUfadaqfqaqabSWdaeaape Gaam4ra8aadaahaaadbeqaa8qacqGHRaWkaaaaleqan8aabaWdbiab gUIiYdaakiaayQW7caaMk8UaamOzaiaadAhapaWaa0baaSqaa8qaca WGobaapaqaa8qacqGHRaWkaaGccaGGGcGaamizaiaadIhacqGHsisl aaa@8962@

0 X β ˜ N u N + tτ, x 1 , l 0 u N tτ, x 1 , l 0 v ˜ N + t, x 1 , l 0  d x 1 ] dt,          20 MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Waaubmaeqal8aabaWdbiaaicdaa8aabaWdbiaadIfaa0WdaeaapeGa ey4kIipaaOGaaGPcVlaayQW7paWaaCbiaeaapeGaeqOSdigal8aabe qaa8qacaGGClaaaOWdamaaCaaaleqabaWdbiaad6eaaaGcdaqadaWd aeaapeGaamyDa8aadaqhaaWcbaWdbiaad6eaa8aabaWdbiabgUcaRa aakmaabmaapaqaa8qacaWG0bGaeyOeI0IaeqiXdqNaaiilaiaadIha paWaaSbaaSqaa8qacaaIXaaapaqabaGcpeGaaiilaiaadYgapaWaaS baaSqaa8qacaaIWaaapaqabaaak8qacaGLOaGaayzkaaGaeyOeI0Ia amyDa8aadaqhaaWcbaWdbiaad6eaa8aabaWdbiabgkHiTaaakmaabm aapaqaa8qacaWG0bGaeyOeI0IaeqiXdqNaaiilaiaadIhapaWaaSba aSqaa8qacaaIXaaapaqabaGcpeGaaiilaiaadYgapaWaaSbaaSqaa8 qacaaIWaaapaqabaaak8qacaGLOaGaayzkaaaacaGLOaGaayzkaaWd amaaxacabaWdbiaadAhaaSWdaeqabaWdbiaacYTaaaGcpaWaa0baaS qaa8qacaWGobaapaqaa8qacqGHRaWkaaGcdaqadaWdaeaapeGaamiD aiaacYcacaWG4bWdamaaBaaaleaapeGaaGymaaWdaeqaaOWdbiaacY cacaWGSbWdamaaBaaaleaapeGaaGimaaWdaeqaaaGcpeGaayjkaiaa wMcaaiaacckacaWGKbGaamiEa8aadaWgaaWcbaWdbiaaigdaa8aabe aak8qacaGGDbGaaiiOaiaadsgacaWG0bGaaiilaiaacckacaGGGcGa aiiOaiaacckacaGGGcGaaiiOaiaacckacaGGGcGaaiiOamaabmaapa qaa8qacaaIYaGaaGimaaGaayjkaiaawMcaaaaa@8498@

0 T [ G u ˜ N v ¯ Nt  dx+ G m=1 2 c m u ˜ N x m x v ˜ N x m x  dx+ MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Waaubmaeqal8aabaWdbiaaicdaa8aabaWdbiaadsfaa0WdaeaapeGa ey4kIipaaOGaaGPcVlaayQW7caGGBbWaaubeaeqal8aabaWdbiaadE eapaWaaWbaaWqabeaapeGaeyOeI0caaaWcbeqdpaqaa8qacqGHRiI8 aaGccaaMk8UaaGPcV=aadaWfGaqaa8qacaWG1baal8aabeqaa8qaca GGClaaaOWdamaaDaaaleaapeGaamOtaaWdaeaapeGaeyOeI0caaOWd aiqadAhagaqeamaaDaaaleaapeGaamOtaiaadshaa8aabaWdbiabgk HiTaaakiaacckacaWGKbGaamiEaiabgUcaRmaavababeWcpaqaa8qa caWGhbWdamaaCaaameqabaWdbiabgkHiTaaaaSqab0WdaeaapeGaey 4kIipaaOGaaGPcVlaayQW7daGfWbqabSWdaeaapeGaamyBaiabg2da 9iaaigdaa8aabaWdbiaaikdaa0WdaeaapeGaeyyeIuoaaOGaaGPcVl aayQW7caWGJbWdamaaBaaaleaapeGaamyBaaWdaeqaaOWaaCbiaeaa peGaamyDaaWcpaqabeaapeGaaii3caaak8aadaqhaaWcbaWdbiaad6 eacaWG4bWdamaaBaaameaapeGaamyBaaWdaeqaaaWcbaWdbiabgkHi Taaakmaabmaapaqaa8qacaWG4baacaGLOaGaayzkaaWdamaaxacaba WdbiaadAhaaSWdaeqabaWdbiaacYTaaaGcpaWaa0baaSqaa8qacaWG obGaamiEa8aadaWgaaadbaWdbiaad2gaa8aabeaaaSqaa8qacqGHsi slaaGcdaqadaWdaeaapeGaamiEaaGaayjkaiaawMcaaiaacckacaWG KbGaamiEaiabgUcaRaaa@7E27@

G b u N +a u N v ˜ N x  dx] dt= 0 T ( 0 X g 0 t, x 1 v N x 1 ,0  d x 1 + MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Waaubeaeqal8aabaWdbiaadEeapaWaaWbaaWqabeaapeGaeyOeI0ca aaWcbeqdpaqaa8qacqGHRiI8aaGccaaMk8UaaGPcVpaabmaapaqaa8 qaceWGIbWdayaalaWdbiabgEGirlaadwhapaWaa0baaSqaa8qacaWG obaapaqaa8qacqGHsislaaGccqGHRaWkcaWGHbGaamyDa8aadaqhaa WcbaWdbiaad6eaa8aabaWdbiabgkHiTaaaaOGaayjkaiaawMcaa8aa daWfGaqaa8qacaWG2baal8aabeqaa8qacaGGClaaaOWdamaaDaaale aapeGaamOtaaWdaeaapeGaeyOeI0caaOWaaeWaa8aabaWdbiaadIha aiaawIcacaGLPaaacaGGGcGaamizaiaadIhacaGGDbGaaiiOaiaads gacaWG0bGaeyypa0JaeyOeI0Yaaubmaeqal8aabaWdbiaaicdaa8aa baWdbiaadsfaa0WdaeaapeGaey4kIipaaOGaaGPcVlaayQW7caGGOa Waaubmaeqal8aabaWdbiaaicdaa8aabaWdbiaadIfaa0WdaeaapeGa ey4kIipaaOGaaGPcVlaayQW7caWGNbWdamaaBaaaleaapeGaaGimaa WdaeqaaOWdbmaabmaapaqaa8qacaWG0bGaaiilaiaadIhapaWaaSba aSqaa8qacaaIXaaapaqabaaak8qacaGLOaGaayzkaaGaamODa8aada qhaaWcbaWdbiaad6eaa8aabaWdbiabgkHiTaaakmaabmaapaqaa8qa caWG4bWdamaaBaaaleaapeGaaGymaaWdaeqaaOWdbiaacYcacaaIWa aacaGLOaGaayzkaaGaaiiOaiaadsgacaWG4bWdamaaBaaaleaapeGa aGymaaWdaeqaaOWdbiabgUcaRaaa@7F25@

0 X g t, x 1 v N x 1 , l 0  d x 1 ) dt+ G u 0N x v ¯ N τ,x  dx+ 0 T [ G f v N  dx+ MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Waaubmaeqal8aabaWdbiaaicdaa8aabaWdbiaadIfaa0WdaeaapeGa ey4kIipaaOGaaGPcVlaayQW7caWGNbWaaeWaa8aabaWdbiaadshaca GGSaGaamiEa8aadaWgaaWcbaWdbiaaigdaa8aabeaaaOWdbiaawIca caGLPaaacaWG2bWdamaaDaaaleaapeGaamOtaaWdaeaapeGaeyOeI0 caaOWaaeWaa8aabaWdbiaadIhapaWaaSbaaSqaa8qacaaIXaaapaqa baGcpeGaaiilaiaadYgapaWaaSbaaSqaa8qacaaIWaaapaqabaaak8 qacaGLOaGaayzkaaGaaiiOaiaadsgacaWG4bWdamaaBaaaleaapeGa aGymaaWdaeqaaOWdbiaacMcacaGGGcGaamizaiaadshacqGHRaWkda qfqaqabSWdaeaapeGaam4ra8aadaahaaadbeqaa8qacqGHsislaaaa leqan8aabaWdbiabgUIiYdaakiaayQW7caaMk8UaamyDa8aadaqhaa WcbaWdbiaaicdacaWGobaapaqaa8qacqGHsislaaGcdaqadaWdaeaa peGaamiEaaGaayjkaiaawMcaaiqadAhagaqea8aadaqhaaWcbaWdbi aad6eaa8aabaWdbiabgkHiTaaakmaabmaapaqaa8qacqaHepaDcaGG SaGaamiEaaGaayjkaiaawMcaaiaacckacaWGKbGaamiEaiabgUcaRm aavadabeWcpaqaa8qacaaIWaaapaqaa8qacaWGubaan8aabaWdbiab gUIiYdaakiaayQW7caaMk8Uaai4wamaavababeWcpaqaa8qacaWGhb WdamaaCaaameqabaWdbiabgkHiTaaaaSqab0WdaeaapeGaey4kIipa aOGaaGPcVlaayQW7caWGMbGaamODa8aadaqhaaWcbaWdbiaad6eaa8 aabaWdbiabgkHiTaaakiaacckacaWGKbGaamiEaiabgUcaRaaa@889D@

0 X β ˜ i N u N tτ, x 1 , l 0 ) + u N tτ, x 1 , l 0 v ˜ N t, x 1 , l 0 dx 1 dt,           21 MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Waaubmaeqal8aabaWdbiaaicdaa8aabaWdbiaabIfaa0WdaeaapeGa ey4kIipaaOGaaGPcVlaayQW7paWaaCbiaeaapeGaaeOSdaWcpaqabe aapeGaaii3caaak8aadaqhaaWcbaWdbiaabMgaa8aabaWdbiaab6ea aaGcdaqcWaWdaeaapeGaaeyDa8aadaWgaaWcbaWdbiaab6eaa8aabe aak8qadaqadaWdaeaapeGaaeiDaiabgkHiTiaabs8acaGGSaGaaeiE a8aadaWgaaWcbaWdbiaaigdaa8aabeaak8qacaGGSaGaaeiBa8aada WgaaWcbaWdbiaaicdaa8aabeaak8qacaGGPaWdamaaCaaaleqabaWd biabgUcaRaaakiabgkHiTiaabwhapaWaa0baaSqaa8qacaqGobaapa qaa8qacqGHsislaaGcdaqadaWdaeaapeGaaeiDaiabgkHiTiaabs8a caGGSaGaaeiEa8aadaWgaaWcbaWdbiaaigdaa8aabeaak8qacaGGSa GaaeiBa8aadaWgaaWcbaWdbiaaicdaa8aabeaaaOWdbiaawIcacaGL PaaaaiaawIcacaGLPaaapaWaaCbiaeaapeGaaeODaaWcpaqabeaape Gaaii3caaak8aadaqhaaWcbaWdbiaab6eaa8aabaWdbiabgkHiTaaa kmaabmaapaqaa8qacaqG0bGaaiilaiaabIhapaWaaSbaaSqaa8qaca aIXaaapaqabaGcpeGaaiilaiaabYgapaWaaSbaaSqaa8qacaaIWaaa paqabaaak8qacaGLOaGaayzkaaGaaeizaiaabIhapaWaaSbaaSqaa8 qacaaIXaaapaqabaaak8qacaGLOaGaayzxaaGaaeizaiaabshacaGG SaGaaeiOaiaabckacaqGGcGaaeiOaiaabckacaqGGcGaaeiOaiaabc kacaqGGcGaaeiOamaabmaapaqaa8qacaaIYaGaaGymaaGaayjkaiaa wMcaaaaa@8382@

где u 0N + = k=1 s C 0 k φ k MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamyDa8aadaqhaaWcbaWdbiaaicdacaWGobaapaqaa8qacqGHRaWk aaGccqGH9aqpdaGfWbqabSWdaeaapeGaam4Aaiabg2da9iaaigdaa8 aabaWdbiaadohaa0WdaeaapeGaeyyeIuoaaOGaaGPcVlaadoeapaWa a0baaSqaa8qacaaIWaaapaqaa8qacaWGRbaaaOGaeqOXdO2damaaBa aaleaapeGaam4AaaWdaeqaaaaa@48CC@ *,  u 0N = k=1+s N C 0 k φ k MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamyDa8aadaqhaaWcbaWdbiaaicdacaWGobaapaqaa8qacqGHsisl aaGccqGH9aqpdaGfWbqabSWdaeaapeGaam4Aaiabg2da9iaaigdacq GHRaWkcaWGZbaapaqaa8qacaWGobaan8aabaWdbiabggHiLdaakiaa yQW7caWGdbWdamaaDaaaleaapeGaaGimaaWdaeaapeGaam4Aaaaaki abeA8aQ9aadaWgaaWcbaWdbiaadUgaa8aabeaaaaa@4A8C@

Предполагаем, что найдутся постоянные c i >0 MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaam4ya8aadaWgaaWcbaWdbiaadMgaa8aabeaak8qacqGH+aGpcaaI Waaaaa@3A19@  такие, что

c 1 u W 2 1 G 2 Au,u c 2 u W 2 1 G 2  u W 2 1 G , u | Γ =0. MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaam4ya8aadaWgaaWcbaWdbiaaigdaa8aabeaatCvAUfeBSn0BKvgu HDwzZbqeg0uySDwDUbYrVrhAPngaiuaak8qacaWFwaIaamyDaiaa=z bipaWaa0baaSqaa8qacaWGxbWdamaaDaaameaapeGaaGOmaaWdaeaa peGaaGymaaaalmaabmaapaqaa8qacaWGhbaacaGLOaGaayzkaaaapa qaa8qacaaIYaaaaOGaeyizIm6aaeWaa8aabaWdbiaadgeacaWG1bGa aiilaiaadwhaaiaawIcacaGLPaaacqGHKjYOcaWGJbWdamaaBaaale aapeGaaGOmaaWdaeqaaOWdbiaa=zbicaWG1bGaa8NfG8aadaqhaaWc baWdbiaadEfapaWaa0baaWqaa8qacaaIYaaapaqaa8qacaaIXaaaaS WaaeWaa8aabaWdbiaadEeaaiaawIcacaGLPaaaa8aabaWdbiaaikda aaGccaqGGcGaeyiaIiIaamyDaiabgIGiolaadEfapaWaa0baaSqaa8 qacaaIYaaapaqaa8qacaaIXaaaaOWaaeWaa8aabaWdbiaadEeaaiaa wIcacaGLPaaacaGGSaGaaeiOaiaadwhacaGG8bWdamaaBaaaleaape Gaeu4KdCeapaqabaGcpeGaeyypa0JaaGimaiaac6caaaa@70F9@

Также предположим, что найдется постоянная C 1 MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaam4qa8aadaWgaaWcbaWdbiaaigdaa8aabeaaaaa@37EA@ , не зависящая от сетки по пространственным переменным и времени, такая, что

max t u ˜ N L 2 G + u ˜ N L 2 0,T; W 2 1 G C 1 ,  β N L 2 0,T; L 2 0,X C 1 . MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaaxababaaeaa aaaaaaa8qacaWGTbGaamyyaiaadIhaaSWdaeaapeGaamiDaaWdaeqa aOWdbiaayQW7tCvAUfeBSn0BKvguHDwzZbqeg0uySDwDUbYrVrhAPn gaiuaacaWFwaYdamaaxacabaWdbiaadwhaaSWdaeqabaWdbiaacYTa aaGcpaWaaSbaaSqaa8qacaWGobaapaqabaGcpeGaa8NfG8aadaWgaa WcbaWdbiaadYeapaWaaSbaaWqaa8qacaaIYaaapaqabaWcpeWaaeWa a8aabaWdbiaadEeaaiaawIcacaGLPaaaa8aabeaak8qacqGHRaWkca WFwaYdamaaxacabaWdbiaadwhaaSWdaeqabaWdbiaacYTaaaGcpaWa aSbaaSqaa8qacaWGobaapaqabaGcpeGaa8NfG8aadaWgaaWcbaWdbi aadYeapaWaaSbaaWqaa8qacaaIYaaapaqabaWcpeWaaeWaa8aabaWd biaaicdacaGGSaGaamivaiaacUdacaWGxbWdamaaDaaameaapeGaaG OmaaWdaeaapeGaaGymaaaalmaabmaapaqaa8qacaWGhbaacaGLOaGa ayzkaaaacaGLOaGaayzkaaaapaqabaGcpeGaeyizImQaam4qa8aada WgaaWcbaWdbiaaigdaa8aabeaak8qacaGGSaGaaeiOaiaa=zbicqaH YoGypaWaaSbaaSqaa8qacaWGobaapaqabaGcpeGaa8NfG8aadaWgaa WcbaWdbiaadYeapaWaaSbaaWqaa8qacaaIYaaapaqabaWcpeWaaeWa a8aabaWdbiaaicdacaGGSaGaamivaiaacUdacaWGmbWdamaaBaaame aapeGaaGOmaaWdaeqaaSWdbmaabmaapaqaa8qacaaIWaGaaiilaiaa dIfaaiaawIcacaGLPaaaaiaawIcacaGLPaaaa8aabeaak8qacqGHKj YOcaWGdbWdamaaBaaaleaapeGaaGymaaWdaeqaaOWdbiaac6caaaa@7F70@ ***TRANSLATION ERROR*** (22)

Считаем, что функции Φ i MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaeuOPdy0damaaBaaaleaapeGaamyAaaWdaeqaaaaa@38CF@  линейно независимы. Тогда найдется постоянная C 2 MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaam4qa8aadaWgaaWcbaWdbiaaikdaa8aabeaaaaa@37EB@ , не зависящая от N MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamOtaaaa@36E0@ , такая, что

τ i=1 M k=1 r | β Ni k | 2 C 2 β N L 2 0,T; L 2 0,X 2 ( C 1 ) 2 C 2 . MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaeqiXdq3aaybCaeqal8aabaWdbiaadMgacqGH9aqpcaaIXaaapaqa a8qacaWGnbaan8aabaWdbiabggHiLdaakiaayQW7caaMk8+aaybCae qal8aabaWdbiaadUgacqGH9aqpcaaIXaaapaqaa8qacaWGYbaan8aa baWdbiabggHiLdaakiaayQW7caaMk8UaaiiFaiabek7aI9aadaqhaa WcbaWdbiaad6eacaWGPbaapaqaa8qacaWGRbaaaOGaaiiFa8aadaah aaWcbeqaa8qacaaIYaaaaOGaeyizImQaam4qa8aadaWgaaWcbaWdbi aaikdaa8aabeaatCvAUfeBSn0BKvguHDwzZbqeg0uySDwDUbYrVrhA Pngaiuaak8qacaWFwaIaeqOSdi2damaaBaaaleaapeGaamOtaaWdae qaaOWdbiaa=zbipaWaa0baaSqaa8qacaWGmbWdamaaBaaameaapeGa aGOmaaWdaeqaaSWdbmaabmaapaqaa8qacaaIWaGaaiilaiaadsfaca GG7aGaamita8aadaWgaaadbaWdbiaaikdaa8aabeaal8qadaqadaWd aeaapeGaaGimaiaacYcacaWGybaacaGLOaGaayzkaaaacaGLOaGaay zkaaaapaqaa8qacaaIYaaaaOGaeyizImQaaiikaiaadoeapaWaaSba aSqaa8qacaaIXaaapaqabaGcpeGaaiyka8aadaahaaWcbeqaa8qaca aIYaaaaOGaam4qa8aadaWgaaWcbaWdbiaaikdaa8aabeaak8qacaGG Uaaaaa@7BE1@  (23)

Поскольку число r фиксировано, то оценка (23) влечет также оценку вида

βNL20,T;W2s0,XC3, (24)

где s MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaae4Caaaa@3703@  определяется из условия Φ i W 2 s 0,X MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaeuOPdy0damaaBaaaleaapeGaamyAaaWdaeqaaOWdbiabgIGiolaa dEfapaWaa0baaSqaa8qacaaIYaaapaqaa8qacaWGZbaaaOWaaeWaa8 aabaWdbiaaicdacaGGSaGaamiwaaGaayjkaiaawMcaaaaa@4161@ . Оценка (22) гарантирует также оценку

u ˜ N t, x 1 , l 0 L 2 0,T; W 2 1/2 0,X C 3 MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamXvP5wqSX2qVr wzqf2zLnharyqtHX2z15gih9gDOL2yaGqbaabaaaaaaaaapeGaa8Nf G8aadaWfGaqaa8qacaWG1baal8aabeqaa8qacaGGClaaaOWdamaaBa aaleaapeGaamOtaaWdaeqaaOWdbmaabmaapaqaa8qacaWG0bGaaiil aiaadIhapaWaaSbaaSqaa8qacaaIXaaapaqabaGcpeGaaiilaiaadY gapaWaaSbaaSqaa8qacaaIWaaapaqabaaak8qacaGLOaGaayzkaaGa a8NfG8aadaWgaaWcbaWdbiaadYeapaWaaSbaaWqaa8qacaaIYaaapa qabaWcpeWaaeWaa8aabaWdbiaaicdacaGGSaGaamivaiaacUdacaWG xbWdamaaDaaameaapeGaaGOmaaWdaeaapeGaaGymaiaac+cacaaIYa aaaSWaaeWaa8aabaWdbiaaicdacaGGSaGaamiwaaGaayjkaiaawMca aaGaayjkaiaawMcaaaWdaeqaaOWdbiabgsMiJkaadoeapaWaaSbaaS qaa8qacaaIZaaapaqabaaaaa@61A3@ . (25)

Фиксируем s>0 MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaam4Caiabg6da+iaaicdaaaa@38C7@  и предположим, что ΦiW2s0,X для всех i. Оценки (22)-(25) влекут, что найдутся подпоследовательности uNk,  βNk такие, что

u~NkuL20,T;W21G, u~NkuL0,T;L2G,u~Nkt,x1,l0ut,x1,l0L20,T;W2120,X,β~Nkβ~L2(0,T;W2s0,X 

слабо, * MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaaeOkaaaa@36BA@  -слабо и по норме.

Если мы дополнительно предположим, что у нас есть оценка вида

u~Nx1,l0W2s00,T;W2s10,XC4 (26)

или вида

βNW2s00,T;W2s10,XC5, (27)

где s 1 MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaam4Ca8aadaWgaaWcbaWdbiaaigdaa8aabeaaaaa@381A@  произвольно (в том числе возможно, что s 1 <0 MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaam4Ca8aadaWgaaWcbaWdbiaaigdaa8aabeaak8qacqGH8aapcaaI Waaaaa@39F2@  ) и s 0 >0 MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaam4Ca8aadaWgaaWcbaWdbiaaicdaa8aabeaak8qacqGH+aGpcaaI Waaaaa@39F5@ , то стандартные утверждения о компактности влекут, что существует подпоследовательность u N k MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamyDa8aadaWgaaWcbaWdbiaad6eapaWaaSbaaWqaa8qacaWGRbaa paqabaaaleqaaaaa@397B@  такая, что u N k t, x 1 , l 0 u t, x 1 , l 0 MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamyDa8aadaWgaaWcbaWdbiaad6eapaWaaSbaaWqaa8qacaWGRbaa paqabaaaleqaaOWdbmaabmaapaqaa8qacaWG0bGaaiilaiaadIhapa WaaSbaaSqaa8qacaaIXaaapaqabaGcpeGaaiilaiaadYgapaWaaSba aSqaa8qacaaIWaaapaqabaaak8qacaGLOaGaayzkaaGaeyOKH4Qaam yDamaabmaapaqaa8qacaWG0bGaaiilaiaadIhapaWaaSbaaSqaa8qa caaIXaaapaqabaGcpeGaaiilaiaadYgapaWaaSbaaSqaa8qacaaIWa aapaqabaaak8qacaGLOaGaayzkaaaaaa@4D14@ , или β N k β ˜ MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaeqOSdi2damaaBaaaleaapeGaamOta8aadaWgaaadbaWdbiaadUga a8aabeaaaSqabaGcpeGaeyOKH46damaaxacabaWdbiabek7aIbWcpa qabeaapeGaaii3caaaaaa@3FB3@  сильно в L 2 0,T; L 2 0,X MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaamita8aadaWgaaWcbaWdbiaaikdaa8aabeaak8qadaqadaWdaeaa peGaaGimaiaacYcacaWGubGaai4oaiaadYeapaWaaSbaaSqaa8qaca aIYaaapaqabaGcpeWaaeWaa8aabaWdbiaaicdacaGGSaGaamiwaaGa ayjkaiaawMcaaaGaayjkaiaawMcaaaaa@42A8@ .

При выполнении этих оценок можно сформулировать следующее утверждение:

Лемма 1. Пусть имеют место оценки (22) MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSn0BKvguHDwzZbqef00uGuvsGC0B0H wAJbWexLMBbXgBcf2CPn2qVrwzqf2zLnharyavP1wzZbItLDhis9wB H5garmWu51MyVXgaryWqVvNCPvMCG4uz3bqee0evGueE0jxyaibaie Ydh9qrpeeu0dXdh9vqqj=hEeeu0xXdbba9arpi0=irpK0dbba91qpK 0=vr0RYxir=dbbc9q8aq0=yqpe0xbba9suk9fr=xfr=xfrpiWZqaai aaciWacmaadaGabiaaeaGaauaaaOqaaGGaaKqzafaeaaaaaaaaa8qa caWFtacaaa@42F3@ (25) и одна из оценок (26), (27). Тогда в равенствах (30), (31) можно перейти к пределу по N MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamOtaaaa@36E0@ , и предельное решение есть обобщенное решение задачи сопряжения из класса

u L 2 Q , u t L 2 0,T, W 2 1 G MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamyDaiabgIGiolaadYeapaWaaSbaaSqaa8qacaaIYaaapaqabaGc peWaaeWaa8aabaWdbiaadgfaaiaawIcacaGLPaaacaGGSaGaamyDa8 aadaWgaaWcbaWdbiaadshaa8aabeaak8qacqGHiiIZcaWGmbWdamaa BaaaleaapeGaaGOmaaWdaeqaaOWdbmaabmaapaqaa8qacaaIWaGaai ilaiaadsfacaGGSaGaam4va8aadaqhaaWcbaWdbiaaikdaa8aabaWd biabgkHiTiaaigdaaaGcdaqadaWdaeaapeGaam4raaGaayjkaiaawM caaaGaayjkaiaawMcaaaaa@4E6A@ , u ± L 2 (0,T; W 2 1 G ± MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamyDa8aadaahaaWcbeqaa8qacqGHXcqSaaGccqGHiiIZcaWGmbWd amaaBaaaleaapeGaaGOmaaWdaeqaaOWdbiaacIcacaaIWaGaaiilai aadsfacaGG7aGaam4va8aadaqhaaWcbaWdbiaaikdaa8aabaWdbiaa igdaaaGcdaqadaWdaeaapeGaam4ra8aadaahaaWcbeqaa8qacqGHXc qSaaaakiaawIcacaGLPaaaaaa@47FE@ .

Доказательство. Рассмотрим равенства (20), (21). Взяв N= N k MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamOtaiabg2da9iaad6eapaWaaSbaaSqaa8qacaWGRbaapaqabaaa aa@3A03@ , фиксировав функции v N ± MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamODa8aadaqhaaWcbaWdbiaad6eaa8aabaWdbiabgglaXcaaaaa@3A34@  и переходя к пределу по k, MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaam4AaiaacYcaaaa@37AD@  получим равенства:

0 T [ G + u + v ¯ Nt +  dx+ G + m=1 2 c m u x m + v ˜ N x m + t,x  dx+ MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Waaubmaeqal8aabaWdbiaaicdaa8aabaWdbiaadsfaa0WdaeaapeGa ey4kIipaaOGaaGPcVlaayQW7caGGBbWaaubeaeqal8aabaWdbiaadE eapaWaaWbaaWqabeaapeGaey4kaScaaaWcbeqdpaqaa8qacqGHRiI8 aaGccaaMk8UaaGPcVlaadwhapaWaaWbaaSqabeaapeGaey4kaScaaO WdaiqadAhagaqeamaaDaaaleaapeGaamOtaiaadshaa8aabaWdbiab gUcaRaaakiaacckacaWGKbGaamiEaiabgUcaRmaavababeWcpaqaa8 qacaWGhbWdamaaCaaameqabaWdbiabgUcaRaaaaSqab0WdaeaapeGa ey4kIipaaOGaaGPcVlaayQW7daGfWbqabSWdaeaapeGaamyBaiabg2 da9iaaigdaa8aabaWdbiaaikdaa0WdaeaapeGaeyyeIuoaaOGaaGPc VlaayQW7caWGJbWdamaaBaaaleaapeGaamyBaaWdaeqaaOWdbiaadw hapaWaa0baaSqaa8qacaWG4bWdamaaBaaameaapeGaamyBaaWdaeqa aaWcbaWdbiabgUcaRaaak8aaceWG2bGbaGaadaqhaaWcbaWdbiaad6 eacaWG4bWdamaaBaaameaapeGaamyBaaWdaeqaaaWcbaWdbiabgUca Raaakmaabmaapaqaa8qacaWG0bGaaiilaiaadIhaaiaawIcacaGLPa aacaGGGcGaamizaiaadIhacqGHRaWkaaa@7579@

G + b u + +a u + v ˜ N + t,x ) dx] dt= G + u 0 + x v ˜ N + τ,x  dx+ MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Waaubeaeqal8aabaWdbiaadEeapaWaaWbaaWqabeaapeGaey4kaSca aaWcbeqdpaqaa8qacqGHRiI8aaGccaaMk8UaaGPcVpaabmaapaqaa8 qaceWGIbWdayaalaWdbiabgEGirlaadwhapaWaaWbaaSqabeaapeGa ey4kaScaaOGaey4kaSIaamyyaiaadwhapaWaaWbaaSqabeaapeGaey 4kaScaaaGccaGLOaGaayzkaaWdaiqadAhagaacamaaDaaaleaapeGa amOtaaWdaeaapeGaey4kaScaaOWaaeWaa8aabaWdbiaadshacaGGSa GaamiEaaGaayjkaiaawMcaaiaacMcacaGGGcGaamizaiaadIhacaGG DbGaaiiOaiaadsgacaWG0bGaeyypa0Zaaubeaeqal8aabaWdbiaadE eapaWaaWbaaWqabeaapeGaey4kaScaaaWcbeqdpaqaa8qacqGHRiI8 aaGccaaMk8UaaGPcVlaadwhapaWaa0baaSqaa8qacaaIWaaapaqaa8 qacqGHRaWkaaGcdaqadaWdaeaapeGaamiEaaGaayjkaiaawMcaa8aa ceWG2bGbaGaadaqhaaWcbaWdbiaad6eaa8aabaWdbiabgUcaRaaakm aabmaapaqaa8qacqaHepaDcaGGSaGaamiEaaGaayjkaiaawMcaaiaa cckacaWGKbGaamiEaiabgUcaRaaa@7199@

0TG+fv~N+ dx0Xβ~u+t,x1,l0ut,x1,l0v~N+t,x1,l0dx1dt, (28)

0 T [ G u ˜ v ¯ Nt  dx+ G m=1 2 c m u ˜ x m x v ˜ Nx m x  dx+ MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Waaubmaeqal8aabaWdbiaaicdaa8aabaWdbiaabsfaa0WdaeaapeGa ey4kIipaaOGaaGPcVlaayQW7caGGBbWaaubeaeqal8aabaWdbiaabE eapaWaaWbaaWqabeaapeGaeyOeI0caaaWcbeqdpaqaa8qacqGHRiI8 aaGccaaMk8UaaGPcV=aadaWfGaqaa8qacaqG1baal8aabeqaa8qaca GGClaaaOWdamaaCaaaleqabaWdbiabgkHiTaaakiqadAhagaqea8aa daqhaaWcbaWdbiaab6eacaqG0baapaqaa8qacqGHsislaaGccaqGGc GaaeizaiaabIhacqGHRaWkdaqfqaqabSWdaeaapeGaae4ra8aadaah aaadbeqaa8qacqGHsislaaaaleqan8aabaWdbiabgUIiYdaakiaayQ W7caaMk8+aaybCaeqal8aabaWdbiaab2gacqGH9aqpcaaIXaaapaqa a8qacaaIYaaan8aabaWdbiabggHiLdaakiaayQW7caaMk8Uaae4ya8 aadaWgaaWcbaWdbiaab2gaa8aabeaakmaaxacabaWdbiaabwhaaSWd aeqabaWdbiaacYTaaaGcpaWaa0baaSqaa8qacaqG4bWdamaaBaaame aapeGaaeyBaaWdaeqaaaWcbaWdbiabgkHiTaaakmaabmaapaqaa8qa caqG4baacaGLOaGaayzkaaWdamaaxacabaWdbiaabAhaaSWdaeqaba WdbiaacYTaaaGcpaWaa0baaSqaa8qacaqGobGaaeiEa8aadaWgaaad baWdbiaab2gaa8aabeaaaSqaa8qacqGHsislaaGcdaqadaWdaeaape GaaeiEaaGaayjkaiaawMcaaiaabckacaqGKbGaaeiEaiabgUcaRaaa @7C34@

Gbu+auv~Nx dx dt=Gu0xv¯Nτ,x dx+0TGfv~N dx+

0Xβ~ut,x1,l0)+ut,x1,l0v~Nt,x1,l0dx1dt. (29)

Далее берем произвольную функцию v W 2 1 Q ± L 2 Q MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamODaiabgIGiolaadEfapaWaa0baaSqaa8qacaaIYaaapaqaa8qa caaIXaaaaOWaaeWaa8aabaWdbiaadgfapaWaaWbaaSqabeaapeGaey ySaelaaaGccaGLOaGaayzkaaGaeyykICSaamita8aadaWgaaWcbaWd biaaikdaa8aabeaak8qadaqadaWdaeaapeGaamyuaaGaayjkaiaawM caaaaa@4633@ , удовлетворяющую однородным условиям Дирихле на боковой поверхности области G MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaam4raaaa@36D9@  и такую, что v | t=T =0 MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamODaiaacYhapaWaaSbaaSqaa8qacaWG0bGaeyypa0JaamivaaWd aeqaaOWdbiabg2da9iaaicdaaaa@3D14@ . Построив приближение функции v MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamODaaaa@3708@  в норме W 2 1 Q ± , MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaam4va8aadaqhaaWcbaWdbiaaikdaa8aabaWdbiaaigdaaaGcdaqa daWdaeaapeGaamyua8aadaahaaWcbeqaa8qacqGHXcqSaaaakiaawI cacaGLPaaacaGGSaaaaa@3E47@  перейдем к пределу и из (28), (29) получим равенства:

0 T [ G + u + v t +  dx+ G + m=1 2 c m u x m + v x m + t,x  dx+ MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Waaubmaeqal8aabaWdbiaaicdaa8aabaWdbiaadsfaa0WdaeaapeGa ey4kIipaaOGaaGPcVlaayQW7caGGBbWaaubeaeqal8aabaWdbiaadE eapaWaaWbaaWqabeaapeGaey4kaScaaaWcbeqdpaqaa8qacqGHRiI8 aaGccaaMk8UaaGPcVlaadwhapaWaaWbaaSqabeaapeGaey4kaScaaO GaamODa8aadaqhaaWcbaWdbiaadshaa8aabaWdbiabgUcaRaaakiaa cckacaWGKbGaamiEaiabgUcaRmaavababeWcpaqaa8qacaWGhbWdam aaCaaameqabaWdbiabgUcaRaaaaSqab0WdaeaapeGaey4kIipaaOGa aGPcVlaayQW7daGfWbqabSWdaeaapeGaamyBaiabg2da9iaaigdaa8 aabaWdbiaaikdaa0WdaeaapeGaeyyeIuoaaOGaaGPcVlaayQW7caWG JbWdamaaBaaaleaapeGaamyBaaWdaeqaaOWdbiaadwhapaWaa0baaS qaa8qacaWG4bWdamaaBaaameaapeGaamyBaaWdaeqaaaWcbaWdbiab gUcaRaaakiaadAhapaWaa0baaSqaa8qacaWG4bWdamaaBaaameaape GaamyBaaWdaeqaaaWcbaWdbiabgUcaRaaakmaabmaapaqaa8qacaWG 0bGaaiilaiaadIhaaiaawIcacaGLPaaacaGGGcGaamizaiaadIhacq GHRaWkaaa@73AC@

G + b u + +a u + v + t,x  dx  dt= G + u 0 + x v + 0,x  dx+ 0 T G + f v +  dx MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Waaubeaeqal8aabaWdbiaadEeapaWaaWbaaWqabeaapeGaey4kaSca aaWcbeqdpaqaa8qacqGHRiI8aaGccaaMk8UaaGPcVpaabmaapaqaa8 qaceWGIbWdayaalaWdbiabgEGirlaadwhapaWaaWbaaSqabeaapeGa ey4kaScaaOGaey4kaSIaamyyaiaadwhapaWaaWbaaSqabeaapeGaey 4kaScaaaGccaGLOaGaayzkaaGaamODa8aadaahaaWcbeqaa8qacqGH RaWkaaGcdaqadaWdaeaapeGaamiDaiaacYcacaWG4baacaGLOaGaay zkaaGaaiiOaiaadsgacaWG4bWaaKWia8aabaWdbiaacckacaWGKbGa amiDaiabg2da9maavababeWcpaqaa8qacaWGhbWdamaaCaaameqaba WdbiabgUcaRaaaaSqab0WdaeaapeGaey4kIipaaOGaaGPcVlaayQW7 caWG1bWdamaaDaaaleaapeGaaGimaaWdaeaapeGaey4kaScaaOWaae Waa8aabaWdbiaadIhaaiaawIcacaGLPaaacaWG2bWdamaaCaaaleqa baWdbiabgUcaRaaakmaabmaapaqaa8qacaaIWaGaaiilaiaadIhaai aawIcacaGLPaaacaGGGcGaamizaiaadIhacqGHRaWkdaqfWaqabSWd aeaapeGaaGimaaWdaeaapeGaamivaaqdpaqaa8qacqGHRiI8aaGcca aMk8UaaGPcVdGaayzxaiaawUfaamaavababeWcpaqaa8qacaWGhbWd amaaCaaameqabaWdbiabgUcaRaaaaSqab0WdaeaapeGaey4kIipaaO GaaGPcVlaayQW7caWGMbGaamODa8aadaahaaWcbeqaa8qacqGHRaWk aaGccaGGGcGaamizaiaadIhacqGHsislaaa@851F@

0Xβ~u+t,x1,l0ut,x1,l0v+t,x1,l0 dx1] dt, (30)

0 T [ G u v t  dx+ G m=1 2 c m u x m x v x m t,x  dx+ MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Waaubmaeqal8aabaWdbiaaicdaa8aabaWdbiaadsfaa0WdaeaapeGa ey4kIipaaOGaaGPcVlaayQW7caGGBbWaaubeaeqal8aabaWdbiaadE eapaWaaWbaaWqabeaapeGaeyOeI0caaaWcbeqdpaqaa8qacqGHRiI8 aaGccaaMk8UaaGPcVlaadwhapaWaaWbaaSqabeaapeGaeyOeI0caaO GaamODa8aadaqhaaWcbaWdbiaadshaa8aabaWdbiabgkHiTaaakiaa cckacaWGKbGaamiEaiabgUcaRmaavababeWcpaqaa8qacaWGhbWdam aaCaaameqabaWdbiabgkHiTaaaaSqab0WdaeaapeGaey4kIipaaOGa aGPcVlaayQW7daGfWbqabSWdaeaapeGaamyBaiabg2da9iaaigdaa8 aabaWdbiaaikdaa0WdaeaapeGaeyyeIuoaaOGaaGPcVlaayQW7caWG JbWdamaaBaaaleaapeGaamyBaaWdaeqaaOWdbiaadwhapaWaa0baaS qaa8qacaWG4bWdamaaBaaameaapeGaamyBaaWdaeqaaaWcbaWdbiab gkHiTaaakmaabmaapaqaa8qacaWG4baacaGLOaGaayzkaaGaamODa8 aadaqhaaWcbaWdbiaadIhapaWaaSbaaWqaa8qacaWGTbaapaqabaaa leaapeGaeyOeI0caaOWaaeWaa8aabaWdbiaadshacaGGSaGaamiEaa GaayjkaiaawMcaaiaacckacaWGKbGaamiEaiabgUcaRaaa@7693@

G b u +a u v t,x  dx  dt= G u 0 x v 0,x  dx+ 0 T G f v  dx+ MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Waaubeaeqal8aabaWdbiaadEeapaWaaWbaaWqabeaapeGaeyOeI0ca aaWcbeqdpaqaa8qacqGHRiI8aaGccaaMk8UaaGPcVpaabmaapaqaa8 qaceWGIbWdayaalaWdbiabgEGirlaadwhapaWaaWbaaSqabeaapeGa eyOeI0caaOGaey4kaSIaamyyaiaadwhapaWaaWbaaSqabeaapeGaey OeI0caaaGccaGLOaGaayzkaaGaamODa8aadaahaaWcbeqaa8qacqGH sislaaGcdaqadaWdaeaapeGaamiDaiaacYcacaWG4baacaGLOaGaay zkaaGaaiiOaiaadsgacaWG4bWaaKWia8aabaWdbiaacckacaWGKbGa amiDaiabg2da9maavababeWcpaqaa8qacaWGhbWdamaaCaaameqaba WdbiabgkHiTaaaaSqab0WdaeaapeGaey4kIipaaOGaaGPcVlaayQW7 caWG1bWdamaaDaaaleaapeGaaGimaaWdaeaapeGaeyOeI0caaOWaae Waa8aabaWdbiaadIhaaiaawIcacaGLPaaacaWG2bWdamaaCaaaleqa baWdbiabgkHiTaaakmaabmaapaqaa8qacaaIWaGaaiilaiaadIhaai aawIcacaGLPaaacaGGGcGaamizaiaadIhacqGHRaWkdaqfWaqabSWd aeaapeGaaGimaaWdaeaapeGaamivaaqdpaqaa8qacqGHRiI8aaGcca aMk8UaaGPcVdGaayzxaiaawUfaamaavababeWcpaqaa8qacaWGhbWd amaaCaaameqabaWdbiabgkHiTaaaaSqab0WdaeaapeGaey4kIipaaO GaaGPcVlaayQW7caWGMbGaamODa8aadaahaaWcbeqaa8qacqGHsisl aaGccaGGGcGaamizaiaadIhacqGHRaWkaaa@8577@

0Xβ~ut,x1,l0)+ut,x1,l0vt,x1,l0dx1dt, (31)

справедливые для всех v ± W 2 1 Q ± MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamODa8aadaahaaWcbeqaa8qacqGHXcqSaaGccqGHiiIZcaWGxbWd amaaDaaaleaapeGaaGOmaaWdaeaapeGaaGymaaaakmaabmaapaqaa8 qacaWGrbWdamaaCaaaleqabaWdbiabgglaXcaaaOGaayjkaiaawMca aaaa@425A@ , таких, что v ± T,x =0 MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamODa8aadaahaaWcbeqaa8qacqGHXcqSaaGcdaqadaWdaeaapeGa amivaiaacYcacaWG4baacaGLOaGaayzkaaGaeyypa0JaaGimaaaa@3F3A@ , и удовлетворяющих условиям Дирихле в (4). Используя определение обобщенной производной, получим, что существуют обобщенные производные u t ± L 2 0,T; W 2 1 G ± MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamyDa8aadaqhaaWcbaWdbiaadshaa8aabaWdbiabgglaXcaakiab gIGiolaadYeapaWaaSbaaSqaa8qacaaIYaaapaqabaGcpeWaaeWaa8 aabaWdbiaaicdacaGGSaGaamivaiaacUdacaWGxbWdamaaDaaaleaa peGaaGOmaaWdaeaapeGaeyOeI0IaaGymaaaakmaabmaapaqaa8qaca WGhbWdamaaCaaaleqabaWdbiabgglaXcaaaOGaayjkaiaawMcaaaGa ayjkaiaawMcaaaaa@4AFF@  и u ± 0,x = u 0 ± x MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamyDa8aadaahaaWcbeqaa8qacqGHXcqSaaGcdaqadaWdaeaapeGa aGimaiaacYcacaWG4baacaGLOaGaayzkaaGaeyypa0JaamyDa8aada qhaaWcbaWdbiaaicdaa8aabaWdbiabgglaXcaakmaabmaapaqaa8qa caWG4baacaGLOaGaayzkaaaaaa@451C@ . Таким образом, мы пришли к определению обобщенного решения задачи сопряжения из класса u L 2 Q , u t L 2 0,T, W 2 1 G MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamyDaiabgIGiolaadYeapaWaaSbaaSqaa8qacaaIYaaapaqabaGc peWaaeWaa8aabaWdbiaadgfaaiaawIcacaGLPaaacaGGSaGaamyDa8 aadaWgaaWcbaWdbiaadshaa8aabeaak8qacqGHiiIZcaWGmbWdamaa BaaaleaapeGaaGOmaaWdaeqaaOWdbmaabmaapaqaa8qacaaIWaGaai ilaiaadsfacaGGSaGaam4va8aadaqhaaWcbaWdbiaaikdaa8aabaWd biabgkHiTiaaigdaaaGcdaqadaWdaeaapeGaam4raaGaayjkaiaawM caaaGaayjkaiaawMcaaaaa@4E6A@ , u ± L 2 (0,T; W 2 1 G ± MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamyDa8aadaahaaWcbeqaa8qacqGHXcqSaaGccqGHiiIZcaWGmbWd amaaBaaaleaapeGaaGOmaaWdaeqaaOWdbiaacIcacaaIWaGaaiilai aadsfacaGG7aGaam4va8aadaqhaaWcbaWdbiaaikdaa8aabaWdbiaa igdaaaGcdaqadaWdaeaapeGaam4ra8aadaahaaWcbeqaa8qacqGHXc qSaaaakiaawIcacaGLPaaaaaa@47FE@ .

РЕЗУЛЬТАТЫ И ЭКСПЕРИМЕНТЫ

Перейдем к рассмотрению численных экспериментов и анализу их результатов. Полученный программный комплекс был зарегистрирован, и получено соответствующее свидетельство. Получаемые результаты вычислений напрямую зависят от характеристик производительности компьютера. Характеристики компьютера, на котором были получены описываемые далее данные, следующие: процессор Intel(R) Core(TM) i5-9500F CPU @ 3.00GHz 3.00GHz, 16.00 GB RAM.

В описываемом эксперименте τ MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaeqiXdqhaaa@37D2@   MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSn0BKvguHDwzZbqef00uGuvsGC0B0H wAJbWexLMBbXgBcf2CPn2qVrwzqf2zLnharyavP1wzZbItLDhis9wB H5garmWu51MyVXgaryWqVvNCPvMCG4uz3bqee0evGueE0jxyaibaie Ydh9qrpeeu0dXdh9vqqj=hEeeu0xXdbba9arpi0=irpK0dbba91qpK 0=vr0RYxir=dbbc9q8aq0=yqpe0xbba9suk9fr=xfr=xfrpiWZqaai aaciWacmaadaGabiaaeaGaauaaaOqaaGGaaKqzafaeaaaaaaaaa8qa caWFtacaaa@42F3@  время выполнения расчета в секундах, ε 0 MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaeqyTdu2damaaBaaaleaapeGaaGimaaWdaeqaaaaa@38C8@   MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSn0BKvguHDwzZbqef00uGuvsGC0B0H wAJbWexLMBbXgBcf2CPn2qVrwzqf2zLnharyavP1wzZbItLDhis9wB H5garmWu51MyVXgaryWqVvNCPvMCG4uz3bqee0evGueE0jxyaibaie Ydh9qrpeeu0dXdh9vqqj=hEeeu0xXdbba9arpi0=irpK0dbba91qpK 0=vr0RYxir=dbbc9q8aq0=yqpe0xbba9suk9fr=xfr=xfrpiWZqaai aaciWacmaadaGabiaaeaGaauaaaOqaaGGaaKqzafaeaaaaaaaaa8qa caWFtacaaa@42F3@  рассчитаная точность полученных вычислений, δ MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaeqiTdqgaaa@37B2@   MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSn0BKvguHDwzZbqef00uGuvsGC0B0H wAJbWexLMBbXgBcf2CPn2qVrwzqf2zLnharyavP1wzZbItLDhis9wB H5garmWu51MyVXgaryWqVvNCPvMCG4uz3bqee0evGueE0jxyaibaie Ydh9qrpeeu0dXdh9vqqj=hEeeu0xXdbba9arpi0=irpK0dbba91qpK 0=vr0RYxir=dbbc9q8aq0=yqpe0xbba9suk9fr=xfr=xfrpiWZqaai aaciWacmaadaGabiaaeaGaauaaaOqaaGGaaKqzafaeaaaaaaaaa8qa caWFtacaaa@42F3@  уровень случайного шума, r MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamOCaaaa@3704@ , функции Φ MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaeuOPdyeaaa@3787@ :

Φ 1 = x 2 2x MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaaeOPd8aadaWgaaWcbaWdbiaaigdaa8aabeaak8qacqGH9aqpcaWG 4bWdamaaCaaaleqabaWdbiaaikdaaaGccqGHsislcaaIYaGaamiEaa aa@3E23@

Φ 2 =x*sin 3* x 4 2 MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaaeOPd8aadaWgaaWcbaWdbiaaikdaa8aabeaak8qacqGH9aqpcaWG 4bGaaiOkaiGacohacaGGPbGaaiOBamaabmaapaqaa8qacaaIZaGaai Okamaalaaapaqaa8qacaWG4baapaqaa8qacaaI0aaaaaGaayjkaiaa wMcaaiabgkHiTiaaikdaaaa@44B7@

Φ 3 = x 3 2 x 2 MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaaeOPd8aadaWgaaWcbaWdbiaaiodaa8aabeaak8qacqGH9aqpcaWG 4bWdamaaCaaaleqabaWdbiaaiodaaaGccqGHsislcaaIYaGaamiEa8 aadaahaaWcbeqaa8qacaaIYaaaaaaa@3F2E@  .

В следующей таблице представлены результаты расчетов при ε= 10 3 MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaeqyTduMaeyypa0JaaGymaiaaicdapaWaaWbaaSqabeaapeGaeyOe I0IaaG4maaaaaaa@3C25@ .

 

Таблица 1. Расчеты при ε= 10 3 MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaaeyTdiabg2da9iaaigdacaaIWaWdamaaCaaaleqabaWdbiabgkHi Tiaaiodaaaaaaa@3BB9@

No exp.

Φ

r

δ

ε_0

τ

1

Φ1

3

0

0,0107

6,95

2

Φ1

4

0

0,0136

7,39

3

Φ1

5

0

0,0166

7,2

4

Φ2

3

0

0,0094

7,15

5

Φ2

4

0

0,0126

6,2

6

Φ2

5

0

0,0165

7,18

7

Φ3

3

0

0,01

6,07

8

Φ3

4

0

0,0135

8,35

9

Φ3

5

0

0,0171

6,07

 

Далее представлены результаты при ε= 10 4 MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaeqyTduMaeyypa0JaaGymaiaaicdapaWaaWbaaSqabeaapeGaeyOe I0IaaGinaaaaaaa@3C26@ .

 

Таблица 2. Расчеты при ε= 10 4 MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaaeyTdiabg2da9iaaigdacaaIWaWdamaaCaaaleqabaWdbiabgkHi Tiaaisdaaaaaaa@3BBA@

No exp.

Φ

r

δ

ε_0

τ

1

Φ1

3

0

0,00102

9,75

2

Φ1

4

0

0,00124

9,79

3

Φ1

5

0

0,00154

10,52

4

Φ2

3

0

0,00099

9,61

5

Φ2

4

0

0,00121

9,71

6

Φ2

5

0

0,00159

8,22

7

Φ3

3

0

0,00091

10,42

8

Φ3

4

0

0,012

9,78

9

Φ3

5

0

0,0164

11,77

 

И результаты при увеличении точности до ε= 10 5 MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaeqyTduMaeyypa0JaaGymaiaaicdapaWaaWbaaSqabeaapeGaeyOe I0IaaGynaaaaaaa@3C27@ .

 

Таблица 3. Расчеты при ε= 10 5 MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaaeyTdiabg2da9iaaigdacaaIWaWdamaaCaaaleqabaWdbiabgkHi Tiaaiwdaaaaaaa@3BBB@

No exp.

Φ

r

δ

ε_0

τ

1

Φ1

3

0

0,000099

13,9

2

Φ1

4

0

0,000135

12,14

3

Φ1

5

0

0,000166

13,32

4

Φ2

3

0

0,000109

11,56

5

Φ2

4

0

0,000121

13,03

6

Φ2

5

0

0,000163

13,53

7

Φ3

3

0

0,0001

12,91

8

Φ3

4

0

0,000119

12,36

9

Φ3

5

0

0,000165

11,96

 

Также для проверки устойчивости решения на условия переопределения накладывались случайные возмущения данных. ψ new 0 =ψ x 1+δ 2σ1 , MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaeqiYdK3damaaBaaaleaapeGaamOBaiaadwgacaWG3baapaqabaGc peWaaeWaa8aabaWdbiaaicdaaiaawIcacaGLPaaacqGH9aqpcqaHip qEdaqadaWdaeaapeGaamiEaaGaayjkaiaawMcaamaabmaapaqaa8qa caaIXaGaey4kaSIaeqiTdq2aaeWaa8aabaWdbiaaikdacqaHdpWCcq GHsislcaaIXaaacaGLOaGaayzkaaaacaGLOaGaayzkaaGaaiilaaaa @4E6C@  где σ 0,1 MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaeq4WdmNaeyicI48aamWaa8aabaWdbiaaicdacaGGSaGaaGymaaGa ay5waiaaw2faaaaa@3D8A@ , а δ MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaeqiTdqgaaa@37B2@  задается пользователем. В ходе экспериментов случайный шум был равен 5 и 10 %, соответственно δ=5 MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaeqiTdqMaeyypa0JaaGynaaaa@3977@  или δ=10 MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaeqiTdqMaeyypa0JaaGymaiaaicdaaaa@3A2D@ .

Далее были произведены расчеты при различных ε MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaeqyTdugaaa@37B4@  при добавлении случайного шума в 5 и 10 %, в таблице 4 приведены расчеты при ε= 10 3 MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaaeyTdiabg2da9iaaigdacaaIWaWdamaaCaaaleqabaWdbiabgkHi Tiaaiodaaaaaaa@3BB9@ .

 

Таблица 4. Расчеты при изменениях δ MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaeqiTdqgaaa@37B2@  при ε= 10 3 MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaaeyTdiabg2da9iaaigdacaaIWaWdamaaCaaaleqabaWdbiabgkHi Tiaaiodaaaaaaa@3BB9@

No exp.

Φ

r

δ

ε_0

τ

2

Φ1

3

5

0,0103

11,04

3

Φ1

3

10

0,015

10,54

5

Φ1

4

5

0,0155

11,98

6

Φ1

4

10

0,015

14,57

8

Φ1

5

5

0,0171

11,63

9

Φ1

5

10

0,0187

18,63

11

Φ2

3

5

0,013

8,8

12

Φ2

3

10

0,0117

11,19

14

Φ2

4

5

0,0153

9,88

15

Φ2

4

10

0,0148

12,79

17

Φ2

5

5

0,0166

11,92

18

Φ2

5

10

0,0186

18,79

20

Φ3

3

5

0,0127

10,69

21

Φ3

3

10

0,0134

13,03

23

Φ3

4

5

0,016

13,46

24

Φ3

4

10

0,0169

12,82

26

Φ3

5

5

0,0186

12,13

27

Φ3

5

10

0,0188

17,16

 

ЗАКЛЮЧЕНИЕ И ВЫВОДЫ

В результате вычислений отчетливо видно увеличение времени работы программы при повышении точности и при достаточно серьезных изменениях входных данных (при увеличении ошибки до 15 и 20 % расчеты могут выполняться с ошибками или занять кратно больше времени). Также стоит отметить, что увеличение времени работы при ε= 10 5 MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaeqyTduMaeyypa0JaaGymaiaaicdapaWaaWbaaSqabeaapeGaeyOe I0IaaGynaaaaaaa@3C27@  не так заметно повышает точность вычислений, соответственно для большей эффективности и дальнейших вычислений и проверки алгоритма было решено остановиться на ε= 10 4 MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaeqyTduMaeyypa0JaaGymaiaaicdapaWaaWbaaSqabeaapeGaeyOe I0IaaGinaaaaaaa@3C26@  в связи с небольшими временными потерями, но достаточно точных вычислениях.

×

About the authors

Sergei N. Shergin

Yugra State University

Author for correspondence.
Email: ssn@ugrasu.ru

Candidate of Physics and Mathematics, Associate Professor of the Engineering School of Digital Technologies

Russian Federation, Khanty-Mansiysk

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