Учебник - Методы решения основных задач уравнений математической физики - Колесникова, страница 5

¥.−ν =ᮮ⢥âáâ¢ã¥â®¤­ á®¡á⢥­­ ï äã­ªæ¨ï,   ᮡá⢥­­ë¥ ä㭪樨, ᮮ⢥âáâ¢ãî騥ࠧ­ë¬á®¡á⢥­­ë¬§­ ç¥­¨ï¬,®à⮣®­ «ì­ë,Rba Xk (x)Xn (x) dx = 0, k 6= n, ¨ ¯à¥¤áâ ¢«ïîâ ᮡ®©®à⮣®­ «ì­ãî ¢ L2 [a, b] á¨á⥬ã.Žç¥­ì ç áâ® ¢ ­ è¨å § ¤ ç å p(x) ≡ 1, q(x) ≡ 0,22L = − d 2 ,   L∗1 ≡ d 2 .dxâ. ¥.¯®«­ãî¨ â®£¤ dx‚ í⮬ á«ãç ¥ § ¤ ç  ¯à¨¬¥â ¢¨¤ −X ′′ (x) = λ2 X(x), a < x < b,c X(a) + d1 X ′ (a) = 0, 1c2 X(b) + d2 X ′ (b) = 0.Žá®¡¥­­®áâì íâ¨å § ¤ ç ­  ¬¥â®¤ ”ãàì¥ á®á⮨⠢ ⮬, ç⮥᫨ ¢ ª ¦¤®¬ ¨§ ª« áá¨ç¥áª¨å ªà ¥¢ëå ãá«®¢¨© ®â«¨ç¥­ ®â0 «¨èì ®¤¨­ ª®íää¨æ¨¥­â, ⮠᢮©á⢠ ¯®«ãç îé¨åáï ᮡá⢥­­ëå ä㭪権 ¨§¢¥áâ­ë ¨§ ¬ â¥¬ â¨ç¥áª®£®  ­ «¨§  2-£®ªãàá .à¨¬¥à 11. ¥è¨â¥ § ¤ ç㠘âãଠ{‹¨ã¢¨««ï◮ −X ′′ (x) = µX(x), x ∈ (0; π),X ′ (0) = 0, ′X (π) = 0.’ ª ª ª ⥮à¨ï ­¥ ¢á¥¬ ¨§¢¥áâ­ ,   § ¤ ç  ¯à®áâ ï, â® à¥-訬 ¥ñ ¯à¨ ¢á¥å ¢®§¬®¦­ëå §­ ç¥­¨ïå ¯ à ¬¥âà µ:2) µ = 0 : X(x) = C1 x + C2 .X ′ (0) = 0 ⇐⇒ C1 = 0, X ′ (π) = 0 ⇐⇒ C1 = 0 ⇒ X0 (x) = 1.1) µ < 0 ⇔ µ = −λ2 < 0 : X(x) = X(x) = C1 ch λx + C2 sh λx.X ′ (0) = 0 ⇐⇒ C2 = 0, X ′ (π) = −C1 λ sh λπ = 0 ⇐⇒2⇐⇒ C1 = 0 ⇒ X(x) ≡ 0.3) µ > 0 ⇐⇒ µ = λ > 0 : X(x) = C1 cos λx + C2 sin λx.X ′ (0) = 0 ⇐⇒ C2 = 0, X ′ (π) = −C1 λ sin λπ = 0 ⇒31⇒ λπ = kπ, k ∈ Z ⇒ Xk (x) = cos kx, k ∈ N.‘¢®©á⢠ ¯®«ã祭­®© á¨á⥬ë ᮡá⢥­­ëå ä㭪権 å®à®è® ¨§¢¥áâ­ë ¨§ ¬ â¥¬ â¨ç¥áª®£®  ­ «¨§  2-£® ªãàá :á¨á-{cos kx}, k = 0, 1, 2, .

. . ®à⮣®­ «ì­  ¨ ¯®«­  ¢ L2 [0; π].Žâ¢¥â. {cos kx}, k ∈ N ∪ {0}.◭ˆâ ª, ­ ©¤¥­ë ᮡá⢥­­ë¥ ä㭪樨 Xk (x) ¨ ᮡá⢥­­ë¥â¥¬ λ2k .T ′ (t)Tk (t): Tk (t) + α = µk = −λ2k ⇐⇒k⇐⇒ Tk′ (t) + (α + λ2k )Tk (t) = 0 ¨ ¯®«ãç ¥¬ ¬­®¦¥á⢮ à¥è¥­¨© ãà ¢­¥­¨ï § ¤ ç¨ uk (x, t) = Tk (t)Xk (x), 㤮¢«¥â¢®àïîé¨å§­ ç¥­¨ï’¥¯¥àì ­ å®¤¨¬ªà ¥¢ë¬ ãá«®¢¨ï¬. Žáâ «®áì 㤮¢«¥â¢®à¨âì ­ ç «ì­ë¬ ãá«®¢¨ï¬.II.’¥¯¥àì 㦥 ¨é¥¬ à¥è¥­¨¥¢á¥©§ ¤ ç¨ ¢ ¢¨¤¥ ä®à¬ «ì-­®£® à鸞u(x, t) =∞XTk (t)Xk (x).k®¤áâ ¢¨¬ ­ ç «ì­ë¥ ãá«®¢¨ï:∞Xu(x, t)|t=0 =Tk (0)Xk (x) = u0 (x).k‚®§­¨ª« ­¥®¡å®¤¨¬®áâìᮡá⢥­­ë¬äã­ªæ¨ï¬à §«®¦¨â짠¤ ç¨:u0 (x)u0 (x) =¢∞Pà樂ãàì¥ak Xk (x).¯®‚®âk§¤¥áì-â® ¨ ¯®âॡ®¢ «¨áì ᢮©á⢠権.â® ¢áïªãîan =u0 (x)¨ ᮡá⢥­­ëå äã­ª-’ ª ª ª ­ è  á¨á⥬  ®à⮣®­ «ì­  ¨ ¯®«­  ¢Rbau0 (x) ∈ L2 [a; b]u0 (x)Xn (x) dx.Rb 2Xn (x) dxa’®£¤ ∞PTk (0)Xk (x) = u0 (x) =k®«ã稫¨ ¤«ï∞Pkak Xk (x) ⇐⇒ Tk (0) = ak .Tk (t) § ¤ ç¨ Š®è¨:½ ′Tk (t) + αTk (t) = −λ2k Tk (t),Tk (0) = ak ,ª®â®àë¥, ª ª ¨§¢¥áâ­®, ¨¬¥îâ ¥¤¨­á⢥­­®¥ à¥è¥­¨¥.32L2 [a; b],¬®¦­® à §«®¦¨âì ¢ àï¤ ”ãàì¥, £¤¥”®à¬ «ì­®¥ à¥è¥­¨¥u(x, t) =∞PTk (t)Xk (x)­ ©¤¥­®, â.

ª.k­ ©¤¥­ë ¢á¥Xk (x), Tk (t).‡   ¬ ¥ ç   ­ ¨ ¥1. Ž¡à â¨â¥ ¢­¨¬ ­¨¥ ­  â®, çâ® § -¤ ç  ˜âãଠ{‹¨ã¢¨««ï ᮮ⢥âáâ¢ã¥â ®¯¥à â®àã L1 ¨ ­¥ § ¢¨á¨â ®â ⮣®, çâ® á⮨â á«¥¢  ¢ ­ è¨å ãà ¢­¥­¨ïå § ¤ ç 8*, 9*.4.2. Š ª ¡ëâì, ¥á«¨ ãà ¢­¥­¨¥ ­¥®¤­®à®¤­®¥¨ ªà ¥¢ë¥ ãá«®¢¨ï ­¥®¤­®à®¤­ë¥? áᬮâਬ ⥯¥àì § ¤ ç¨ 8 ¨ 9.’¥¯¥àì à¥è¥­¨¥ § ¤ ç á®á⮨⠨§I. å®¤¨¬ äã­ªæ¨îw0 (x, t),âàñåíâ ¯®¢.ª®â®à ï 㤮¢«¥â¢®àï¥â § ¤ ­-­ë¬ ªà ¥¢ë¬ ãá«®¢¨ï¬.Ž¡é¨å ¯à ¢¨« ¤«ï ­ å®¦¤¥­¨ïw0 (x, t)­¥ áãé¥áâ¢ã¥â. ®-í⮬ã à áᬮâਬ ­¥áª®«ìª® ¯à¨¬¥à®¢.

 ¯à¨¬¥à, )u|x=a = ϕ1 (t), ux |x=b = ϕ2 (t). ’®£¤  ¢ ª ç¥á⢥ w0 (x, t)¬®¦­® ¢§ïâì äã­ªæ¨î w0 (x, t) = ϕ1 (t)+(x−a)ϕ2 (t),   ¬®¦­®¨ «î¡ãî ¤àã£ãî | «¨èì ¡ë ®­  㤮¢«¥â¢®àï«  § ¤ ­­ë¬ªà ¥¢ë¬ ãá«®¢¨ï¬.¡)(x − b)2ux |x=a = ϕ1 (t), ux |x=b = ϕ2 (t) ⇒ w0 (x, t) = 2(a − b) ϕ1 (t) +(x − a)2¢)+ 2(b − a) ϕ2 (t).u|x=a = ϕ1 (t), u|x=b = ϕ2 (t).’®£¤  ¢ ஫¨w0 (x, t)¬®¦­®¢§ïâì äã­ªæ¨îw0 (x, t) =¨ â. ¤.(x − a)(x − b)ϕ1 (t) +ϕ2 (t)a−bb−a’¥¯¥àì ­¥®¡å®¤¨¬® ᤥ« âì ᤢ¨£, ç⮡ë ᢥá⨠ª § ¤ ç¥ ᮤ­®à®¤­ë¬¨ ªà ¥¢ë¬¨ ãá«®¢¨ï¬¨:v(x, t) = u(x, t) − w0 (x, t) ⇐⇒ u(x, t) = v(x, t) + w0 (x, t) ⇒vtt + αvt + βv = L1 v + g(x, t), t > 0, a < x < b,v|t=0 = u0 (x) − w0 (x, t)|t=0 , vt |t=0 = u1 (x) − w0t (x, t)|t=0 ,⇒a 6 x 6 b;(c1 v + d1 vx )|x=a = 0, (c2 v + d2 vx )|x=b = 0, t > 0.33‚¨¤­®, çâ® ¨§¬¥­¨«áï, ¢®®¡é¥ £®¢®àï, ᢮¡®¤­ë© ç«¥­g(x, t) ≡ −w0tt − αw0t − βw0 + L1 w0 + f (x, t)¨ ­ ç «ì­ë¥ ãá«®-¢¨ï.

¥à¥®¡®§­ ç¨¬ ¨å ¤«ï 㤮¡á⢠ ¨ ¯®«ã稬 § ¤ çãII. vtt + αvt + βv = L1 v(x, t) + g(x, t), t > 0, a < x < b,v|t=0 = v0 (x), vt |t=0 = v1 (x), a 6 x 6 b;(c1 v + d1 vx )|x=a = 0, (c2 v + d2 vx )|x=b = 0, t > 0.’¥¯¥àì ­ ©¤ñ¬ ᮡá⢥­­ë¥ ä㭪樨 § ¤ ç¨.ᮮ⢥âáâ¢ãî饬 ®¤®¤­®à®¤­ëå ªà ¥¢ëå ãá«®¢¨ïå. ãáâìv = T (t)X(x). ®¤áâ ¢«ï¥¬ ¢ ᮮ⢥âáâ¢ãî饥 ®¤­®à®¤­®¥ãà ¢­¥­¨¥ ¨ ®¤­®à®¤­ë¥ ªà ¥¢ë¥ ãá«®¢¨ï:„«ï í⮣® à §¤¥«¨¬ ¯¥à¥¬¥­­ë¥ ¢­®à®¤­®¬ ãà ¢­¥­¨¨ ¨T ′′ (t)X(x) + αT ′ (t)X(x) + βT (t)X(x) = T (t)L∗1 X(x) ⇐⇒T ′′ (t) + αT ′ (t)L∗ X(x)⇐⇒+β = 1= const = µ,T (t)X(x)c1 T (t)X(a) + d1 T (t)X ′ (a) = 0 ⇐⇒ c1 X(a) + d1 X ′ (a) = 0,c2 T (t)X(b) + d2 T (t)X ′ (b) = 0 ⇐⇒ c2 X(b) + d2 X ′ (b) = 0.…᫨−L∗1 ≡ L| ®¯¥à â®à ˜âãଠ{‹¨ã¢¨««ï, â® ¯®«ã-ç ¥¬ § ¤ ç㠘âãଠ{‹¨ã¢¨««ï: LX(x) = λ2 X(x),c X(a) + d1 X ′ (a) = 0, 1c2 X(b) + d2 X ′ (b) = 0.(’¥¯¥àì, ¢ ®â«¨ç¨¥ ®â § ¤ ç 8* ¨ 9*, ãà ¢­¥­¨¥ ¤«ïTà¥-è âì ­¥ ¡ã¤¥¬, â.

ª. ¤¥«¨«¨ ¯¥à¥¬¥­­ë¥ ¢ ýç㦮¬þ ®¤­®à®¤­®¬ ãà ¢­¥­¨¨.)Ž¡é¥£®  ­ «¨â¨ç¥áª®£® à¥è¥­¨ï â ª®© § ¤ ç¨ ­¥ áãé¥áâ¢ã¥â. ®í⮬㠢 ­ è¨å § ¤ ­¨ïå ç é¥ ¢á¥£® ¢áâà¥ç ¥âáï ýà¥è ¡¥«ì­ë©þ ¢ à¨ ­â, £¤¥2L ≡ − d 2,dxâ. ¥. § ¤ ç  ¨¬¥¥â ¢¨¤ −X ′′ (x) = λ2 X(x),c X(a) + d1 X ′ (a) = 0, 1c2 X(b) + d2 X ′ (b) = 0.¥è ¥¬ § ¤ çã ¤«ï34λ=0¨λ2 > 0,­ å®¤¨¬Xn (x).III.’¥¯¥àì ¨é¥¬ à¥è¥­¨¥ § ¤ ç¨ ¢ ¢¨¤¥ à鸞 ¯® ¢á¥¬ ᮡáâ-¢¥­­ë¬ äã­ªæ¨ï¬ á ª®íää¨æ¨¥­â ¬¨, § ¢¨áï騬¨ ®âv(x, t) =∞Xt:Tk (t)Xk (x).k®¤áâ ¢«ï¥¬ ¢ ãà ¢­¥­¨¥∞XTk′′ (t)Xk (x) + α∞XTk′ (t)Xk (x) + β=−à¨¤ñâáï à §«®¦¨âìäã­ªæ¨ï¬:g(x, t) =∞PTk (t)Xk (x) =kkk∞Xg(x, t)∞XTk (t)λ2k Xk (x) + g(x, t).k¢ àï¤ ”ãàì¥ ¯® ᮡá⢥­­ë¬ak (t)Xk (x), an (t) =k®¤áâ ¢¨¬ àï¤ ¢¬¥á⮯ਠ«¨­¥©­® ­¥§ ¢¨á¨¬ëåg(x, t)Xk :Rbag(x, t)Xn (x) dx.Rb 2Xn (x)dxa¨ ¯à¨à ¢­ï¥¬ ª®íää¨æ¨¥­âëTk′′ (t) + αTk′ (t) + (β + λ2k )Tk (t) = ak (t).‚ë室¨¬ ­  ­ ç «ì­ë¥ ãá«®¢¨ï, ª®â®àë¥ â®¦¥ ¯à¨å®¤¨âáïà §« £ âì ¢ àï¤ ”ãàì¥ ¯® ᮡá⢥­­ë¬ äã­ªæ¨ï¬ § ¤ ç¨:∞Xk∞XTk (0)Xk (x) = v0 (x) =Tk′ (0)Xk (x) = v1 (x) =∞Xk∞Xkk¥è ¥¬ § ¤ ç¨ Š®è¨:bk Xk (x) ⇐⇒ Tk (0) = bk ,ck Xk (x) ⇐⇒ Tk′ (0) = ck . ′′ Tk (t) + αTk′ (t) + (β + λ2k )Tk (t) = ak (t),T (0) = bk , k′Tk (0) = ck¨ § ¯¨á뢠¥¬ ®â¢¥â.à¨¬¥à 12.t(3x2 + 2)+ x cos t, t > 0, 0 < x < π2 , utt = uxx + 3u −π2u|t=0 = cos 2x, ut |t=0 = xπ , 0 6 x 6 π2 ,ux |x=0 = 0, ux |x= π2 = t, t > 0.35◮I.æ¨î‚ ª ç¥á⢥ ä㭪樨2w0 (x, t) = xπ t ,w0 (x, t) ¢®§ì¬ñ¬, ­ ¯à¨¬¥à, äã­ª-㤮¢«¥â¢®àïîéãî ªà ¥¢ë¬ ãá«®¢¨ï¬.

’¥-¯¥àì ¤¥« ¥¬ ᤢ¨£v =u−x2 tx2 t⇐⇒ u = v +⇒ππv − 3v = vxx + x cos t, t > 0, 0 < x < π2 , tthi⇒ v|t=0 = cos 2x, vt |t=0 = 0, x ∈ 0; π2 ,vx |x=0 = 0, vx |x= π2 = 0, t > 0.¨ ¯¥à¥å®¤¨¬ ª® ¢â®à®¬ã ¯ã­ªâã.II.„¥«¨¬ãà ¢­¥­¨¨ ¨¯¥à¥¬¥­­ë¥®¤­®à®¤­ëåvtt − 3v = vxx ,¢á®®â¢¥âáâ¢ãî饬®¤­®à®¤­®¬ªà ¥¢ëå ãá«®¢¨ïå:vx |x=π = 0 ⇒T ′′X ′′⇒ XT ′′ − 3XT = X ′′ T ⇐⇒−3== −λ2 ,TX³π ´³π ´X ′ (0)T (t) = X ′T (t) = 0 ⇐⇒ X ′ (0) = X ′=0⇒2 2 ′′2 −X (x) = λ X(x),′= 0,⇒ X (0)³ ´π′X2 = 0.¥è ¥¬vx |x=0 = 0,¢®§­¨ªèã áá¨ç¥áªã¤ çã˜âãଠ{‹¨ã-¢¨««ï:λ = 0 : X(x) = Ax + B ⇒ X ′ (0) = 0 ⇒⇒ A = 0 ⇒ X′λ2 > 0 : X(x) = A cos λx + B sin λx ⇒³π ´2= 0 ⇒ X0 = 1,³π ´⇒ X ′ (0) = 0 ⇒ B = 0, X ′=0⇒2ππ⇒ −λA sin λ = 0 ⇐⇒ λ = kπ, k ∈ Z ⇒22Xk (x) = cos 2kx,‘¨á⥬  ­ ©¤¥­­ëå ᮡá⢥­­ëå ä㭪権∈ N ∪ {0}36®à⮣®­ «ì­  ¨ ¯®«­  ¢hL2 0; π2i.k ∈ N.{cos 2kx}, k ∈III.’¥¯¥àì ¨é¥¬ à¥è¥­¨¥ § ¤ ç¨ ¢ ¢¨¤¥ à鸞 ¯® ¢á¥¬ ᮡáâ-¢¥­­ë¬ äã­ªæ¨ï¬ á ª®íää¨æ¨¥­â ¬¨, § ¢¨áï騬¨ ®â t:v(x, t) =∞XTk (t) cos 2kx.k=0®¬­¨¬, ç⮠᢮¡®¤­ë© ç«¥­ ¨ ­ ç «ì­ë¥ ãá«®¢¨ï (¢ ¤ ­-u0 (x) = X1 (x))­®¬ á«ãç ¥­ ¤® à §«®¦¨âì ¢ àï¤ë ”ãàì¥ ¯®á®¡á⢥­­ë¬ äã­ªæ¨ï¬ § ¤ ç¨:x=∞Xk=0ak cos 2kx ⇒π2R π2x cos 2mx dxπ((−1)m − 1)⇒ a0 = R π,= , am = R0 π=4πm2222 2mx dxdxcos00∞∞XXTk (t) cos 2kx =Tk′′ (t) cos 2kx − 3R0x dxk=0k=0=−∞X24Tk (t)k cos 2kx + cos tk=0k=0ak cos 2kx ⇐⇒⇐⇒ Tk′′ (t) + (4k 2 − 3)Tk (t) = ak cos t,v|t=0 = cos 2x ⇒ T1 (0) = 1,vt |t=0 = 0 ⇒∞XTk′ (0)Tk (0) = 0,= 0,k = 0, 2, 3, .

. .k = 0, 1, 2, . . .®«ã稫¨áì § ¤ ç¨ Š®è¨:—â® ′′ Tk (t) + (4k 2 − 3)Tk (t) = ak cos t,T (0) = 1, Tk (0) = 0, k = 0, 2, 3, . . . 1′Tk (0) = 0, k = 0, 1, 2, . . ..®á®¡¥­­®¥ ¦¤ñâ ­ á ¯à¨ à¥è¥­¨¨ § ¤ ç Š®è¨?1.  ¤® ®¡à â¨âì ¢­¨¬ ­¨¥ ­  ª®íää¨æ¨¥­â ¯à¨®ª § âìáï, çâ® ¯à¨ ­¥ª®â®àëå §­ ç¥­¨ïåkTk .Œ®¦¥â®­ ¬®¦¥â ¡ëâ쯮«®¦¨â¥«ì­ë¬, ¯à¨ ¤àã£¨å ®âà¨æ â¥«ì­ë¬ ¨«¨ ­ã«¥¢ë¬, ¨à §­ë¬¨ ä®à¬ã« ¬¨.१®­ ­á.3.

 ç «ì­ë¥ ãá«®¢¨ï ¬®£ãâ ¡ëâì à §­ë¬¨ ¤«ï à §­ëå Tk .à¥è¥­¨ï ¯à¨ í⮬ ¡ã¤ãâ ¢ëà ¦ âìáï2. à¨ ­¥ª®â®àëå §­ ç¥­¨ïåk¬®¦¥â ¡ëâì37‚ ­ è¥¬ á«ãç ¥ ′′ T0 (t) − 3T0 (t) = a0 cos t,k=0:⇐⇒T (0) = 0, 0′T0 (0) = 0,√√a T0 (t) = A0 ch 3t + B0 sh 3t − 40 cos t,⇐⇒ T0 (0) = 0,⇐⇒ ′T0 (0) = 0,√a0(ch 3t − cos t).⇐⇒ T0 (t) =4 ′′ T1 (t) + T1 (t) = a1 cos t,T (0) = 1,⇐⇒k=1: 1′T1 (0) = 0,a T1 (t) = A1 cos t + B1 sin t + 21 t sin t,⇐⇒ T1 (0) = 1,⇐⇒ ′T1 (0) = 0,a1⇐⇒ T1 (t) = cos t +t sin t2| §¤¥áì १®­ ­á (ç áâ­®¥ à¥è¥­¨¥ ¨é¥âáï ¢ ¢¨¤¥ t(a cos t ++ b sin t)). ′′ Tk (t) + (4k 2 − 3)Tk (t) = ak cos t,⇐⇒k>2:T (t) = 0, k′Tk (t) = 0,√√Tk (t) = Ak cos t 4k 2 − 3 + Bk sin t 4k 2 − 3++ a2k cos t ,(4k − 3) − 1 ⇐⇒⇐⇒T(t)=0,kTk′ (t) = 0,pak(cost−cos4k 2 − 3t).⇐⇒ Tk (t) =(4k 2 − 3) − 1‡ ¯¨á뢠¥¬ Žâ¢¥â.µ¶√x2 tπt sin tu(x, t) =+(ch 3t − cos t) + 3 cos t −cos 2x+π16π∞³´pX(−1)k − 12 − 3 cos 2kx.4kcost−cost+π ((4k 2 − 3) − 1) k 2k=238à¨¬¥ç ­¨¥.¥è¥­¨¥¨áª «®á좢¨¤¥ä®à-¬ «ì­®£® à鸞, ¢®¯à®á ® ¯®ç«¥­­®¬ ¤¨ää¥à¥­æ¨à®¢ ­¨¨ ª®â®à®£® ®áâ ¢ «áï ®âªàëâë¬.’¥¯¥àì, ª®£¤  ¯®«ã祭 ª®­ªà¥â-­ë© àï¤, ¬®¦­® ¢ëïá­¨âì, ï¥âáï «¨ ®­ ¤¢ ¦¤ë ¯®ç«¥­­®¤¨ää¥à¥­æ¨à㥬ë¬,   §­ ç¨â, ª« áá¨ç¥áª¨¬ à¥è¥­¨¥¬, ¨«¨ ­¥ï¢«ï¥âáï,   §­ ç¨â, ¡ã¤¥â ®¡®¡éñ­­ë¬ à¥è¥­¨¥¬.Š ª ¢¨¤­® ¢ ­ è¥¬ á«ãç ¥, à¥è¥­¨¥ ï¥âáï ª« áá¨ç¥áª¨¬, â.

ª. àï¤ ¬®¦­® ¯®ç«¥­­® ¤¨ää¥à¥­æ¨à®¢ âì ¤¢  à § .◭à¨¬¥à 13. ut = uxx , t > 0, 0 < x < l,u|t=0 = u0 (x), 0 6 x 6 l;(ux − hu)|x=0 = 0, (ux + hu)|x=l = 0, t > 0, h > 0.à¨¬¥à®â«¨ç ¥âáï®â ¯à¥¤ë¤ã饣® ⥬, çâ® ¢ ªà ¥¢ëåãá«®¢¨ïå ®¡  ª®íää¨æ¨¥­â  ®â«¨ç­ë ®â 0,   ¯®â®¬ã ᮡá⢥­­ë¥ §­ ç¥­¨ï ® ­¥ ®¯à¥¤¥«ïîâáï,   § ¤ îâáï âà ­á業¤¥­â­ë¬ ãà ¢­¥­¨¥¬,   ®à⮣®­ «ì­®áâì ᮡá⢥­­ëå ä㭪権 ­¥â ª ®ç¥¢¨¤­  | ¯à¨å®¤¨âáï ®¡à â¨âìáï ª ⥮ਨ.◮II.’ ª ª ª ¢ § ¤ ç¥ ãà ¢­¥­¨¥ ¨ ªà ¥¢ë¥ ãá«®¢¨ï |­®à®¤­ë¥,®¤-â® ¬®¦­® ¯à¨áâ㯠âì áࠧ㠪® ¢â®à®¬ã ¯ã­ªâã |¨é¥¬ à¥è¥­¨¥ ãà ¢­¥­¨ï ¢ ¢¨¤¥T (t)X(x) ⇒T ′ (t)X ′′ (x)== −λ2 ,T (t)X(x)T (t)X ′ (0) − hT (t)X(0) = 0 ⇐⇒ X ′ (0) − hX(0) = 0,⇒T (t)X ′ (l) + hT (t)X(l) = 0 ⇐⇒ X ′ (l) + hX(l) = 0, ⇒½ ′′T ′ (t)X (x) = −λ2 X(x),= −λ2 .¨⇒′′X (0) − hX(0) = 0, X (l) + hX(l) = 0T (t)¥è ¥¬ ª« áá¨ç¥áªãî § ¤ ç㠘âãଠ{‹¨ã¢¨««ï:λ = 0,X = Ax + B ⇒½A − hB = 0,⇐⇒ X(x) ≡ 0,A + h(Al + B) = 0λ2 > 0, X(x) = A cos λx + B sin λx ⇒½Bλ − hA = 0,⇐⇒⇒−Aλ sin λl + Bλ cos λl + Ah cos λl + Bh sin λl = 039 Bλ − hA = 0,⇐⇒ (Bλ + Ah) cos λl + (Bh − Aλ) sin λl = 02 ⇐⇒2−h .⇐⇒ ctg λl = λ 2hλλn l = µn ⇐⇒ λn = µln , ⮣¤  µn³´1 µn − hl , n ∈| ¯®«®¦¨â¥«ì­ë¥ ª®à­¨ ãà ¢­¥­¨ï ctg µn =µn2 hl∈ N (â.

ª. «¥¢ ï ¨ ¯à ¢ ï ç á⨠ãà ¢­¥­¨ï ­¥çñâ­ë¥ ä㭪樨),2µ xµ x− µn t¨ X (x) = µ cos n + hl sin n ⇒ T = C e ( l ) .Ž¡®§­ ç¨¬, ¤«ï 㤮¡á⢠,nnllnn‚ § ¤ ç¥ ᮡá⢥­­ë¥ ç¨á«  㤮¢«¥â¢®àïîâ ¤®¢®«ì­® á«®¦-­®¬ã ãà ¢­¥­¨î. ®í⮬㠢®á¯®«ì§ã¥¬áï á«¥¤ãî騬¨ ¨§ ⥮ਨ ᢮©á⢠¬¨ ᮡá⢥­­ëå §­ ç¥­¨© (¨å áçñâ­®¥ ¬­®¦¥á⢮)¨ ᮡá⢥­­ëå ä㭪権 (á¨á⥬  ®à⮣®­ «ì­  ¨ ¯®«­ ) § ¤ ç¨ ˜âãଠ{‹¨ã¢¨««ï. ®í⮬ãu0 (x) =∞X³µn x ´µn xan µn cos+ hl sin⇐⇒ll1³´Rlµk xµk xu(x)µcos+hlsindx0k0ll⇐⇒ ak = R ³=´2lµk xµk xµcosdx+hlsink0ll³´Rlµµk x2kxu(x)µcos+hlsindx0kl 0ll.=((µk )2 + (hl)2 + 2hl)®¤áâ ¢¨¬ àï¤ ¢ ­ ç «ì­ë¥ ãá«®¢¨ï:∞X1³µn xµn x ´Cn µn cos+ hl sin=ll∞³Xµn xµn x ´an µn cos= u0 (x) =+ hl sin⇐⇒ Cn = an .ll1¥è ¥¬ § ¤ ç¨ Š®è¨:−(⇒ Tn (t) = an eµn 2tl) ⇒³Tn = Cn e−(µn 2tl,)Tn (0) = an ⇒´2 R l u (x) µ cos µk x + hl sin µk x dx0k0l¡¢ lŽâ¢¥â.×22(µ)+(hl)+2hlk1´³µn 2× e−( l ) t µn cos µnl x + hl sin µnl x .∞Pl40◭4.3. —â® ¤¥« âì, ¥á«¨ ®¯¥à â®à−L∗1­¥ ï¥âáﮯ¥à â®à®¬ ˜âãଠ{‹¨ã¢¨««ï?…᫨−L∗1­¥ ï¥âáï ®¯¥à â®à®¬ ˜âãଠ{‹¨ã¢¨««ï, â®ã¬­®¦¥­¨¥¬ ®¡¥¨å ç á⥩ ãà ¢­¥­¨ï ­  ­¥ª®â®àãî äã­ªæ¨îg(x), g(x) > 0¢á¥£¤  ¬®¦­® ¯à¨¢¥á⨠«¥¢ãî ç áâì ãà ¢­¥­¨ïª ¢¨¤ãLX(x) : g(x)L∗1 X(x) ≡ LX(x) = λ2 g(x)X(x),c X(a) + d1 X ′ (a) = 0, 1c2 X(b) + d2 X ′ (b) = 0.‡ ¬¥â¨¬, çâ® ¯à¨ í⮬ à¥è¥­¨¥ § ¤ ç¨ ®áâ ­¥âáï ¯à¥¦­¨¬ (¯®â®¬ã çâ® ¢áñ ¡ë«® ®¤­®à®¤­ë¬).