Учебник - Методы решения основных задач уравнений математической физики - Колесникова, страница 6

®! ˆ§¬¥­¨« áì ¯à ¢ ï ç áâì ãà ¢­¥­¨ï.—â® íâ® ¤ ñâ?«¨áì ᢮©á⢠ ᮡá⢥­­ëå ä㭪権.Žª §ë¢ ¥âáï, ¨§¬¥­¨‘®¡á⢥­­ëå §­ ç¥­¨©¯®-¯à¥¦­¥¬ã áçñâ­®¥ ¬­®¦¥á⢮, á¨á⥬  ᮡá⢥­­ëå ä㭪権 ®áâ ñâáï ¯®«­®©, ­® ⥯¥àì ®­¨ ­¥ ¯à®áâ® ®à⮣®­ «ì­ë,  ®à⮣®­ «ì­ë 㦥ýá ¢¥á®¬þ g(x), â. ¥.= 0, k 6= n.à¨¬¥à 14.Rbag(x)Xk (x)Xn (x) dx = utt − 7ut = uxx + 2ux − 2t − 7x + f (x, t), t > 0, 0 < x < π;u|= 0, ut |t=0 = x, 0 < x < π; t=0u|x=0 = 0, u|x=π = πt, t > 0.‚í⮬¯à¨¬¥à¥®¯¥à â®à¢ãà ¢­¥­¨¨­ á®¡á⢥­­ë¥ä㭪樨 ­¥ ï¥âáï ®¯¥à â®à®¬ ˜âãଠ{‹¨ã¢¨««ï!¥è ¥¬ § ¤ çã ¯® ¯ã­ªâ ¬.◮I.®¤¡¨à ¥¬ãá«®¢¨ï¬:+ xt ⇒II.w0 = xt.äã­ªæ¨î,㤮¢«¥â¢®àïîéã« ¥¬ ᤢ¨£:ªà ¥¢ë¬v = u − xt ⇐⇒ u = v + vtt − 7vt = vxx + 2vx + f (x, t), t > 0, 0 < x < π;⇒ v|t=0 = 0, vt |t=0 = 0, 0 < x < π;v|x=0 = 0, v|x=π = 0, t > 0. ©¤ñ¬ ᮡá⢥­­ë¥ ä㭪樨 § ¤ ç¨.

„¥«¨¬, ª ª ¢á¥£¤ ,¯¥à¥¬¥­­ë¥ ¢ ᮮ⢥âáâ¢ãî饬 ®¤­®à®¤­®¬ ãà ¢­¥­¨¨:T ′′ (t)X(x) − 7T ′ (t)X(x) = X ′′ (x)T + 2T (t)X ′ (x) ⇐⇒41X ′′ (x) + 2X ′ (x)T ′′ (t) − 7T ′ (t)== const,T (t)X(x)T (t)X(0) = T (t)X(π) = 0 ⇐⇒ X(0) = X(π) = 0.⇐⇒®«ã稫 áì § ¤ ç  ˜âãଠ{‹¨ã¢¨««ï á ­¥§­ ª®¬ë¬ ®¯¥à â®à®¬, ¯®í⮬ã à¥è ¥¬ § ¤ çã¤«ï ¢á¥åλ.½−X ′′ (x) − 2X ′ (x) = λX(x),X(0) = X(π) = 0λ = 0 : X0′′ (x) + 2X0′ (x) = 0 ⇐⇒½C1 + C2 = 0,−2x⇒ X0 (x) = 0,⇐⇒ X0 (x) = C1 + C2 e⇒C1 + C2 e−2π = 0λ = −µ2 < 0 : Xµ′′ (x) + 2Xµ′ (x) − µ2 Xµ (x) = 0 ⇐⇒√√³´22⇐⇒ Xµ (x) = e−x Aµ ex 1+µ + Bµ e−x 1+µ ⇒½Aµ +√Bµ = 0,√⇒ Xµ (x) = 0,⇒22Aµ eπ 1+µ + Bµ e−π 1+µ = 0λ = µ2 > 0 : Xµ′′ (x) + 2Xµ′ (x) + µ2 Xµ (x) = 0 ⇒ Xµ (x) = eνx ⇒p⇒ ν = −1 ± 1 − µ2 ⇒√√´³221) 1 − µ2 > 0 : Xµ (x) = e−x Aµ ex 1−µ + Bµ e−x 1−µ ⇐⇒2−x2) 1 − µ = 0 : X±1 (x) = e2⇐⇒ Xµ (x) = 0.(A + Bx) ⇐⇒ Xµ (x) = 0.3) 1 − µ < 0 : Xµ (x) =´³ p´´³³ p= e−x Aµ sin x µ2 − 1 + Bµ cos x µ2 − 1 ⇒´ Xµ (0) = 0 ⇐⇒ Bµ = 0,³ p⇒ Xµ (π) = 0 ⇐⇒ Aµ sin π µ2 − 1 = 0 ⇒p⇒ π µ2 − 1 = πk ⇐⇒ µ2k = 1 + k 2 , k ∈ N.ˆâ ª,Xk (x) = e−x sin kx.„«ï ­ á íâ® ­¥§­ ª®¬ ï á¨á⥬ .Š ª®¢ë ¥ñ ᢮©á⢠?‚®â §¤¥áì 㦥 ¯à¨¤ñâáï ᢥá⨠­ èã § ¤ çã ª ª« áá¨ç¥áª®©§ ¤ ç¥ ˜âãଠ{‹¨ã¢¨««ï, ã ª®â®à®© ᢮©á⢠ ¨§¢¥áâ­ë.42“¬­®¦¨¬ ®¡¥ ç á⨠ãà ¢­¥­¨ï ­ g(x) = e2x :−e2x X ′′ (x) − 2e2x X ′ (x) = λe2x X(x) ⇐⇒¡¢′⇐⇒ − e2x X ′ (x) = λe2x X(x).‘«¥¢  á⮨⠮¯¥à â®à ˜âãଠ{‹¨ã¢¨««ï, á¯à ¢  ¯®ï¢¨«áïg(x) = e2x ,¬­®¦¨â¥«ì¢¥­­ëåä㭪権Rπí⮩  íâ® ®§­ ç ¥â,§ ¤ ç¨e2x Xm (x)Xn (x) dx = 0,Râ.π¥.

2x 0 −xsin kx) (e−x sin mx) dx0 e (eL2 [0; π].¨ ¯®«­  ¢III.çâ®®à⮣®­ «ì­ á¨á⥬  ᮡáâýᢥᮬþe2x ,n 6=R πm, ¨«¨ ¢ ­ è¥¬ á«ãç ¥,= 0 sin kx sin mx dx = π2 δkm ,„ «ìè¥ § ¤ çã à¥è âì ¬®¦­® ¯® ®¡ëç­®© á奬¥, à §«®-¦¨¢ ᢮¡®¤­ë© ç«¥­ ¢ àï¤ ”ãàì¥:f (x, t) =∞X1⇒ an (t) =ak (t)e−x sin kx ⇒Rπ0e2x f (x, t)e−x sin nx dxRπ0e2x (e−x sin nx)2 dx2=πZπf (x, t)ex sin nx dx.

◭0f (x, t) ≡ e−x sin 3x. vtt − 7vt = vxx + 2vx − e−x sin 3x, t > 0, 0 < x < π;v|= 0, vt |t=0 = 0, 0 < x < π; t=0v|x=0 = 0, v|x=π = 0, t > 0.à¨¬¥à 15. ¥è¨âì ¯à¨¬¥à 14, ¥á«¨◮’ ª ª ªe−x sin 3x| ᮡá⢥­­ ï äã­ªæ¨ï, â® à¥è¥­¨¥ § -¤ ç¨ ¬®¦­® ¨áª âì ¢ ¢¨¤¥v = f (t)e−x sin 3x,Šà ¥¢ë¥ãá«®¢¨ïf (0) = 0,¢ë¯®«­¥­ë|f ′ (0) = 0.®áâ «®áì㤮¢«¥â¢®à¨âìãà ¢­¥­¨î ¨ ­ ç «ì­ë¬ ãá«®¢¨ï¬.®¤áâ ¢¨¬ ¢ ãà ¢­¥­¨¥:′′f (t)e−x sin 3x − 7f ′ (t)e−x sin 3x =¡¢= f (t) −6e−x cos 3x − 8e−x sin 3x +¡¢+2 −e−x sin 3x + 3e−x cos 3x f (t) − e−x sin 3x ⇐⇒⇐⇒ f ′′ (t) − 7f ′ (t) + 10f (t) = −1 ⇒1⇒⇒ f (t) = C1 e5t + C2 e2t −1043µ¶1 2t1 5t1e −e −⇒ v(t, x) =e−x sin 3x.61510´³1 e5t − 1 e−x sin 3x.◭v(x, t) = 61 e2t − 1510Žâ¢¥â.§ 5.Œ¥â®¤ ”ãàì¥ ­  ªà㣥.

”㭪樨 ¥áá¥«ï‚ í⮬ ¯ à £à ä¥ ᮡá⢥­­ë¥ ä㭪樨 § ¤ ç | í⮠ᮡá⢥­­ë¥ ä㭪樨 ®¯¥à â®à  ‹ ¯« á  ¢ ªà㣥 ¯à¨ ãá«®¢¨¨, çâ®u|r=r0 = 0.Ãνnk = JnŽ­¨ ¨¬¥îâ ¢¨¤!(n)µkr (An cos nϕ + Bn sin nϕ),r0n = 0, 1, 2, . . . ,£¤¥ ¢ á¢®î ®ç¥à¥¤ìAn cos nϕ + Bn sin nϕk = 1, 2, . . . ,| ᮡá⢥­­ë¥ äã­ª-樨 § ¤ ç¨ ˜âãଠ{‹¨ã¢¨««ï:Jnµ(n)µkr0 r«ï:(½−Φ′′ (ϕ) = λΦ(ϕ),Φ(ϕ) = Φ(ϕ + 2π), λn = n2 , n ∈ N ∪ {0},¶| ᮡá⢥­­ë¥ ä㭪樨 § ¤ ç¨ ˜âãଠ{‹¨ã¢¨«-(r))′2+ nr Rn (r) = λ2 rRn (r),−(rRnRn (r0 ) = 0,(n)|Rn (0)| < ∞,µk | ¯®«®¦¨â¥«ì­ë¥Jn (µk ) = 0, k ∈ N.λ2k =Ã(n)µkr0!2,ª®à­¨ ãà ¢­¥­¨ï ¥áá¥«ï ¯®à浪 n:’ ª ª ª á¯à ¢  ¢ ãà ¢­¥­¨¨ ˜âãଠ{‹¨ã¢¨««ï á⮨⠭¥λ2 Rn (r),λ2 rRn (r), ⮠ᮡá⢥­­ë¥ ä㭪樨, ᮮ⢥âáâ¢ãî騥 à §­ë¬ k (¯à¨ 䨪á¨à®¢ ­­®¬ n), ­¥ ¯à®áâ® ®à⮣®R r0k Rn dr = 0, k 6= m.­ «ì­ë,   ®à⮣®­ «ì­ë á ¢¥á®¬ r : 0 rRnn ¥è¥­¨¥ § ¤ ç ¬®¦­® ¨áª âì ¢ ¢¨¤¥u(t, r, ϕ) =∞ X∞Xn=0 k=144Tnk (t)vnk (r, ϕ).‘®¡á⢥­­ë¥ ä㭪樨 § ¤ ç 㤮¢«¥â¢®àïîâ ãà ¢­¥­¨î∆vnk = −µ(n)µkr0¶2vnk .‘ä®à¬ã«¨à㥬 § ¤ çã ® ª®«¥¡ ­¨¨ ªà㣫®© ¬¥¬¡à ­ë, § ªà¥¯«ñ­­®© ¯® ªà î.‡ ¤ ç  11*.p2 utt + but + cu = a ∆u + f (t, x, y), px2 + y 2 < r0 , t > 0,u|t=0 = u0 (x, y), ut |t=0 = u1 (x, y), x2 + y 2 6 r0 , u|√ 2 2= 0.x +y =r0’ ª ª ª § ¤ ç  ¯®áâ ¢«¥­  ¢ ªà㣥, ¯¥à¥©¤ñ¬ ª ¯®«ïà­ë¬ª®®à¤¨­ â ¬:x = r cos ϕ, y = r sin ϕ.’ ª ª ª â®çª (r, ϕ)¨(r, ϕ+2πk) | ®¤­  ¨ â  ¦¥ â®çª  ­ è¥© ¬¥¬¡à ­ë, â® à¥è¥­¨¥¤®«¦­® 㤮¢«¥â¢®àïâì ãá«®¢¨î u(t, r, ϕ) = u(t, r, ϕ + 2π).

Š ª¨§¢¥áâ­®, â ª ï § ¬¥­  ¯¥à¥¬¥­­ëå ¤ ¦¥ «®ª «ì­® ­¥ ¢áî¤ãï¥âáï ¢§ ¨¬­® ®¤­®§­ ç­®© | 类¡¨ ­ ¢ â®çª¥r = 0à -¢¥­ 0.Ž¯¥à â®à ‹ ¯« á  ¨¬¥¥â ¢¨¤1 ∂∆=r ∂rµ¶∂21 ∂1 ∂2∂1 ∂2≡++,r+ 2∂rr ∂ϕ2∂r2 r ∂r r2 ∂ϕ2r = 0.ãà ¢­¥­¨¥ ¨¬¥¥â ®á®¡¥­­®áâì ¢ â®çª¥®í⮬㠬®¦­®®¦¨¤ âì ­¥ª®â®àëå ®á®¡¥­­®á⥩ à¥è¥­¨ï § ¤ ç¨ ¢ í⮩ â®çª¥.‚ ¯à¨¬¥à¥ 9 ¬ë 㦥 á⮫ª­ã«¨áì á ⥬, çâ® ¯¥à¥å®¤ ®â¤¥ª à⮢ëå ª®®à¤¨­ â ª ªà¨¢®«¨­¥©­ë¬ (áä¥à¨ç¥áª¨¬) ¨§¬¥­¨« § ¤ çã | ¯¥à¥¢¥« § ¤ ç㠊®è¨ ¢R3¢ ®¤­®¬¥à­ãî § ¤ çã® ª®«¥¡ ­¨¨ ¯®«ã¡¥áª®­¥ç­®© áâàã­ë.ˆâ ª, ᤥ« ¥¬ § ¬¥­ã ¯¥à¥¬¥­­ëå.‡ ¤ ç  11**.utt + but + cu = a2 ∆u + f (t, r, ϕ),2∂ + 1 ∂2 ,r < r0 , t > 0, ∆ = ∂ 2 + 1r ∂r∂rr2 ∂ϕ2u(t, r, ϕ) = u(t, r, ϕ + 2π),u|= u0 (r, ϕ), ut |t=0 = u1 (r, ϕ), r 6 r0 , t=0u|r=r0 = 0.45◮I.¥à¢ë© ¯ã­ªâ á奬ë à¥è¥­¨ï ¯® ¬¥â®¤ã ”ãàì¥ ¢ë-¯®«­¥­ | ªà ¥¢ë¥ ãá«®¢¨ï ®¤­®à®¤­ë¥.

à¨áâ㯠¥¬ áࠧ㠪®¢â®à®¬ã.II.„¥«¨¬ ¯¥à¥¬¥­­ë¥ ¢ ᮮ⢥âáâ¢ãî饬 ®¤­®à®¤­®¬ ãà ¢-­¥­¨¨ ¨ ®¤­®à®¤­ëå ªà ¥¢ëå ãá«®¢¨ïå.Žâ¤¥«¨¬ ¯à®áâà ­á⢥­­ë¥ ¯¥à¥¬¥­­ë¥ ®â ¢à¥¬¥­¨:u(r, ϕ, t) = T (t)v(r, ϕ) ⇒⇒ T ′′ (t)v(r, ϕ) + bT ′ (t)v(r, ϕ) + cT (t)v(r, ϕ) = a2 T (t)∆v(r, ϕ) ⇔T ′′ (t)T ′ (t)c∆v(r, ϕ)⇐⇒ 2+b 2+ 2 == const = µ,a T (t)a T (t) av(r, ϕ)u|r=r0 = 0 ⇐⇒ T (t)v(r0 , ϕ) = 0 ⇐⇒ v(r0 , ϕ) = 0,T (t)v(r, ϕ + 2π) = T (t)v(r, ϕ) ⇐⇒ v(r, ϕ + 2π) = v(r, ϕ).„«ï ãà ¢­¥­¨ï∆v(r, ϕ) = µv(r, ϕ)¯®«ã稫 áì ªà ¥¢ ï § -¤ ç  ˜âãଠ{‹¨ã¢¨««ï.‡¤¥áì ¬ë ¤®«¦­ë ¯®¢¥à¨âì, çâ® § ¤ ç ½−∆v = µv, x ∈ D,v|∂D = 0¨¬¥¥â ¯®«®¦¨â¥«ì­ë¥ ᮡá⢥­­ë¥ §­ ç¥­¨ïµ = λ2 > 0.à®¤®«¦¨¬ ¤¥«¥­¨¥ ¯¥à¥¬¥­­ëåv(r, ϕ) = R(r)Φ(ϕ),â.

ª. ãà ¢­¥­¨¥ ¨ ªà ¥¢ë¥ ãá«®¢¨ï ¯®-¯à¥¦­¥¬ã ®¤­®à®¤­ë.1R(r)µ¶1 1 ′′1 ′Φ (ϕ) = −λ2 ⇐⇒R (r) + R (r) +rΦ(ϕ) r2Φ′′ (ϕ)1(r2 R′′ (r) + rR′ (r)) + λ2 r2 = −= ν,⇐⇒R(r)Φ(ϕ)R(r0 )Φ(ϕ) = 0 ⇐⇒ R(r0 ) = 0,′′R(r)Φ(ϕ) = R(r)Φ(ϕ + 2π) ⇐⇒ Φ(ϕ) = Φ(ϕ + 2π).®«ã稫¨áì ¤¢¥ § ¤ ç¨ ˜âãଠ{‹¨ã¢¨««ï.½−Φ′′ (ϕ) = νΦ(ϕ),Φ(ϕ) = Φ(ϕ + 2π),­¥§­ ª®¬ë¬­ ¬ªà ¥¢ë¬ ãá«®¢¨¥¬ ¯¥à¨®¤¨ç­®á⨠à¥è¥­¨ï, ¨ ¢â®à ï, ¤«ïR(r),¥à¢ ï46§ ¤ ç ,á¡®«¥¥ á«®¦­ ï ¨ ­¥§­ ª®¬ ï:½r2 R′′ (r) + rR′ (r) + (λ2 r2 − ν)R(r) = 0, r < r0 ,R(r0 ) = 0.¥à¢ãî á¨á⥬㠫¥£ª® à¥è¨âì:ν = 0 : Φ(ϕ) = C1 ϕ + C2 ⇒ Φ0 (ϕ) = 1,ν > 0 ⇐⇒ ν = µ2 > 0 : Φ(ϕ) = Φ(ϕ + 2π) ⇐⇒A cos µϕ + B sin µϕ = A cos µ(ϕ + 2π) + B sin µ(ϕ + 2π) ⇐⇒⇐⇒ (−A sin µ(ϕ + π) + B cos µ(ϕ + π)) sin µπ = 0 ⇒ µ = n,n = ±1, ±2, .

. .2ν < 0 ⇐⇒ ν = −µ < 0 : Φ(ϕ) = Φ(ϕ + 2π) ⇐⇒A ch µϕ + B sh µϕ = A ch µ(ϕ + 2π) + B sh µ(ϕ + 2π) ⇐⇒ ∅.Žâá á«¥¤ã¥â, ç⮠ᮡá⢥­­ë¥ §­ ç¥­¨ïν = n2­¥®âà¨-n = 0 ᮮ⢥âáâ¢ã¥â ®¤­  ᮡá⢥­­ ï äã­ªæ¨ï Φ0 (ϕ) = 1,   «î¡®¬ã n > 1 ᮮ⢥âáâ¢ãîâ ¤¢¥ ᮡá⢥­­ë¥ä㭪樨: cos nϕ, sin nϕ, µ = n, n ∈ N. ˆ­®£¤  ¯¨èãâ â ª:½Φn (ϕ) = cos nϕ, n = 0, 1, 2, . . .;Φm (ϕ) = sin |m|ϕ, m = −1, −2, . . .æ â¥«ì­ë, ¯à¨çñ¬ˆâ ª,−Φ′′ (ϕ) = n2 Φ(ϕ),⇐⇒Φ(ϕ) = Φ(ϕ + 2π)⇔ Φ0 (ϕ) = 1, Φn (ϕ) = An cos nϕ + Bn sin nϕ, µ = n, n ∈ N ⇐⇒½⇐⇒ {1, cos nϕ, sin nϕ},’¥¯¥à짠©¬ñ¬áï¢â®à®©§ ¤ ç¥©.¥à¥¯¨è¥¬n ∈ N.¥ñ¯®-¤à㣮¬ã:½−r2 Rn′′ (r) − rRn′ (r) + n2 Rn (r) = λ2 r2 Rn (r), r < r0 ,Rn (r0 ) = 0.®«ã稫¨áì ªà ¥¢ë¥ § ¤ ç¨ ¤«ï­ëåλ.n ∈ N ∪ {0}¨ ­¥¨§¢¥áâ-‘â®ï騩 á«¥¢  ®¯¥à â®à ­¥ ï¥âáï ®¯¥à â®à®¬ ˜âãଠ{‹¨ã¢¨««ï. ‘¤¥« ¥¬ ¥£® â ª®¢ë¬, à §¤¥«¨¢ ®¡¥ ç á⨠­ −rRn′′ (r) − Rn′ (r) + n2r:Rn (r)= λ2 rRn (r) ⇐⇒r47⇐⇒ − (rRn′ (r))′ +â®­®¢ ïn2Rn (r) = λ2 rRn (r).r¤«ï ­ á § ¤ ç  ˜âãଠ{‹¨ã¢¨««ï (¯à®¨§¢®¤-­ë¥ ýᢥà­ã«¨áìþ ¢ ®¤­®ç«¥­) ­  ᮡá⢥­­ë¥ ä㭪樨, ­®â®«ìª® á ®¤­¨¬ ®¤­®à®¤­ë¬ ªà ¥¢ë¬ ãá«®¢¨¥¬(2−(rRn′ (r))′ + nr Rn (r) = λ2 rRn (r),Rn (r0 ) = 0.Rn (r0 ) = 0:r < r0 ,Ž¤­ ª® § ¬¥â¨¬, çâ® ®¯¥à â®à ˜âãଠ{‹¨ã¢¨««ï«ï¥âá磌 áá¨ç¥áª¨¬ | ãá«®¢¨¥p(r) = 0 ¯à¨ r = 0.Žâáâ㯫¥­¨¥*.