Учебник - Методы решения основных задач уравнений математической физики - Колесникова (1188241), страница 6
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®! §¬¥¨« áì ¯à ¢ ï ç áâì ãà ¢¥¨ï.â® íâ® ¤ ñâ?«¨áì ᢮©á⢠ᮡá⢥ëå äãªæ¨©.ª §ë¢ ¥âáï, ¨§¬¥¨®¡á⢥ëå § 票©¯®-¯à¥¦¥¬ã áçñ⮥ ¬®¦¥á⢮, á¨á⥬ ᮡá⢥ëå äãªæ¨© ®áâ ñâáï ¯®«®©, ® ⥯¥àì ®¨ ¥ ¯à®áâ® ®à⮣® «ìë, ®à⮣® «ìë 㦥ýá ¢¥á®¬þ g(x), â. ¥.= 0, k 6= n.ਬ¥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⢥ë¥äãªæ¨¨ ¥ 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¬ ᮡáâ¢¥ë¥ äãªæ¨¨ § ¤ 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.â ª,Xk (x) = e−x sin kx.«ï á íâ® ¥§ ª®¬ ï á¨á⥬ . 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¢¥ëåäãªæ¨©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, ¨«¨ ¢ 襬 á«ãç ¥,= 0 sin kx sin mx dx = π2 δkm , «ìè¥ § ¤ çã à¥è âì ¬®¦® ¯® ®¡ë箩 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.ਬ¥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,à ¥¢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 − 1510⢥â.§ 5.¥â®¤ ãàì¥ ªà㣥.
ãªæ¨¨ ¥áᥫï í⮬ ¯ à £à ä¥ á®¡áâ¢¥ë¥ äãªæ¨¨ § ¤ ç | í⮠ᮡáâ¢¥ë¥ äãªæ¨¨ ®¯¥à â®à ¯« á ¢ ªà㣥 ¯à¨ ãá«®¢¨¨, çâ®u|r=r0 = 0.Ãνnk = Jn¨ ¨¬¥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, . . . ,| ᮡáâ¢¥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},¶| ᮡáâ¢¥ë¥ äãªæ¨¨ § ¤ 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), ⮠ᮡáâ¢¥ë¥ äãªæ¨¨, 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).®¡áâ¢¥ë¥ äãªæ¨¨ § ¤ 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 ¯®«ïà묪®®à¤¨ â ¬:x = r cos ϕ, y = r sin ϕ. ª ª ª â®çª (r, ϕ)¨(r, ϕ+2πk) | ®¤ ¨ â ¦¥ â®çª 襩 ¬¥¬¡à ë, â® à¥è¥¨¥¤®«¦® 㤮¢«¥â¢®àïâì ãá«®¢¨î u(t, r, ϕ) = u(t, r, ϕ + 2π).
ª¨§¢¥áâ®, â ª ï § ¬¥ ¯¥à¥¬¥ëå ¤ ¦¥ «®ª «ì® ¥ ¢áî¤ãï¥âáï ¢§ ¨¬® ®¤®§ 箩 | 类¡¨ ¢ â®çª¥r = 0à -¢¥ 0.¯¥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¥¢¥« § ¤ çã ®è¨ ¢R3¢ ®¤®¬¥àãî § ¤ çã® ª®«¥¡ ¨¨ ¯®«ã¡¥áª®¥ç®© áâàãë.â 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¥¬ áà §ã ª®¢â®à®¬ã.II.¥«¨¬ ¯¥à¥¬¥ë¥ ¢ ᮮ⢥âáâ¢ãî饬 ®¤®à®¤®¬ ãà ¢-¥¨¨ ¨ ®¤®à®¤ëå ªà ¥¢ëå ãá«®¢¨ïå.⤥«¨¬ ¯à®áâà áâ¢¥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®¡áâ¢¥ë¥ § ç¥¨ïµ = λ2 > 0.த®«¦¨¬ ¤¥«¥¨¥ ¯¥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) ⇐⇒ ∅.âáî¤ á«¥¤ã¥â, ç⮠ᮡáâ¢¥ë¥ § 票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¥äãªæ¨¨: cos nϕ, sin nϕ, µ = n, n ∈ N. ®£¤ ¯¨èãâ â ª:½Φn (ϕ) = cos nϕ, n = 0, 1, 2, . . .;Φm (ϕ) = sin |m|ϕ, m = −1, −2, . . .æ ⥫ìë, ¯à¨çñ¬â 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},¥¯¥àì§ ©¬ñ¬áï¢â®à®©§ ¤ 祩.¥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¥ ýᢥà㫨áìþ ¢ ®¤®ç«¥) ᮡáâ¢¥ë¥ äãªæ¨¨, ®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 ,¤ ª® § ¬¥â¨¬, çâ® ®¯¥à â®à âãଠ{¨ã¢¨««ï«ï¥âá磌 áá¨ç¥áª¨¬ | ãá«®¢¨¥p(r) = 0 ¯à¨ r = 0.âáâ㯫¥¨¥*.
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