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WuChong-shi ( )§17.1 ( ) !#$%&u(x,t)=v(x,t)+w(x,t)’(∂2u∂t2−a2∂2u∂x2=f(x,t)u x=0=0u x=l=0u t=0=0∂u∂t t=0=0=∂2v∂t2−a2∂2v∂x2=f(x,t)v x=0=0v x=l=0+∂2w∂t2−a2∂2w∂x2=0w x=0=0w x=l=0w t=0=−v t=0∂w∂t t=0=−∂v∂t t=0)*+,-./0f(x,t)1234567’89:;-./+,1=’?9@ABC1=DE FGHIJKLMNOPQRSTU{Xn(x),n=1,2,3,···}’VWXY Z[\]^_‘’ab’cdefgu(x,t)hijklmnijkof(x,t)pqRSTUrsu(x,t)=∞Xn=1Tn(t)Xn(x),f(x,t)=∞Xn=1gn(t)Xn(x),tuvLMwxTn(t)ydEz{Tn(t)KP|TU’}~ nK lm(Q)’ d wg lm ERSTUQ{Xn(x)}n n MK {Xn(x)} jk g nRSTU’y~ zjk lm jk ∂2u∂t2−a2∂2u∂x2=0,0xl,t0,u x=0=0,u x=l=0,t≥0 OnRS X00n(x)+λnXn(x)=0,Xn(0)=0,Xn(l)=0.WuChong-shi§17.1¡¢£⁄¥ƒ§¤'“(«)‹2›fiu(x,t) f(x,t)nrsfl – lm’†‡o ·’∞Xn=1T00n(t)Xn(x)−a2∞Xn=1Tn(t)X00n(x)=∞Xn=1gn(t)Xn(x). ¶Xn(x)•~ n lm’‚„”∞Xn=1T00n(t)Xn(x)+a2∞Xn=1λnTn(t)Xn(x)=∞Xn=1gn(t)Xn(x).v»…RSTUn‰ ¿’c OTn(t)•~ n lmT00n(t)+λna2Tn(t)=gn(t). `’fu(x,t)nrsfl –´ˆ ’˜d O∞Xn=1Tn(0)Xn(x)=0,∞Xn=1T0n(0)Xn(0)=0.»…RSTUn‰ ¿’y ¯xTn(0)=0,T0n(0)=0.¶gijk lmn U ˘M’˙¨¶Laplace ’cdewxTn(t)=lnπaZt0gn(τ)sinnπla(t−τ)dτ.˚¸gM’ q jk nRSTUrsMEv¶˚¸lMwg˝16.2Gn g ∂2u∂t2−a2∂2u∂x2=A0sinωt,0xl,t0,u x=0=0,u x=l=0,t≥0,u t=0=0,∂u∂t t=0=0,0≤x≤l. jk nRSTU˛ˇ15.1—G x’ dLu(x,t)=∞Xn=1Tn(t)sinnπlx,fijkoA0sinωt˜q˚PQRSTUrs’A0sinωt=2A0π∞Xn=11−(−1)nnsinnπlxsinωt, –lm ´ˆ ’c OT00(t)+nπla2Tn(t)=2A0π1−(−1)nnsinωt,T(0)=0,T0(0)=0.g y Tn(t)=2A0l2π1−(−1)nn1(nπa)2−(ωl)2sinωtWuChong-shi ( )‹3›−2A0ωl3π2a1−(−1)nn21(nπa)2−(ωl)2sinnπlat. ‚dewx˝16.3n P¸ flngu(x,t)=4A0l2π∞Xn=012n+11[(2n+1)πa]2−(ωl)2sin2n+1lπxsinωt−4A0ωl3π2a∞Xn=01(2n+1)21[(2n+1)πa]2−(ωl)2sin2n+1lπxsin2n+1lπat.Æ{ ’˝ª’Poissonlmn P ∂2u∂x2+∂2u∂xy2=f(x,y),0xa,0yb,u x=0=0,u x=a=0,0≤y≤b,u y=0=0,u y=b=0,0≤x≤a, t˜d¶q jk RSTUrsn MwgE˝ª’dLu(x,y)=∞Xn=1Yn(y)sinnπax,f(x,y)=∞Xn=1gn(y)sinnπax. –lm ’d Y00n(y)−nπa2Yn(y)=gn(y),Yn(0)=0,Yn(b)=0,wxYn(y)’˜c xŁgu(x,y)EØŒº ’˜deLu(x,y)=∞Xm=1Xm(x)sinmπby,f(x,y)=∞Xm=1hm(x)sinmπby, u¯xXm(x)~ nijk lm X00m(x)−mπb2Xm(x)=hm(x),Xm(0)=0,Xm(a)=0,wxXm(x)ydE æD ı Ełø1œß -./0gn(y) hm(x)1 23? œß’ Yn(y) Xm(x)1-./ +, :=EWuChong-shi§17.1¡¢£⁄¥ƒ§¤'“(«)‹4› de P n M’yfu(x,y) f(x,y) qRSTU{Xn(x)} ‚qRSTU{Ym(y)}rs( U)u(x,y)=∞Xn=1∞Xm=1cnmsinnπaxsinmπby,f(x,y)=∞Xn=1∞Xm=1dnmsinnπaxsinmπby,rs Ucnm wE f(x,y)K˛ TU’•ecnm˜K˛ nEˇ Urs ’ t˛ Ł Ef nrsfl –lm’y −∞Xn=1∞Xm=1cnmnπa2+mπb2sinnπaxsinmπby=∞Xn=1∞Xm=1dnmsinnπaxsinmπby.»…RSTUn‰ ¿’ ! U’y −cnmnπa2+mπb2=dnm.{Kcnm=−dnmnπa2+mπb2. u’w gu(x,y)=−∞Xn=1∞Xm=1dnmnπa2+mπb2sinnπaxsinmπby.˚¸ Mn#KØŒ$%Łwgijk lmEWuChong-shi ( )‹5›§17.2&’()*+, ’(-./012’34 5678 9ø : ;=A 6?@A BC -./19D’EFG ø:;= ./1E1HI;=JK ./1L•-./;=œ MNO•P QR./+, ./;=1=?@STUV QR./+, ./;=•WXYZ1[ \].Z^ 1_‘a-./;=)bcdLeefglmn g ˝E Łhxijk n#i’j lm ´ˆ kKjknE∂2u∂t2−a2∂2u∂x2=0,0xl,t0,u x=0=μ(t),u x=l=ν(t),t≥0,u t=0=0,∂u∂t t=0=0,0≤x≤l. Ł ¶ M’lm ’V nfijk jk„’you(x,t)=v(x,t)+w(x,t),p v(x,t)’q ~ v(x,t) x=0=μ(t),v(x,t) x=l=ν(t).˚`’w(x,t) tcP ~ jk w(x,t) x=0=0,w(x,t) x=l=0.Prs ’w(x,t)•~ nlm ´ˆ kfKijkn’∂2w∂t2−a2∂2w∂x2=−∂2v∂t2−a2∂2v∂x2,w t=0=−v t=0,∂w∂t t=0=−∂v∂t t=0.t¶ 16.2—˙Ru 1—n M’cdewxw(x,t)’˜c xŁgu(x,t)EFvwxy#$%[\v(x,t)L zWwv(x,t)~ v(x,t) x=0=μ(t),v(x,t) x=l=ν(t),•e {n | EWuChong-shi§17.2}~ ƒ~ ‹6›ª fit ”K U’˚cVWwˇ(x,y) n y=v(x,t) n (0,μ(t)) (l,ν(t))ydE˝ª’d v(x,t)=A(t)x+B(t), – ’yd xB(t)=μ(t),A(t)=1lν(t)−μ(t).˜d v(x,t)=A(t)x2+B(t),A(t)=1l2ν(t)−μ(t),B(t)=μ(t),˙v(x,t)=A(t)(l−x)2+B(t)x2,A(t)=1l2μ(t),B(t)=1l2ν(t). 17.1wg g ∂u∂t−κ∂2u∂x2=0,0xl,t0,u x=0=Asinωt,u x=l=0,t≥0,u t=0=0,0≤x≤l. Oijk n fl’dLjk„TU v(x,t)=A1−xlsinωt.{Kou(x,t)=A1−xlsinωt+w(x,t), w(x,t)~ g ∂w∂t−κ∂2w∂x2=−Aω1−xlcosωt,0xl,t0,w x=0=0,w x=l=0,t≥0,w t=0=0,0≤x≤l.fw(x,t) lmnijko1−x/lkq jk nRSTUrs’ w(x,t)=∞Xn=1Tn(t)sinnπlx,1−xl=∞Xn=12nπsinnπlx.»…Tn(t)• ~ ijkP lmT0n(t)+κnπl2Tn(t)=−2Aωnπcosωt´ˆ Tn(0)=0, ˘wxTn(t)=2Aωl2nπ1κ2(nπ)4+ω2l4κ(nπ)2exph−nπl2κti−κ(nπ)2cosωt−ωl2sinωt.˚`cw Łw(x,t)’v ’c O g ngu(x,t)EWuChong-shi ( )‹7› œß1./ v(x,t)’ 1w(x,t)16=78 ¡¢œß’: 1w(x,t)¢œßEW ’6=781=1£ ⁄ a’¥ƒ4X§¤ 1u(x,t) 6 'ß1’“«‹›3123? fiœßE˚`cdeflxP –nWw †pnjk„TUv(x,t)’qw(x,t)•~ n g ‡d ·E iJn ¶’ tcK •‚„ u(x,t)nlmK•Kjkn’ ”w(x,t)nlmKjknEc n g »’˚…‰ Wwjk„TUv(x,t)˜Klmng’∂2v∂t2−a2∂2v∂x2=0.Æ{¿· `nμ(t) ν(t)’Kde O˚P nE•‚„ nlmK•Kjkn’´ˆfi˚¸lMk flm jk„E 17.2wg g ∂2u∂t2−a2∂2u∂x2=0,0xl,t0,u x=0=0,∂u∂x x=l=Asinωt,t≥0,u t=0=0,∂u∂t t=0=0,0≤x≤l. ˜ˇc¯˘NOP jk„TU’flm jk„E ’Lu(x,t)=v(x,t)+w(x,t)’ Oijk n TU fl’d jk„TUv(x,t) v(x,t)=f(x)sinωt, f(x)K˙¨ lm f00(x)+ωa2f(x)=0,f(0)=0,f0(l)=Ang’f(x)=Aaω1cosωlasinωax.w(x,t)•~ n g K∂2w∂t2−a2∂2w∂x2=0,0xl,t0,w x=0=0,∂w∂x x=l=0,t≥0,WuChong-shi§17.2}~ ƒ~ ‹8›w t=0=0,∂w∂t t=0=−Aacosωlasinωax,0≤x≤l. Prg w(x,t)=∞Xn=0Cnsin2n+12lπat+Dncos2n+12lπatsin2n+12lπx.»…´ˆ ’de xCn=−4Aπcosωla12n+1Zl0sinωaxsin2n+12lπxdx=(−)n4Aω(2n+1)πa1ωa2−2n+12lπ2,Dn=0.f˚`w nv(x,t) w(x,t) ˚’c u xŁgu(x,t)E