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WuChong-shi ( )§16.1 ∂u∂t−κ∂2u∂x2+∂2u∂y2=0,0xa,0yb,t0,∂u∂x x=0=0,∂u∂x x=a=0,0≤y≤b,t≥0,∂u∂y y=0=0,∂u∂y y=b=0,0≤x≤a,t≥0,u t=0=φ(x,y),0≤x≤a,0≤y≤b. u(x,y,t)=v(x,y)T(t) !# v(x,y)T0(t)−κ∂2v∂x2+∂2v∂y2T(t)=0=⇒∂2v∂x2+∂2v∂y2v(x,y)=1κT0(t)T(t)=−λ$∂2v∂x2+∂2v∂y2+λv(x,y)=0,T0(t)+λκT(t)=0,%&λ’!#()*+,-./01(23456 789$∂v∂x x=0=0∂v∂x x=a=0∂v∂y y=0=0∂v∂y y=b=0:;v(x,y)=X(x)Y(y) *=!# X00(x)Y(y)+X(x)Y00(y)+λX(x)Y(y)=0=⇒X00(x)X(x)+λ=−Y00(y)Y(y)=ν$X00(x)+μX(x)=0Y00(y)+νY(y)=0?@)*A./μ Bμ,νCλ&DEFG’HI+ JKLMNOμ+ν=λ0:P3456!# Q8$RX0(0)=0,X0(a)=0CY0(0)=0,Y0(b)=0.S TUX(x)VWXY X00(x)+μX(x)=0WuChong-shi§16.1Z[\]^_‘abcdef2gX0(0)=0,X0(a)=0Fhμ=0( .i +jk’X(x)=A0x+B0. (lm)3456 $A0=0,B0no.pqλ=0’Grst rsu/8vwX(x)=1.xyz{|}~~ ~μ=0 μ=0 X(x)=B0,B0 0Fhμ6=0( .i +jk’X(x)=Asin√μx+Bcos√μx. (lm)3456 Q$RA=0,√μsin√μa=0. √μa=nπ rstμn=nπa2,n=1,2,3,···rsu/Xn(x)=cosnπax. μ=0Cμ0+ 8 rstμn=nπa2,n=0,1,2,3,···,rsu/Xn(x)=cosnπax.1¡8 k$¢£Y(y)+rst⁄¥Y00(y)+νY(y)=0Y0(0)=0,Y0(b)=0+kwrstνm=mπb2,m=0,1,2,3,···,rsu/Ym(x)=cosmπby.WuChong-shiƒ§¤'“«_‘‹(›)f3gfi£fl-+nCm :*= –T00(t)=A00,n=m=0,Tnm(t)=Anme−λnmκt,%†‡·,78 +· Tnm(t)=Anme−λnmκt,n=0,1,2,3,···,m=0,1,2,3,···,λnm=μn+νm=nπa2+mπb2.¶• $‚G-k⁄¥+„kunm(x,y,t)=Xn(x)Ym(y)Tnm(t)=Anmcosnπaxcosmπbye−λnmκtC”ku(x,y,t)=∞Xn=0∞Xm=0unm(x,y,t)=∞Xn=0∞Xm=0Anmcosnπaxcosmπbye−λnmκt=∞Xn=0∞Xm=0Anmcosnπaxcosmπbyexp−nπa2+mπb2κt. »…56 Eu(x,y,t) t=0=∞Xn=0∞Xm=0Anmcosnπaxcosmπby=φ(x,y).‰= h¿ rsu/+`´ˆ-–˜¯˘/0˙¨ ˚¸R{Xn(x),n=0,1,2,···}+`´ˆ Q˚¸R{Ym(y),m=0,1,2,···}+`´ˆ ˝80%m ˛ˇRJK+`´—ˆZa0Xn(x)Xn0(x)dx=a2(1+δn0)δnn0,Zb0Ym(y)Ym0(y)dy=b2(1+δm0)δmm0. & ˚ n=0 n6=0Cm=0 m6=0+‡·0 + ’Anm=4ab1(1+δn0)(1+δm0)Za0Zb0φ(x,y)cosnπaxcosmπbydxdy.WuChong-shi§16.2 Æ ª4 §16.2 ŁØŒº æ x ı łøœß æ x ı ~ łøœ 0 ~æ x ı | ~ łøœß ∂2u∂t2−a2∂2u∂x2=f(x,t),0xl,t0,u x=0=0,u x=l=0,t≥0,u t=0=0,∂u∂t t=0=0,0≤x≤l.wA –fi£ lm + ? 2 ) +F - +!# +»$%C»&’(w00)W *+,-./01234567u(x,t)=v(x,t)+w(x,t),¨P lm lm8+1( LM9:;E+lm3456˝0k+¢= ¨£ $„kv(x,t)0¸£f(x,t)· ?@AB+‡·0 CD k lm +EF G $ lm +G„kv(x,t) ∂2v∂t2−a2∂2v∂x2=f(x,t). H Iu(x,t)=v(x,t)+w(x,t) J∂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=0KLM NO v(x,t) P MQw(x,t)~KR w(x,t)=∞Xn=1Cnsinnπlat+Dncosnπlatsinnπlx,WuChong-shiƒ§¤'“«_‘‹(›)f5gS u(x,t)=v(x,t)+∞Xn=1Cnsinnπlat+Dncosnπlatsinnπlx,TUVWı ∞Xn=1Dnsinnπlx=−v(x,t) t=0,∞Xn=1Cnnπalsinnπlx=−∂v(x,t)∂t t=0,X Y ~Z[\K] Q^_‘ Cn=−2nπaZl0∂v(x,t)∂t t=0sinnπlxdx,Dn=−2lZl0v(x,0)sinnπlxdx.• a ßb æ x ı ~} c0• d æ c~} efghi ~ ı |ø0• ß~ P jM O v(x,t)0k jf(x,t)lmnopq~rl0• VWı ~st uv0w16.1 k-k⁄¥∂2u∂t2−a2∂2u∂x2=f(x),0xl,t0,u x=0=0,u x=l=0,t≥0,u t=0=0,∂u∂t t=0=0,0≤x≤l,%&f(x)wxyu/0 Dfl–k¥+z˚{|02£ + lm D’x+u/ 8 lm8u/7vwD’x+u/ u(x,t)=v(x)+w(x,t),%&v(x)NO.i +3t⁄¥v00(x)=−1a2f(x),v(0)=0,v(l)=0;}w(x,t)~NO-k⁄¥∂2w∂t2−a2∂2w∂x2=0,0xl,t0,w x=0=0,w x=l=0,t≥0,w t=0=−v(x),∂w∂t t=0=0,0≤x≤l.WuChong-shi§16.2\ b a f6gw16.2 k-k⁄¥∂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,%&a,A0 ω(wxy./0 u(x,t)=v(x,t)+w(x,t),˛ˇR lm + · 8Plm8u/v(x,t)vwv(x,t)=f(x)sinωt. $v(x,t)NO lm lm3456 ∂2v∂t2−a2∂2v∂x2=A0sinωt,0xl,t0,v x=0=0,v x=l=0,t≥0,7 ’ f(x) $−ω2f(x)−a2f00(x)=A0,f(0)=0,f(l)=0.G lm.i +jkwf(x)=−A0ω2+Asinωax+Bcosωax. lm34568 -–B=A0ω2,A=A0ω2tanωl2a.£’f(x)=−A0ω21−cosωax−tanωl2asinωax=−A0ω21−cos(ω(x−l/2)/a)cos(ωl/2a).¡ –w(x,t) NO+-k⁄¥ ∂2w∂t2−a2∂2w∂x2=0,0xl,t0,w x=0=0,w x=l=0,t≥0,w t=0=0,∂w∂t t=0=−ωf(x),0≤x≤l.J+”kww(x,t)=∞Xn=1hCnsinnπlat+Dncosnπlatisinnπlx.WuChong-shiƒ§¤'“«_‘‹(›)f7g ¸ +»…56 8 -–Dn=0,Cn=−2ωnπaZl0f(x)sinnπlxdx=−2A0ωl3π2a1−(−)nn21(nπa)2−(ωl)2.DEn= /( Cn ˝w00¡ –Aw(x,t)=−4A0ωl3π2a∞Xn=01(2n+1)21[(2n+1)πa]2−(ωl)2sin2n+1lπxsin2n+1lπatCu(x,t)=−A0ω21−cosω(x−l/2)/acos(ωl/2a)sinωt−4A0ωl3π2a∞Xn=01(2n+1)21[(2n+1)πa]2−(ωl)2sin2n+1lπxsin2n+1lπat. ! + ω` ’ + E ω=(2k+1)πa/l,kw G -+ ‚/ ¨! +¡¸‰¢£⁄¥˙ƒ0w16.3 k-k⁄¥∂2u∂x2+∂2u∂y2=xy,0xa,0yb,u x=0=0,u x=a=0,0≤y≤b,u y=0=φ(x),u y=b=ψ(x),0≤x≤a. §¤ – +jk16x3y+f(x+iy)+g(x−iy).h u/fCg '“ f(x+iy)+g(x−iy)=−a224i(x+iy)2−(x−iy)2=−16a2xy, $R+kv(x,y)=16 x2−a2xyNOlm3456v(x,y) x=0=0,v(x,y) x=a=0.;u(x,y)=v(x,y)+w(x,y),WuChong-shi§16.2\ b a f8g 8 –w(x,t) NO+-k⁄¥ ∂2w∂x2+∂2w∂y2=0,0xa,0yb,w x=0=0,w x=a=0,0≤y≤b,w y=0=φ(x),w y=b=ψ(x)−b6 x2−a2x,0≤x≤a.%&+ Cfi3456«’lm+ ¶• ‹§¤ k0