搜索
WuChong-shi Green ( )§30.1 Green ! # $%∂2u(x,t)∂t2−a2∂2u(x,t)∂x2=f(x,t),0xl,t0,u(x,t) x=0=μ(t),u(x,t) x=l=ν(t),t0,u(x,t) t=0=φ(x),∂u(x,t)∂t t=0=ψ(x),0xl.& ’( )* Green+,G(x,t;x0,t0)*-%./(01234!/5)6(01237846)9 h∂2∂t2−a2∂2∂x2iG(x,t;x0,t0)=δ(x−x0)δ(t−t0),0x,x0l,t,t002:; #=G(x,t;x0,t0) x=0=0,G(x,t;x0,t0) x=l=0,t,t00,G(x,t;x0,t0) tt0=0,∂G(x,t;x0,t0)∂t tt0=0,0x,x0l # ?@AB= CDEF%GHI JK L M%2t=t0/5NO P 2Q R ST! UVWX Y! !Z O2[\ ]^ J!Green+,G(x,t;x0,t0) _‘abcdeGreen+,fgh=f(x,t),μ(t),ν(t)Yφ(x),ψ(x)i # #u(x,t)jkNl]cdmNGreen+,WuChong-shi§30.1noGreenpqr2sGreentuvwxy(Green+,2z{|}~ {|} ) Q N 3Green+,G(x,−t;x00,−t00) # ∂2∂t2−a2∂2∂x2G(x,−t;x00,−t00)=δ(x−x00)δ(t−t00),0x,x00l,t,t000,G(x,−t;x00,−t00) x=0=0,G(x,−t;x00,−t00) x=l=0,t,t000,G(x,−t;x00,−t00) −t−t00=0,∂G(x,−t;x00,−t00)∂t −t−t00=0,0x,x00l.i ^ Green+,G(x,−t;x00,−t00)YG(x,t;x0,t0) ) 2 8[0,l]Y[0,∞) _xYt G(x0,−t0;x00,−t00)−G(x00,t00;x0,t0)=Zl0dxZ∞0G(x,−t;x00,−t00)∂2G(x,t;x0,t0)∂t2−G(x,t;x0,t0)∂2G(x,−t;x00,−t00)∂t2dt−Z∞0dtZl0G(x,−t;x00,−t00)∂2G(x,t;x0,t0)∂x2−G(x,t;x0,t0)∂2G(x,−t;x00,−t00)∂x2dx=Zl0G(x,−t;x00,−t00)∂G(x,t;x0,t0)∂t−G(x,t;x0,t0)∂G(x,−t;x00,−t00)∂t∞0dx−Z∞0G(x,−t;x00,−t00)∂G(x,t;x0,t0)∂x−G(x,t;x0,t0)∂G(x,−t;x00,−t00)∂xl0dt, =YAB= & N 0 P $ N Green+,278 _‘a /8 a G(x00,t00;x0,t0)=G(x0,−t0;x00,−t00), ix00Yt00 xYt G(x,t;x0,t0)=G(x0,−t0;x,−t).2?^ ¡¢£ itYt0_⁄¥ƒ/NO §¤ '“U« /8 ‹›;fifl –†$‡ ·3K ¶ \m WuChong-shi•‚„”Greenpq(‚)r3s»Greentu…‰ ¿ f(x,t),μ(t),ν(t)`φ(x),ψ(x)´ˆ˜¯˘v˜u(x,t)˙¨ ˚ Q i # £ ¸ x0Yt0 ∂2u(x0,t0)∂t02−a2∂2u(x0,t0)∂x02=f(x0,t0),0x0l,t00,u(x0,t0) x0=0=μ(t0),u(x0,t0) x0=l=ν(t0),t00,u(x0,t0) t0=0=φ(x0),∂u(x0,t0)∂t0 t0=0=ψ(x0),0x0l. NGreen+, # ∂2∂(−t0)2−a2∂2∂x02#G(x0,−t0;x,−t)=δ(x−x0)δ(t−t0),0x,x0l,t,t00,G(x0,−t0;x,−t) x0=0=0,G(x0,−t0;x,−t) x0=l=0,t,t00,G(x0,−t0;x,−t) −t0−t=0,∂G(x0,−t0;x,−t)∂t −t0−t=0,0x,x0l.˝eGreen+, _‘a a ¡ ˛& ∂2∂t02−a2∂2∂x02G(x,t;x0,t0)=δ(x−x0)δ(t−t0),0x,x0l,t,t00,G(x,t;x0,t0) x0=0=0,G(x,t;x0,t0) x0=l=0,t,t00,G(x,t;x0,t0) t0t=0,∂G(x,t;x0,t0)∂t t0t=0,0x,x0l.i ^ G(x,t;x0,t0)Yu(x0,t0) ) Zl0dx0Z∞0G(x,t;x0,t0)f(x0,t0)dt0−u(x,t)=Zl0dx0Z∞0G(x,t;x0,t0)∂2u(x0,t0)∂t02−u(x0,t0)∂2G(x,t;x0,t0)∂t02dt0−a2Z∞0dt0Zl0G(x,t;x0,t0)∂2u(x0,t0)∂x02−u(x0,t0)∂2G(x,t;x0,t0)∂x02dx0. =YAB= $& ˇ— u(x,t)=Zl0dx0Z∞0G(x,t;x0,t0)f(x0,t0)dt0−Zl0G(x,t;x0,t0)∂u(x0,t0)∂t0−u(x0,t0)∂G(x,t;x0,t0)∂t0∞0dx0+a2Z∞0G(x,t;x0,t0)∂u(x0,t0)∂x0−u(x0,t0)∂G(x,t;x0,t0)∂x0l0dt0=Zl0dx0Zt0G(x,t;x0,t0)f(x0,t0)dt0−Zl0G(x,t;x0,0)ψ(x0)−φ(x0)∂G(x,t;x0,t0)∂t0 t0=0dx0−a2Zt0ν(t0)∂G(x,t;x0,t0)∂x0 x0=l−μ(t0)∂G(x,t;x0,t0)∂x0 x0=0dt0.WuChong-shi§30.1noGreenpqr4s Greentuv h∂2∂t2−a2∂2∂x2iG(x,t;x0,t0)=δ(x−x0)δ(t−t0),0x,x0l,t,t00,G(x,t;x0,t0) x=0=0,G(x,t;x0,t0) x=l=0,t,t00,G(x,t;x0,t0) tt0=0,∂G(x,t;x0,t0)∂t tt0=0,0x,x0l )*:; +, G(x,t;x0,t0)=∞Xn=1Tn(t)sinnπlx, / iδ+,˛ - +, δ(x−x0)=2l∞Xn=1sinnπlx0sinnπlx,3% Tn(t)$ AÆ T00(t)+nπal2Tn(t)=2lsinnπlx0δ(t−t0),Tn(tt0)=0,T0n(tt0)=0.# Tn(t)=2nπasinnπlx0sinnπla(t−t0)η(t−t0).P Green+,G(x,t;x0,t0)$%G(x,t;x0,t0)=2πa∞Xn=11nsinnπlx0sinnπlxsinnπla(t−t0)η(t−t0).WuChong-shi•‚„”Greenpq(‚)r5s !^]ª78 ?/ Green+,G(r,t;r0,t0) # h∂2∂t2−a2∇2iG(r,t;r0,t0)=δ(r−r0)δ(t−t0),t,t00,G(r,t;r0,t0) tt0=0,∂G(r,t;r0,t0)∂t tt0=0. Fourier ⁄g(r,ω;r0,t0)=1√2πZ∞−∞G(r,t;r0,t0)eiωtdt,3%$i # ˇ h(−iω)2−a2∇2ig(r,ω;r0,t0)=1√2πeiωt0δ(r−r0) h∇2+ωa2ig(r,ω;r0,t0)=−1√2πa2eiωt0δ(r−r0), 27.2Ł Ø & Œg(r,ω;r0,t0)=1√2πa2eiωt014π|r−r0|ei(ω/a)|r−r0|. º ⁄ $ G(r,t;r0,t0)=1√2πZ∞−∞g(r,ω;r0,t0)e−iωtdω=14πa21|r−r0|12πZ∞−∞e−iω(t−t0)·ei(ω/a)|r−r0|dω=14πa21|r−r0|δ|r−r0|a−(t−t0)=14πa1|r−r0|δ(|r−r0|−a(t−t0)).CDEFG Jt0/52r0 ¤ t/5! Œæ r06 a(t−t0) ?^Green+, ST$& Œ]ªı 78£ AÆ ∂2u(r,t)∂t2−a2∇2u(r,t)=f(r,t),t0,u(r,t) t=0=φ(r),∂u(r,t)∂t t=0=ψ(r) # u(r,t)=14πa2ZZZ|r0−r|atf(r0,t−|r0−r|/a)|r0−r|dr0+14πaZZΣ0ψ(r0)|r0−r|dΣ0+∂∂tZZΣ0φ(r0)|r0−r|dΣ0#, £Σ0% r6 łat øœ |r0−r|=at