第三十讲 Green函数(三 第3页 用Gren函数及已知条件f(x,t),p(t),v(1)和o(x),v(x) 将定解问题的解u(x,t)表示出来 为此,将定解问题中的自变量改写成x′和t 2u(x,t)202u(x,t) 0<x′<l,t'>0. u(x,t)=0=(),u(x,t)l==1=v( t>0 u(x,t)l=0=(x) v(x),0<x<l 再写出Gren函数的定解问题 0<x,x′<l,t,t>0 (_t) a2zG(x,-t1;x,-1)=6x-m1)t-t) G(x,-t;x,-)1=1=0 t,t>0 G(x,-t;x,-1)<=0 aG(r,-t';r, -t) 0<x,x′<l 利用 Green函数的对称性与倒易性关系,也可以改写成 0-dx2(u,右x,1)=6(x-m1)(-1),0<xx<,t>0 G(a, t; a,t')l x,t;x,t)=0, t,t>0, G(x,tx,):=0, 0<xx'<l 将两个方程分别乘以G(x,tx,1)和(,),相减,再积分 (a, t; r,t)f(ar, tdt'-u(a, t) 2(x,t) /(cax;x,)2 (x,t) a-G(r, t; I', t) a-u(ar,t) a-G(, t; a', i axl 代入边界条件和初始条件,就可以化简为 G(a, t; I, t) a',t) u(r,t G(r, t; r, t') o')f(ar, t')da G(r, t;r, O)v(ar)-o(r) aG(a, t; a', t') aG(a, t; I', t) x’=lWu Chong-shi ➲➳➵➸ Green ♣q (➳ ) r 3 s ➺ Green t✉➻ ➼➽➾➚ f(x, t), µ(t), ν(t) ➪ φ(x), ψ(x) ➶➹➘ ➴➷✈ ➘ u(x, t) ➬➮ ➱✃ ✎ ◗ ✗ ✐ ✔★✣✤ ↔✢ ❐ ➦❒➒➓➔ x 0 ❨ t 0 ✗ ∂ 2u(x 0 , t0 ) ∂t02 − a 2 ∂ 2u(x 0 , t0 ) ∂x02 = f(x 0 , t0 ), 0 < x0 < l, t0 > 0, u(x 0 , t0 )   x0=0 = µ(t 0 ), u(x 0 , t0 )   x0=l = ν(t 0 ), t0 > 0, u(x 0 , t0 )   t 0=0 = φ(x 0 ), ∂u(x 0 , t0 ) ∂t0    t 0=0 = ψ(x 0 ), 0 < x0 < l. ❹➓◆ Green ✰✱✢✔★✣✤ " ∂ 2 ∂(−t 0) 2 − a 2 ∂ 2 ∂x02 # G(x 0 , −t 0 ; x, −t) = δ(x − x 0 )δ(t − t 0 ), 0 < x, x0 < l, t, t0 > 0, G(x 0 , −t 0 ; x, −t)   x0=0 = 0, G(x 0 , −t 0 ; x, −t)   x0=l = 0, t, t0 > 0, G(x 0 , −t 0 ; x, −t)   −t 0<−t = 0, ∂G(x 0 , −t 0 ; x, −t) ∂t    −t 0<−t = 0, 0 < x, x0 < l. ❮❡ Green ✰✱✢❴❵❛➍➎➏❛❻→✗❰✫✡➒➓➔  ∂ 2 ∂t02 − a 2 ∂ 2 ∂x02  G(x, t; x 0 , t0 ) = δ(x − x 0 )δ(t − t 0 ), 0 < x, x0 < l, t, t0 > 0, G(x, t; x 0 , t0 )   x0=0 = 0, G(x, t; x 0 , t0 )   x0=l = 0, t, t0 > 0, G(x, t; x 0 , t0 )   t 0>t = 0, ∂G(x, t; x 0 , t0 ) ∂t    t 0>t = 0, 0 < x, x0 < l. ✐❼❫✌✍❽❾❿✡ G(x, t; x 0 , t0 ) ❨ u(x 0 , t0 ) ✗✮➀✗❹➃ ❽✗ Z l 0 dx 0 Z ∞ 0 G(x, t; x 0 , t0 )f(x 0 , t0 )dt 0 − u(x, t) = Z l 0 dx 0 Z ∞ 0  G(x, t; x 0 , t0 ) ∂ 2u(x 0 , t0 ) ∂t02 − u(x 0 , t0 ) ∂ 2G(x, t; x 0 , t0 ) ∂t02  dt 0 − a 2 Z ∞ 0 dt 0 Z l 0  G(x, t; x 0 , t0 ) ∂ 2u(x 0 , t0 ) ∂x02 − u(x 0 , t0 ) ∂ 2G(x, t; x 0 , t0 ) ∂x02  dx 0 . ➆➇➈✛❁❂❨❆❇❁❂✗✩✫✡ÏÐ✎ u(x, t) = Z l 0 dx 0 Z ∞ 0 G(x, t; x 0 , t0 )f(x 0 , t0 )dt 0 − Z l 0  G(x, t; x 0 , t0 ) ∂u(x 0 , t0 ) ∂t0 − u(x 0 , t0 ) ∂G(x, t; x 0 , t0 ) ∂t0 ∞ 0 dx 0 + a 2 Z ∞ 0  G(x, t; x 0 , t0 ) ∂u(x 0 , t0 ) ∂x0 − u(x 0 , t0 ) ∂G(x, t; x 0 , t0 ) ∂x0 l 0 dt 0 = Z l 0 dx 0 Z t 0 G(x, t; x 0 , t0 )f(x 0 , t0 )dt 0 − Z l 0  G(x, t; x 0 , 0)ψ(x 0 ) − φ(x 0 ) ∂G(x, t; x 0 , t0 ) ∂t0     t 0=0  dx 0 − a 2 Z t 0  ν(t 0 ) ∂G(x, t; x 0 , t0 ) ∂x0     x0=l − µ(t 0 ) ∂G(x, t; x 0 , t0 ) ∂x0     x0=0  dt 0