用Gren函数及已知条件f(x,t),p(1),(t)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 02au(x,t)202u(x,t) f(ar, t), 0<x<l,t'>0. u(x,t)2=0=(t),u(x,t)=1=v(t), u(x2,t)1=0=() du(r, t) =v( 0<x<l 再写出Gren函数的定解间题 6(x-x)6(t-t") 0<x,x<l,t,t>0, (x,-t;x,-t)l-=0=0, G(a t. t G(r,-t;x,-t)_t<=:=0 G(x,-t';x,-t) 0<r.r'<l 利用 Green函数的对称性与倒易性关系,也可以改写成 _a2G(x,t;z,t=6(x-x)6(t-t1 0<x,x<l,t,t>0. G(x,tx,t)=0=0,G(x,t;x,t)=0, t,t>0, G(a, t; r, t')l t≈0.aG(x,右;x,t 0<ar<I 将两个方程分别乘以G(x,t;x,t)和u(x,t),相减,再积分, dz/G(r, t;r, t")f(a, t)dt'-u(r, t) drG(, t; r, t'y 02u(r,t) u(r, t') a-G(r,t:r, t) dt G(,t; 2, t)gu(a 0x/2-u(,t)22G(,t:; r','dr ar, t) 2 代入边界条件和初始条件,就可以化简为 dr/G(r, t; I, t'f(, edt'-G(a, t; t, 2u r,2-u(ar, t"G(z, ti t',21 dr'/G(r, t; r,, t')f(r, t,)dt lL(t' aG(a, t; z,t) G(a, t;I',t)Wu 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