求出Gren函数的具体形式 02202 ](x,门=0-)(-,0<x,<t>0 =0=0,G(x,t;x,tO=1=0 t,t>0 按相应齐次问题的本征函数展开 同时,将δ函数也按该组本征函数展开 于是,Tn(t)就满足常微分方程的初值问题 T"(t +(nna Tn(t)=T 6(t-t) Tn(t<t) Tn(t<t) 解之即得 Tn(t) 所以,Gren函数G(x,t;x,t)就是Wu Chong-shi §30.1 ♥♦ Green ♣q r 4 s Ñ ➱ Green t✉✈ÒÓÔÕ h ∂ 2 ∂t2 − a 2 ∂ 2 ∂x2 i 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 )   x=0 = 0, G(x, t; x 0 , t0 )   x=l = 0, t, t0 > 0, G(x, t; x 0 , t0 )   t<t0 = 0, ∂G(x, t; x 0 , t0 ) ∂t    t<t0 = 0, 0 < x, x0 < l Ö✮✯✿❀✣✤✢×Ø✰✱ÙÚ✗ G(x, t; x 0 , t0 ) = X∞ n=1 Tn(t) sin nπ l x, Û✴✗✐ δ ✰✱❰Ö✲Ü×Ø✰✱ÙÚ✗ δ(x − x 0 ) = 2 l X∞ n=1 sin nπ l x 0 sin nπ l x, ✸ ✪✗ Tn(t) ✩ÝÞßà❽✌✍✢❆á✣✤ T 00(t) + nπa l 2 Tn(t) = 2 l sin nπ l x 0 δ(t − t 0 ), Tn(t < t0 ) = 0, T 0 n(t < t0 ) = 0. ★â➄➅ Tn(t) = 2 nπa sin nπ l x 0 sin nπ l a(t − t 0 ) η(t − t 0 ). P✡✗ Green ✰✱ G(x, t; x 0 , t0 ) ✩✪ G(x, t; x 0 , t0 ) = 2 πa X∞ n=1 1 n sin nπ l x 0 sin nπ l x sin nπ l a(t − t 0 ) η(t − t 0 )