§4.4*拉普拉斯变换法 n几 1.P178 T 7()=f() 325){7(0)=0 327)7()=0 记(=7(L[(=F(p) 则p27(p)p70)-7(0) 7(p)= Fp SIn n p
( ) T ( )t f ( )t l n a T t ÷ = ø ö ç è æ ¢¢ + 2 p T ¢(0) = 0 T (0) = 0 L[T(t)] T ( p) L[f (t)] F(p) ~ , ~ 记 = = P178 (3.2.5) (3.2.7) 1. ( ) ( ) ( ) T( )t F( ) p l n a p T p pT T ~ 0 0 ~ 2 2 ÷ = ø ö ç è æ - - ¢ + p 则 ( ) ( ) ( ) ú û ù ê ë é = * ÷ ø ö ç è æ + \ = t l n a n a l L f t l n a p F p T p p p p sin ~ 2 2 § 4.4* 拉普拉斯变换法
7()2()im (t-rdt n几 2.(傅氏变换主要用于解无界问题),拉氏适合解混 合问题 L.=a2l0<x<0t>0 n(0)=f()im(x)=0(≥0 x→00 n(:0)=0.,(x0)=0
( ) ( ) ( ) ò \ = - t t d l n a f n a l T t 0 sin t t p t p 2.(傅氏变换主要用于解无界问题),拉氏适合解混 合问题 ,0 , 0 2 utt = a uxx < x < ¥ t > (0, ) = ( ), lim ( , ) = 0 ( ³ 0) ®¥ u t f t u x t t x u(x,0) = 0, u (x,0) = 0 t
仍选t作变换∵二阶微商的变换要涉及 函数的在t=0,和一阶导数的值而关于 x的边界条件未给出 项=(cp)()=f() 则 p2u(x, p)-pu(x,0)u, (x,)=au(x,p) 10p)=F0) limu(x, p)=0 x→00
的边界条件未给出 函数的在 和一阶导数的值 而关于 仍选 作变换 二阶微商的变换要涉及 x 0, , t t = Q L[u(t)] u(x p) L[f (t)] F(p) ~ , , ~ = = ( ) ( ) ( ) u (x p) x p u x p pu x u x a t , ~ , ,0 ,0 ~ 2 2 2 2 ¶ ¶ - - = ( ) ( ) ( , ) 0 ~ , lim ~ 0, ~ = = ®¥ u p F p u x p x 则
2l-2=0 即 n(0)=F limu=o u =c,ple a tc plea 由0)=F设:c(p)+c()=F()
0 ~ ~ 2 2 2 2 - u = a p u dx d u ( ) F ~ 0 ~ = 0 ~ lim = ®¥ u x 即 ( ) ( ) x a p x a p u c p e c p e 1 2 ~ = + - u ( ) F c ( p) c (p) F(p) ~ : ~ 0 ~ 由 = 设 1 + 2 =
由i(x)=0c(=0c2=0 c1(p)=F(p) (x, p)=f(pe a n(c)=|(p)e n(:)=L2∠
( ) 0 ( ) 0 0 ~u ¥ = c2 p e = \ c2 = x a p 由 c ( p ) F ( p ) ~ 1 = ( ) ( ) x a p u x p F p e - = ~ , ~ ( ) ( ) ú û ù ê ë é = × - - x a p u x t L F p e ~ , 1 ( ) ú û ù ê ë é ÷ ø ö ç è æ = - - a x u x t L L f t 1 , ÷ ø ö ç è æ = - a x f t