复习上次课 1建立了定义 U()小=/(xk==G(o) FGo)2∫Go-o=/() 2.(常用)性质 F[/(x)=(o)[/(x F‖f(x 小=1F[(x F[=F小*F/
复习上次课 1.建立了定义 [ ( )] ( ) ( ) ò ¥ -¥ - = = w w F f x f x e dx G i x î í ì [ ( )] ( ) ( ) ò ¥ -¥ - F G = G e d = f x i x w w w 1 w 2.(常用)性质 ( ) F [f (x)] (i ) F[f (x)] n n = w [ ( )] F [f ( ) x ] i F f x x x w 1 0 = ò [ ] [ ] [ ] 1 2 1 2 F f × f = F f * F f
§42傅里叶变换法 用傅氏变换解数理方程 一、波动问题 a2u=0.-∞<x<∞、t>0 (1) (2) Lo=y() (3) 曾由行波法求得 )=2l 四{x+a)+(x-c)+ ara v(a do
§4.2 傅里叶变换法 用傅氏变换解数理方程 一、波动问题 u tt - a 2 u xx = 0,-¥ < x < ¥ , t > 0 （1） u t = 0 = j (x ) （2） u t t = 0 = y (x ) （3） 曾由行波法求得 ( ) [ ( ) ( )] ( ) ò + - = + + - + x at x at d a u x t j x at j x at y a a 2 1 2 1
现用傅氏变换法求解 为此对定解问题各项施行傅氏变换 ∫n(x,t 02u 10X dx-a e1o dx=0 ax ∫(.0kd=」.o(x)kd !n(x0)k“h=(x)
现用傅氏变换法求解 为此对定解问题各项施行傅氏变换 ( , ) 0 2 2 2 2 2 = ¶ ¶ - ¶ ¶ ò ò ¥ - ¥ - ¥ - ¥ - e dx x u u x t e dx a t iw x iw x ( ) ( ) ò ò ¥ - ¥ ¥ - ¥ - - u x e dx = x e dx iw x iw x ,0 j ( ) ( ) ò ò ¥ - ¥ - ¥ - ¥ - = ¶ ¶ u x e dx x e dx t iw x iw x ,0 y
(r, t e -io dx=io, t) 记 p(x=0(0 y(x e-ioxdx=o) 则 tabula, )=0(4) i(o,t)=0(o) (5) i,(o,0)=V()
记 u (x t )e dx u ( t ) i x , ~ , w w ò = ¥ - ¥ - j ( ) j (w ) w ~ ò = ¥ - ¥ - x e dx i x y ( ) y (w ) w ~ ò = ¥ - ¥ - x e dx i x 则 ( ) ( , ) 0 ( ) 4 , ~ ~ 2 2 2 + a u t = dt d u t w w w ( ) ( ) (5 ) ~ , ~u w t = j w ( ) ( ) (6 ) ~ ,0 ~u t w = y w
解(4得 i0,)=40)cd0+80ao() (5)代入(7) A(o)=0(0) (,1)=0o)0sa0+B(o) sin aot(7) (6)代入(7): Blo ao=vla ∴B()=-v(o)
解(4)得 ( , ) ( )cos ( )sin (7) ~ u w t = A w awt + B w awt (5)代入(7): (w) j(w) ~ A = ( ) ( )cos ( )sin (7 ) ~ , ~ \u w t = j w awt + B w awt ¢ (6)代入(7): (w) w y(w) ~ B a = ( ) y( ) w w w 1 ~ a \B =