1. By using the method of characteristics, find an explicit local solution
1 2 2
to u + ---[ ( u ) + x ] = 0, if t>0, x in R, with u(x,0) = (x^2)/2
t 2 x
2. solve the wave equation for infinite vibrating string
2 [ c_1 , x<0
u = c (x)u , where c(x) =[
tt xx [ c_2 , x>0.
Let a wave u(x,t)=f(x-c_1t) come in from the left, thus the initial
conditions are u(x,0) = f(x) and u (x,0) = -c_1f'(x).
t
Assume that u(x,t) and u (x,t) are continuous everywhere. Also give
x
an interpretation for the solution you find.
===========================================================================
第二題由D'Alembert method 可以知道解的長相為
u(x,t) = F(x-c_1t)+G(x+c_1t) for x<0 and all t
= H(x-c_2t)+K(x+c_2t) for x>0 and all t.
然後接下來我就不知道要怎麼做了
ans : u(x,t) =f(x-c_1t) + [(c_2-c_1)/(c_2+c_1)]f(-c_1t-x) for x<0
c_1
= [(2c_2)/(c_2+c_1)]f((---(x-c_2t)) for x>0
c_2
--
※ 發信站: 批踢踢實業坊(ptt.cc)
◆ From: 115.43.192.87