- #1
wahoo2000
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I have made an attempt on this one, but I'm not quite sure that I have done it correctly so far..? I am now heading a (for me massive partial integration, and therefore I think it's better to ask before I start.
Find the solution u(x,t) of the inhomogenous wave equation
[tex]u_{tt}-u_{xx}=x, 0<x<1, t>0[/tex]
[tex]u(0,t)=1, u_{x}(1,t)=0[/tex]
[tex]u(x,0)=1, u_{t}(x,0)=0[/tex]
Let u(x,t)=S(x)+w(x,t) where S''(x)=x, S(0)=S'(1)=0
then
S'(x)= X^2+C
S(x)=x^3/6+C*x+D
S(0)=0 ==> D=0
S'(1)=0 ==> C=-1/2
==>
S(x)=x^3/6-x/2
Then the equation for w is:
[tex]
w_{tt}-w_{xx}=0, 0<x<1, t>0[/tex]
[tex]w(0,t)=1, w_{x}(1,t)=0[/tex]
[tex]w(x,0)=u(x,0)-S(x)=1-\frac{x^{3}}{6}+\frac{x}{2}, w_{t}(x,0)=0
[/tex]
I will try to send the rest as a reply to this post... I get database error when I try to send more than this.. :/
Homework Statement
Find the solution u(x,t) of the inhomogenous wave equation
[tex]u_{tt}-u_{xx}=x, 0<x<1, t>0[/tex]
[tex]u(0,t)=1, u_{x}(1,t)=0[/tex]
[tex]u(x,0)=1, u_{t}(x,0)=0[/tex]
The Attempt at a Solution
Let u(x,t)=S(x)+w(x,t) where S''(x)=x, S(0)=S'(1)=0
then
S'(x)= X^2+C
S(x)=x^3/6+C*x+D
S(0)=0 ==> D=0
S'(1)=0 ==> C=-1/2
==>
S(x)=x^3/6-x/2
Then the equation for w is:
[tex]
w_{tt}-w_{xx}=0, 0<x<1, t>0[/tex]
[tex]w(0,t)=1, w_{x}(1,t)=0[/tex]
[tex]w(x,0)=u(x,0)-S(x)=1-\frac{x^{3}}{6}+\frac{x}{2}, w_{t}(x,0)=0
[/tex]
I will try to send the rest as a reply to this post... I get database error when I try to send more than this.. :/
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