Solve the wave equation subject to the boundary conditions of u(0,t) - 0 for t>=0, and u(L,t)=0, for t>=0.
u(x,0) = f(x) for 0<=x<=L
du/dt (x,0) = g(x)
f(x) = sin (2pix/L) , g(x) = 0
Note: Sorry but I don't even understand what the problem is asking, I have a really bad professor for this course (Applied Math), and to be honest, I had him before and I had to learn everything on my own. But then was Calculus I, and now this is difficult, I don't know even where to start, please help!
Thank you.
I'm still quite lost. So far we still haven't learn this general solution of
Is this given or you actually compute it?
My professor did one example, and man, he didn't even get to finish it because he tried to do it in the last few seconds of the class. He has f(x) = 0, g(x) = sin(pi*x/L). Then he took the du/dt, and then he stopped there...
So is that a way to solve it without using the general solution? Like the separation of variables as the last post suggested?
Thank you.
Given the equation,
.
With homogenous boundary value problem,
And initial value problems,
The solution is,
Where,
In this problem thus, for all .
And, .
Thus,
Use the following result:
Theorem: The set is orthogonal on .
And furthermore,
.
(Where is the Kronecker Delta.)
Which means,
This tells us that .
Hence the solution is,
NOTE: Topsquark used something called d'Alembert's General Solution.