- #1
mizzcriss
- 4
- 0
Homework Statement
Show that the boundary-value problem $$u_{tt}=u_{xx}\qquad u(x,0)=2f(x)\qquad u_t(x,0)=2g(x)$$ has the solution $$u(x,t)=f(x+t)+f(x-t)+G(x+t)-G(x-t)$$ where ##G## is an antiderivative/indefinite integral of ##g##. Here, we assume that ##-\infty<x<\infty## and ##t\geq 0##
Homework Equations
I know that the solution for a wave equation of the form ##u_{tt}=a^2u_{xx}## is ##u(x,t)=f(x+at)+g(x-at)##
The Attempt at a Solution
Using the "knowledge" above, I tried to take the first partial derivative of ##u(x,t)=f(x+t)+g(x-t)## with respect to ##t## since ##a=1##. I got ##u_t(x,t)=f'(x+t)-g'(x-t)## Then using the conditions ##u(x,0)=2f(x)## and ##u_t(x,0)=2g(x)##, I substituted 0 for ##t## and set them equal to ##2f(x)## and ##2g(x)##.
I came up with ##f(x)=g(x)=\frac{1}{2}(f'(x)-g'(x))##, which I thought could also be ##\frac{1}{2}(f'(x)-G(x))##. But then I got stuck because I have no idea how to get to $$u(x,t)=f(x+t)+f(x-t)+G(x+t)-G(x-t)$$
Part of my confusion is that I only have initial conditions and no boundary conditions, and all the examples I can find have boundary conditions as well, not just initial.
Thanks so much if you can help