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Consider $$u_{tt}-a^2u_{xx}+u_t+a u_x=0,\quad 0<x<\infty,\quad t>0,(*)$$ where $u_t=\frac{\partial u}{\partial t}$ and etc. It is not so hard to use the method of characteristics to solve it as $$u(x,t)=e^\frac{x-a t}{2a}[g(x+at)-\frac{1}{2a} \int_0^{x-at} f(\tau)e^{-\frac{\tau}{2a}}\mathrm{d} \tau].$$

My question is how to solve (*) under the inital-boundary value: $$u|_{t=0}=\varphi(x),\ u_t|_{t=0}=\psi(x),\ 0\leq x<\infty;\\ u|_{x=0}=0,\ t\geq 0.$$

I could not find the solution, since I do not know how to determine the $f,g$...

My attempt is as follows: Take $x=0$, then $g(at)=\frac{1}{2a}\int_0^{-at} f(\tau)e^{-\frac{\tau}{2a}}\mathrm{d}\tau$, and thus ??? $u(x,t)=e^\frac{x-at}{2a}[g(x+at)-g(-x+at)$ Right?? Then there exists only one function, which should satisfy $u=\varphi$, $u=\psi$ on $t=0$...I was troubled.

My brower always appear the following error:

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What is the message? How to solve it?

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  • $\begingroup$ What happens if you plug the lengthy formula into the equations of the boundary conditions? $\endgroup$ – user66081 Jan 31 '15 at 11:57
  • $\begingroup$ This is called the 'Telegraph problem'.. Fourier Transforming is the best way to solve it. $\endgroup$ – Mattos Jan 31 '15 at 14:26

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