PDE: a traveling wave solution to the diffusion equation

[bobcat817]

Homework Statement

Consider a traveling wave u(x,t) =f(x - at) where f is a given function of one variable.
(a) If it is a solution of the wave equation, show that the speed must be $$a = \pm c$$ (unless f is a linear function).
(b) If it is a solution of the diffusion equation, find f and show that the speed a is arbitrary.

Homework Equations

Wave: u(x,t) = f(x+ct) + g(x-ct) and $$u_{tt} = c^2u_{xx}$$

Diffusion: $$u_{t} = ku_{xx}$$ and $$u(x,t) = \frac{1}{\sqrt{4kt\pi}}\int^{\infty}_{-\infty}e^{-(x-y)^2/4kt}\phi(y)dy$$

The Attempt at a Solution

(a) The wave equation is $$u_{tt} = c^2u_{xx}$$ with the general solutions u(x,t) = f(x + ct) + g(x - ct), so for the solution to be u(x,t) = f(x-at), $$a = \pm c$$. If f is linear, then $$u_{tt} =u_{xx}=0$$, so it doesn't matter what a equals.

(b) $$u(x,t) = f(x-at) = \frac{1}{\sqrt{4kt\pi}}\int^{\infty}_{-\infty}e^{-(x-y)^2/4kt}\phi(y)dy$$ I don't know where to go from here.

 

[yyat]

Plug f(x-at) into the diffusion equation. You will get a simple ODE for f', from which you can obtain all the possible f so that f(x-at) satisfies the PDE.