Thread: diffusion equation

Good morning all, I am stuck on this question.

Given u(x,t)=f(x-at) is a solution to the diffusion equation, find f and show that the speed is arbitrary.
Thanks for any help at all guys and girls.

Did you mean wave equation? If so, I'm not sure you can determine f uniquely. You're actually exhibiting one of the two d'Alembert solutions, where f is arbitrary, but a is not. The other d'Alembert solution looks like f(x+at).

Originally Posted by Ackbeet

Did you mean wave equation? If so, I'm not sure you can determine f uniquely. You're actually exhibiting one of the two d'Alembert solutions, where f is arbitrary, but a is not. The other d'Alembert solution looks like f(x+at).

No, I mean the diffusion equation . That's why this question confuses me.

Hmm. Well, I'm not sure I see my way through to the end of the problem, but plug in your f into the diffusion equation. What do you get?

I get ,
so

I think I am confusing myself.

So now suppose s = x - at. You have an ODE for f, right? I agree with your first equation, but not the second. How did you get that?

Sorry, that second was a mistake, makes no sense.

-kf''(s)-af'(s)=0
kf''(s)+af'(s)=0

So the characteristic equation is kz^2+az=0
which gives z=0 or z=-a/k

Is this correct?

Originally Posted by kittycat485

Sorry, that second was a mistake, makes no sense.

-kf''(s)-af'(s)=0
kf''(s)+af'(s)=0

So the characteristic equation is kz^2+az=0
which gives z=0 or z=-a/k

Is this correct?

Yes, so far.

So the solution is...

Originally Posted by Ackbeet

Yes, so far.

So the solution is...

got it. thanks.

You're very welcome. Have a good one!