Diffusion equation with Dirichlet BC

[obnoxiousris]

Homework Statement

Solve u_t=ku_xx
u(x,0)=0 (ψ(x))
u(0,t)=1
on the half line 0<x<infinity
(exercise 2, 3.1, Strauss)

Homework Equations

The Attempt at a Solution

There are two bits I don't get:

First, I know I have to make an odd or even extension to the whole line. But for both even and odd extensions, the formulae are:
u(x,t)=1/(√4∏kt)(∫(e^-(x-y)2/4kt±e^-(x+y)2/4kt)ψ(y)dy)
this means for u(x,0)=0, everything is multiplied by 0, which clearly isn't right.

Second, I don't know which extension to apply, it seems like both should work: I can either make an odd extension about the line x=1 or an even extension about the t axis, both will intercept the t axis at 1, hence satisfying the BC.

Any help would be greatly appreciated!

 

[mighty2000]

Thank you for your question. I believe I can provide some insights to help you solve this problem.

Firstly, for the odd or even extension, you are correct in saying that both should work. This is because the boundary condition u(0,t)=1 is symmetric, meaning it is satisfied by both even and odd functions. Therefore, you can choose whichever extension method you are more comfortable with.

Now, for the u(x,0)=0 boundary condition, you are correct in saying that the integral will become 0. However, this does not mean that the solution will be 0. This is because the integral is multiplied by 1/√4∏kt, which means that the solution will still have a non-zero value. Additionally, the integral also includes the function ψ(y), which will have an impact on the solution.

I hope this helps clarify things for you. If you have any further questions, please don't hesitate to ask. Good luck with your problem solving!