I need to solve the heat equation:
$$d_tu = \nabla^2u$$ with $0 < x < \infty$ and $0 < y < \pi$.
Boundary conditions: $u(0,y,t) = u(x,0,t) = u(x,\pi,t) = 0$ and $|u(x,y,t)| \rightarrow 0$ as $x$ approaches $\infty$.
I used separation of variables and got the eigenfunction values for $Y$. But now I'm on the $X$ part with
$$X''(x) + \lambda X(x) = 0$$ But this doesn't seem to work because of the infinity boundary condition. Am I approaching this the wrong way?
Thanks!