Hi,

I have the question

Use eigenfunction expansion to solve

$\displaystyle u_t = u_{xx} + q(x,t)$

with IC: $\displaystyle u(x,0) = f(x)$ and BC's: $\displaystyle u(0,t) = u_0$ and $\displaystyle u(\pi,t) = u_{\pi}$ where $\displaystyle u_0$ and $\displaystyle u_{\pi}$ are given constants.

With this method, I am supposed to start with a trial solution based on

$\displaystyle u_t = u_{xx}$ which based on the boundary conditions I would guess to be

$\displaystyle \displaystyle u(x,t) = \sum_1^{\infty} b_n (t) \sin (nx) $

but when checking this against the boundary conditions I find that $\displaystyle u(0,t) = u(\pi,t) = 0$ and do not meet the boundary conditions. What am I missing here?

Thanks