Heat equation with sin initial condition

Question

How do i find the analytical solution of the heat equation:

$$U_t = U_{xx} + \sin{\pi x}$$ subject to $u(0,t) = u(1,t) = 0$ and $u(x,0) = \sin(\pi x).$

I appreciate its a pretty common/general question but i have searched and cannot find any articles explaining this very well. So either an explanation or a link to a good explanation will be appreciated

Thanks, James.

Answer 1

Note that the steady-state solution of the inhomogeneous equation with your boundary conditions is $u_p(x) = \sin(\pi x)/\pi^2$.

If $u_h(x,t)$ solves the homogeneous equation with your boundary conditions and initial condition $u(x,0) = (what?)$, then $u(x,t) = u_p(x) + u_h(x,t)$ is your desired solution.

Fortunately the homogeneous equation has some nice solutions of the form $ u_h(x,t) = \sin(\pi x) T(t)$ (what is $T(t)$?).