How to solve these kind nonhomogeneous heat equation

Question

Solve $$\begin{cases} u_{t}-u_{xx}=\frac{1}{2}xt \ \ &\text{for} \ \ 0\lt x\lt \pi, t\gt0\\ u(0,t)=u(\pi,t)=0 \ \ \\ u(x,0) = \sin x \ \ \\ \end{cases}$$

How to transform it to the normal case?

Answer 1

Try to find a solution of the form

$$ u(x,t) = \sum_n T_n(t)X_n(x) $$

where

\begin{cases} X'' = -\lambda^2 X \\ X(0) = X(\pi) = 0 \end{cases}

You'll get

$$ u(x,t) = \sum_{n=1}^\infty T_n(t)\sin(nx) $$

and the PDE becomes

$$ \sum_{n=1}^\infty\big[T_n'(t) + n^2T_n(t)\big]\sin(nx) = \frac12 tx $$

Decompose the RHS into the corresponding Fourier series

$$ \frac12 tx = \sum_{n=1}^\infty a_n(t) \sin(nx) $$

You'll get a family of IVPs

\begin{cases} T_n''(t) + n^2 T_n(t) = a_n(t) \\ T_1(0) = 1 \\ T_{n>1}(0) = 0 \end{cases}