I have the pde $U_t(x,t)=U_{xx}(x,t)-b(t)U(x,t)+q_0$, where $b(t)$ is a function of only $t$ and $q_0$ is a constant, $0<x<\pi$, $t>0$. I also have some boundary conditions: $U(x,0)=0$ and $U(0,t)=0=U(\pi,t)$. The problem solution using variation of parameters is $$U(x,t)=\frac{4q_0}{\pi a(t)}\sum\limits_{n=1}^\infty \frac{\sin((2n-1)x)}{2n-1}\int\limits_0^t e^{-(2n-1)^2(t-\tau)}a(\tau)d\tau$$ where $$a(t)=e^{\int\limits_0^t b(\sigma)d\sigma)}.$$
What I would like to know is how I can solve this using maple. And, if possible use it to solve a slightly modified version of the above: $U_t(x,t)=U_{xx}(x,t)-bU(x,t)+q(t)$, where $b$ is now a constant and $q$ is a function of only $t$, $0<x<\pi$, $t>0$.
If someone could give me some commands to get started or a maple worksheet, that would be awesome. Thanks! ^_^
The best place for Maple questions is Maple Primes but to answer your question:
$$pde := diff(U(x, t), t)-(diff(U(x, t), x, x))+b*U(x, t)+q(t);$$ $$pdsolve(pde);$$
I suspect Maple can not handle all your initial conditions. Maple returns an answer for pdsolve({pde,U(x,0)=0}) but if you try pdsolve({pde,U(x,0)=0,U(0,t)=0}) you receive an empty answer (either there is no solution or Maple can not compute it).