- #1
Panphobia
- 435
- 13
Homework Statement
$$u_{t}=u_{xx}+f(x) \\ u(0,t)=50 \\ u(\pi , t)=0 \\ u(x,0)=g(x)$$
$$0<x<\pi \\ t>0$$
$$f(x)=\begin{cases}
50 & 0<x<\frac{\pi}{2} \\
0 & \frac{\pi}{2}\leq x< \pi
\end{cases}$$
$$g(x)=\begin{cases}
0 & 0<x<\frac{\pi}{2} \\
50 & \frac{\pi}{2}\leq x< \pi
\end{cases}$$
So what I tried to do here is use the principle of superposition to split this problem up into two different problems ##m(x,t),n(x,t)##.
$$m_t=m_{xx} \\ m(0,t)=50 \\ m(\pi,t)=0 \\ m(x,0) = g(x)$$
and
$$n_t=n_{xx}+f(x) \\ n(0,t)=0 \\ n(\pi,t)=0 \\ n(x,0) =0$$I know how solve the first PDE easily, but the second one is giving me some trouble. I know that you are supposed to do a change of variables and then solve it that way, but how do you take care of the piecewise function ##f(x)## when you are transforming back from the change of variables? Will you just have a piecewise solution in the end?