The first thing I should say is that pdepe is designed to solve systems of parabolic and elliptic PDEs. Practically this means that your b term should contain DuDx. Your b term is zero so your equations are hyperbolic. Getting a good solution from pdepe for equations of this type is very challenging.
But that is not the cause of the error you are getting. The error is due to problems with your boundary conditions. A set of boundary conditions consistent with your initial conditions is
pl = ul-[100 390]';
ql = [0;0];
pr = ur-[100 390]';
qr = [0;0];
Probably this is not what you want but pdepe will at least run to completion. The solution is obviously wrong due to the problem I mentioned first.
If you are sufficiently motivated to experiment, you can try the following:
- Get your boundary condition problem sorted out. You probably don't want to specify Dirichlet boundary conditions at both ends.
- Add a small DuDx term to b. This is generally referred to as "artificial diffusion."
- Refine the mesh.