Nonlinear time dependent 2D system of PDE's

I am wondering if it is possible to solve the following system of time-dependent, nonlinear, 2D PDE's:

where and are constants. I have tried with the pde toolbox, but it is not able to produce a solution. I am implementing this with a rectangular geometry, and boundary conditions:

on all four boundaries of the geometry. Initial conditions are .

For some background, I am trying to implement the H-formulation, which is a numerical method for modeling high temperature superconductors. I am trying to model the magnetic field through a cross sectional area of superconductor.

Answer 1

[Torsten]

I doubt there is a software capable to handle this kind of PDE system automatically.

You will have to discretize the spatial derivatives and solve the resulting system of ordinary differential equations with values for u1 and u2 in the grid points by using MATLAB's "ode15s".

Look up "method-of-lines" for more details.