Help solving this 2d pde

I'am trying to solve this 2d pde on

note that in the rhs u is evaluated at which we can consider as a deviation

where . I used a uniform discretization and got:

As an example we can take and BC

I don't see how to generate the system of equations and corresponding matrix.

Any help or other method to investigate is appreciated.

Answer 1

[Bjorn Gustavsson]

You are in luck! Matlab provides you with a (rather hidden) method to give you the 2-D Laplacian in matrix form. Have a look at numgrid.

qwe = delsq(numgrid('S',7)); 
imagesc(full(qwe))          

Also have a look at delsqdemo.

For the nonlinear part you might have to look at the different iterative solvers, pcg, qmr, etc (these tools are outside of my commonly used functions...)

HTH

[Clifford]

Thanks for your answer...

The l.h.s is linear so we can consider it as where A is a matrix,

the r.h.s is nonlinear, it reads .

Now we have to solve ,

do You think that it will be possible to solve it using matlab fsolve ?

or any more suitable numerical method (Newton method for example) ?