Parabolic and system of PDE (4th order PDE)

Hello everyone! I’m interested in solving nonlinear Gradient Flow problems using the built-in parabolic function, even if the problem is 1D at this step. Let consider the following functional:

whose gradient flow takes the form:

While developing this toy model, I'm using a manufactured solution, then, whenever the solution u(x) is known, the forcing term is:

In order to solve this problem, in Matlab I set the following coefficients:

d = 1;
c = 1;
a = 0;
f = ‘fun(x,u)’;

and

function R = fun(x,u)
R = -f(x) - exp(u);

Moreover, Dirichlet boundary conditions are imposed at the edges:

The code works very well.

Let now consider another functional:

whose Gradient Flow equation is:

Also in this case, a manufactured solution can be used by setting:

In 1D, I would write:

whereas in 2D I would write:

or in a form suitable for the parabolic function:

and in matrix form:

For a system of PDE Matlab wants the form:

then I set the parameters as:

d = [1;0;0;1];
c = [0;0;1;0;...
     0;0;0;1;...
     -1;0;0;0;...
     0;-1;0;0];
a = [0;0;0;1];
f = char('fun(x)','0');

and impose Dirichlet BCs at edge on both functions:

and

But something is wrong, and I cannot understand what. The attached zip file contains all the codes.

Hope someone can help me. Thanks in advance for all your precious suggestions.

Best,

Pietro

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No ideas?

[Alan Weiss]

I cannot see any of your equations. They all show up as missing figures.

[Pietro Pollaci]

Dear Alan,

thanks for your reply. The images were on my dropbox public folder in pdf format, now they are png.

Unfortunately, now they look very huge. Sorry for the inconvenience.