Parabolic and system of PDE (4th order PDE)
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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
==============
No ideas?
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