can FEniCS solve integro-differential equations?

Question

I have a huge set of coupled nonlinear integro-partial differential equations. After a long while trying to simplify the equations and solve them at least semi-analytically I have come to conclude there has been left no way for me but an efficient numerical method. Finite element seems most amenable as is based on Galerkin method which gives a weak form solution, so a great hope that it might finally solve the equations. But at the same time I am so new to this field to write the codes all from the scratch. Then I found FEniCS but everywhere I only read about FEniCS solving PDEs. Now I am interested if FEniCS/Dolphin can also solve integro-differential equations?

Here are the equations in my simplest case, just as example:

wherein, $i=1,2,3$ and $j_1,j_2=1,2,3,4$, thus, the number of unknowns and equations is: $$\overbrace{4}^{\text{components of }U^0}+\overbrace{4^2}^{\text{components of }U^1}+\overbrace{4^3}^{\text{components of }U^2}=84$$

Comments

Good question. Please post an example of your equation.

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@Jan, sorry I couldn't compile my multiline math equations, it doesn't recogize the {align} or {split} environments.

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If you cannot get it to work, you can at least post a .png screenshot.

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@Jan, done, thanks for the suggestion

Answer 1

There was a couple of similar questions on this forum and on deprecated forums.

Number of unknowns is not limited in principle but the dimension of integration domain is effectively limited to 3. This makes FEniCS unsuitable to many variational problems which possess non-local behaviour expressed by convolutions. I believe that these problems often come from integro-differential equations.

There is a chance that you succeed rewriting or approximating the problem to the way that only dimension$\le$3 integration domains are involved. Fixed-point iteration may help sometimes.

Comments

Thanks,
I understand, so Fenics Cannot be a quick solution for me, maybe meanwhile I will try some simple Galerkin formulation. I know I will lose some accuracy but at least it will be a quick solution.