I am facing a quite complicated PDE for 
in the form over a 2D disk. I am using the PDEtoolbox and, because of the next step is an even more complicated equation, to obtain the corefficients d,a,c,f I wrote the symbolic expression of my system and then I used the pdeCoefficients
symEq1 = -diff(u1,t) + a*u1 - 2*b*((u1)^2+(u2)^2)*u1 + kappa*(diff(u1,x,x)+diff(u1,y,y))+...
((diff(u1,x)^2) - (diff(u1,y)^2) + diff(u2,x)*diff(u1,y) + diff(u1,x)*diff(u2,y));
symEq2 = -diff(u2,t) + a*u2 - 2*b*((u1)^2+(u2)^2)*u2 + kappa*(diff(u2,x,x)+diff(u2,y,y))+...
(diff(u1,x)*diff(u2,x) - diff(u1,y)*diff(u2,y) + 2*diff(u2,x)*diff(u2,y));
symCoeffs = pdeCoefficients(pdeeq,[u1 u2],'Symbolic',true)
structfun(@disp,symCoeffs);
coeffs = pdeCoefficientsToDouble(symCoeffs,[u1 u2])
specifyCoefficients(model,'m',0,'d',coeffs.d,'c',coeffs.c,'a',coeffs.a,'f',coeffs.f)
When i run the code I receive the warning:
Warning: After extracting m, d, and c, some gradients remain. Writing all remaining terms to f.
> In sym/pdeCoefficients>extractAF (line 122)
In sym/pdeCoefficients (line 96)
In eqL2 (line 79)
Reading the documentation I found that this warning make the toolbox inable to solve the equation so I have some questions:
- Are the finite element matrixes reliable even if I receive this warning? Could I extract and use them to solve the PDE in a different way?
- In the first page of documentation I found that "The coefficients m, d, c, a, and f can be functions of location (x, y, and, in 3-D, z), and, except for eigenvalue problems, they also can be functions of the solution u or its gradient." so I thought that my case was fine given I only have gradients of u. But in another page I read "The coefficients a, c, and f are functions of position (x, y, z) and possibly of the solution u." where nothing is said about the gradient. Which one is correct? Is there a way to solve this problem with the toolbox?
Here the links of the two doc pages I referred:
https://it.mathworks.com/help/symbolic/sym.pdecoefficients.html
https://it.mathworks.com/help/pde/ug/equations-you-can-solve.html
https://it.mathworks.com/help/pde/ug/put-equations-in-divergence-form.html
