I'm trying to solve a parabolic PDE, which the coefficients may depend on location, time and its partial derivatives.
According to the standard parabolic form in Matlab, the coefficients c, a, f and d need to be prepared. From the Matlab document http://www.mathworks.co.uk/help/pde/ug/scalar-coefficients-in-string-form.html, the coefficients can be written in the "string form" with their dependences:
c = 'pde_coeff_c(x,y,t)';
a = 'pde_coeff_a(x,y,t)';
f = 'pde_coeff_f(x,y,t,u,ux,uy)';
d = 1;
with all the corresponding functions.
Boundary condition and geometry are exported from PDE toolbox GUI as myBoundaryMatrix and myGeomMatrix. Therefore, the mesh is generated as:
[p,e,t] = initmesh(myGeomMatrix, 'hmax', 0.03);
Initial condition u0 is initialised as:
u0 = 0;
and time sequence as:
tlist = 0:0.1:1;
Finally, call the parabolic solver:
u = parabolic(u0,tlist,myBoundaryMatrix,p,e,t,c,a,f,d);
With these configurations, I can successfully use the x, y and t variables to organise my coefficients like c and a. However, I cannot use the u, ux or uy while constructing coefficient f. The u, ux and uy are always empty matrices.
I believe the parabolic coefficients should be able to use u, ux and uy, as mentioned in http://www.mathworks.co.uk/help/pde/ug/pde-coefficients.html:
In all cases, the coefficients d, c, a, and f can be functions of position (x and y) and the subdomain index. For all cases except eigenvalue, the coefficients can also depend on the solution u and its gradient. And for parabolic and hyperbolic equations, the coefficients can also depend on time.
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