# Non constant coefficient in heat equation

I have to solve the following heat equation over a cylindrical domain. In cylindrical coordinates the PDE Equation reads:

 pde = 1/r D[k*r*D[T[t, r, z], {r, 1}], {r, 1}] +
D[k*D[T[t, r, z], {z, 1}], {z, 1}] - \rho*c*D[T[t, r, z], {t}]


The problem is that coefficients k (konductivity), \rho (material density) and c (heat capacity) are in genereral all temperature dependent quantities. According to the FEM Tutorial it should be possible to solve such equations if the PDE is given in the coefficient form Furthermore it has been mentioned in some posts that the inactive form of the pde is needed to make things easier for the FEM solver...

My question is: What is the appropriate form of the obove pde? Do I need to formulate it in cartesian coordinates first instead? I am using Mathematica version 12 by the way. Please Help and thanks in advance. Greets from Germany!

• Have you seen this answer? Have you tried anything. The trouble is going to be (if you use the FEM) that the mass matrices (m or d) can not be time dependent and one would have to write a solver by hand. Not impossible but one would need more information then given here. I can help you do that if you want to. Jul 22 '19 at 5:17