Strong heat equation problem

Hello, I am trying to solve the following problem in a rectangle with Dirichlet conditions at the boundary. I know that the PDE solver exists in matlab, but I would like to know if anyone has an implementation of their own since I am interested in how this works. I have found this implementation, but it is not the same problem and in general I only find implementations for the Poisson problem, which is like a particular case where k = 1 and I would like to get an implementation to solve the general problem. First of all, Thanks