Fully discrete Tau solution for some types of non-local heat transport equations

In this paper, orthogonal functions are constructed based on orthogonal polynomials using Kronecker product. In this regard, we present a general formulation for the two-dimensional orthogonal functions and their derivative matrices. These matrices are used in the fully discrete Tau method, on both space and time variables, to reduce the solution of the parabolic partial differential equation (heat conduction) subject to given initial and non-local boundary conditions to the solution of a system of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the presented technique.