Application of Chebyshev collocation method for solving two classes of non-classical parabolic PDEs

This article contributes a numerical scheme for finding approximate solutions of one-dimensional parabolic partial differential equations (PDEs) under non-classical boundary conditions. This scheme is based on the direct Chebyshev collocation method that has been frequently used for problems of ordinary differential equations (ODEs). In fact, the approximate solution of the problem in the truncated Chebyshev series form is obtained by this method. By substituting Chebyshev series solution into the considered problems and by using the matrix operations and the collocation points, the suggested scheme reduces the problems into the associated linear algebraic systems. By solving this system of equations, the unknown Chebyshev coefficients can be determined. To show the accuracy and the efficiency of the method, two numerical examples are implemented and the comparisons are given by a new collocation method.