Shifted Chebyshev polynomials based solution of partial differential equations

Abstract

This paper presents an efficient numerical method based on shifted Chebyshev polynomials for solving Partial Differential Equations (PDEs). In this method, a power series solution in terms of shifted Chebyshev polynomials has been chosen such that it satisfies the given conditions. Plugging this series solution into the given PDE and using appropriate collocation points a system of linear equations with unknown Chebyshev coefficients is obtained. Then, unknowns are found with the help of Gauss elimination or Newton’s method. Next, different discretization patterns have also been considered to understand the behavior of the results depending upon the collocation points in the domain. These are the two main modifications and novelty of the procedure and this small contribution in the assumption of power series solution in terms of shifted Chebyshev polynomials results in obtaining the approximate solution with less number of terms with good accuracy. Several numerical examples are provided to confirm the reliability and effectiveness of the proposed method.