A Comparative Study on Polynomial Expansion Method and Polynomial Method of Particular Solutions

Abstract

In this study, the polynomial expansion method (PEM) and the polynomialmethod of particular solutions (PMPS) are applied to solve a class of linear ellipticpartial differential equations (PDEs) in two dimensions with constant coefficients. Inthe solution procedure, the sought solution is approximated by the Pascal polynomialsand their particular solutions for the PEM and PMPS, respectively. The multiple-scaletechnique is applied to improve the conditioning of the resulted linear equations andthe accuracy of numerical results for both of the PEM and PMPS. Somemathematicalstatements are provided to demonstrate the equivalence of the PEM and PMPS basesas they are both bases of a certain polynomial vector space. Then, some numericalexperiments were conducted to validate the implementation of the PEM and PMPS.Numerical results demonstrated that the PEM is more accurate and well-conditionedthan the PMPS and the multiple-scale technique is essential in these polynomial methods.