Advances In Hybrid Polynomial Basis Functions: Novel Numerical Approach for Partial Differential Equations

Abstract

This scholarly investigation concentrates on the formulation of an optimized numerical methodology predicated upon the amalgamation of two polynomial basis functions, aimed at resolving a specific subset of linear and nonlinear Partial Differential Equations (PDEs). In this methodological framework, a solution derived from power series is utilized, incorporating Chebyshev and Hermite polynomials to satisfy the particular requirements imposed by the PDE. Subsequently, by substituting the series solution into the designated PDE and employing appropriate collocation points, a linear system of algebraic equations featuring indeterminate hybridization coefficients was established, with its resolution achieved through the Gauss elimination method (GEM) utilizing numerical computation software. Moreover, various discretization schemes were investigated to elucidate the manner in which the results fluctuate in response to variations in the allocation of collocation points throughout the domain. Two illustrative cases were scrutinized employing the numerical technique to assess the method’s efficacy in terms of reliability, effectiveness, and precision. The findings obtained were subjected to benchmarking and validation against established results within the existing body of literature.