Hybrid Polynomial Basis Functions for Solving Linear Partial Differential Equations

This study focuses on creating a numerical method that combines two polynomial basis functions to solve certain linear Partial Differential Equations (PDEs). We use a power series solution that includes Legendre and Hermite polynomials, which meet the needs of these PDEs. Then, by plugging this series solution into the PDE and using specific collocation points, we set up a system of linear equations with some unknown coefficients. We solved this using the Gauss elimination method with a computer program. We also looked at different ways to choose these collocation points to see how they affect the results. We tested our method on two cases to check its reliability, effectiveness, and accuracy. Finally, we compared our findings with established results found in other studies.