Generalized Lucas Tau Method for the Numerical Treatment of the One and Two-Dimensional Partial Differential Heat Equation

Abstract

This paper is dedicated to proposing two numerical algorithms for solving the one- and two-dimensional heat partial differential equations ( $\text{PDEs}$ ). In these algorithms, generalized Lucas polynomials ( $\text{GLPs}$ ) involving two parameters are utilized as basis functions. The two proposed numerical schemes in one and two- dimensions are based on solving the corresponding integral equation to the heat equation, and after that employing, respectively, the tau and collocation methods to convert the heat equations subject to their underlying conditions into systems of linear algebraic equations that can be treated efficiently via suitable numerical procedures. In this article, the convergence analysis is examined for the proposed generalized Lucas expansion. Five illustrative problems are numerically solved via the two proposed numerical schemes to show the applicability and accuracy of the presented algorithms. Our obtained results compare favourably with the exact solutions.