Research Article Generalized Lucas Tau Method for the Numerical Treatment of the One and Two-Dimensional Partial Differential Heat Equation Y. H. Youssri , 1 W. M. Abd-Elhameed , 1 and S. M. Sayed 2 1 Department of Mathematics, Faculty of Science, Cairo University, Giza 12613, Egypt 2 Department of Mathematics, Faculty of Science, Helwan University, Cairo 11795, Egypt Correspondence should be addressed to Y. H. Youssri; youssri@aucegypt.edu Received 20 January 2022; Revised 27 March 2022; Accepted 30 March 2022; Published 25 April 2022 Academic Editor: Baowei Feng Copyright © 2022 Y. H. Youssri et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. This paper is dedicated to proposing two numerical algorithms for solving the one- and two-dimensional heat partial differential equations (PDEs). In these algorithms, generalized Lucas polynomials (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. 1. Introduction Many mathematical models of real-world problems give rise to partial differential equations (PDEs) of initial and bound- ary conditions. PDEs are frequently represented as mathe- matical equations that connect various amounts and their derivatives, e.g., heat transition, a particle’s movement in a straight line, the movement of a rocket, a molecule’s vibra- tion, and a change in a substance’s molecular composition, etc. Every one of these issues is represented by hyperbolic, elliptic, or parabolic partial differential equation (PPDE) and might be homogeneous, in one, two, or three dimen- sions, with non-local boundary conditions in addition to the initial conditions found in the prose. A parabolic PDE is used to solve a variety of scientific problems, including ocean acoustic propagation as well as heat diffusion. The hyperbolic PDE indicates the wave transformation and sound waves of an elastic string, whereas the elliptic PDE describes the Laplace equation. Fibonacci and Lucas polynomial sequences are crucial and they play vital roles in various disciplines. These sequences are employed to find approximate solutions of different types of DEs. For instance, Fibonacci polynomials were used to treat multi-term fractional DEs in [1]. In [2], Lucas polynomials are employed for the numerical treat- ment of sinh-Gordon equation. The authors in [3] devel- oped a matrix method using Fibonacci polynomials for the treatment of the generalized pantograph equations with functional arguments. Another approach based on mixed Fibonacci and Lucas polynomials is followed in [4] to obtain numerical solutions of Sobolev equation in two dimensions. Lucas polynomials are employed in [5] to obtain numerical solutions of multidimensional Burgers-type equations. Lucas polynomials were also employed in [6] to solve the fractional-order electro-hydrodynamics flow model. The Fibonacci and Lucas sequences can be generalized. For example, the authors in [7, 8] introduced two general- ized families of Fibonacci and Lucas polynomials. In addi- tion, they employed such generalized sequences to treat some fractional differential equations. It is well-known that the heat equation is a parabolic PDE that describes the distribution of heat. There are two types of heat equations: non-homogeneous and homoge- neous. Non-homogeneous heat equations have source terms Hindawi Journal of Function Spaces Volume 2022, Article ID 3128586, 13 pages https://doi.org/10.1155/2022/3128586