A Bivariate Chebyshev Integral Collocation Method for Solving One-Dimensional Heat Equation with Initial and Dirichlet Boundary Conditions

Abstract

Partial differential equations (PDEs) play a vital role in modeling diverse physical and engineering processes, such as heat conduction and diffusion. However, obtaining analytical solutions is often difficult or impossible, which motivates the development of efficient numerical techniques. This paper presents a novel bivariate Chebyshev integral collocation method (BCICM) for solving the one-dimensional heat equation on the domain with initial and Dirichlet boundary conditions. In the proposed method, the second-order spatial derivative in the governing PDE is approximated using a truncated bivariate shifted Chebyshev series of the first kind. By performing twofold integration, the approximate solution is reassembled while arbitrary integration functions are determined through boundary conditions. The time derivative is then evaluated analytically and substituted back into the heat equation, yielding an algebraic system for the unknown Chebyshev coefficients. Chebyshev-Gauss (CG) nodes in space and Chebyshev–Gauss–Lobatto (CGL) nodes in time are selected as collocation points to ensure the initial condition is imposed at The entire procedure is implemented using MATLAB to efficiently compute the Chebyshev coefficients and approximate solutions. Three Numerical examples are presented to demonstrate the efficiency, rapid convergence, and accuracy of the proposed method.