Modified Chebyshev Spectral Algorithm for Solving Heat-Transfer Equations in Heterogeneous Media

Abstract

Heat-transfer equations represent fundamental models in the study of heat and mass transfer in physical and engineering media, especially in cases characterized by the heterogeneity of the medium due to spatial changes in material properties such as thermal conductivity. Traditional numerical methods—such as finite difference and finite element methods—face significant challenges when dealing with such media, due to low accuracy and difficulty in representing sharp changes in parameters. This study aims to develop a modified spectral algorithm based on modified Chebyshev polynomials to improve the accuracy and efficiency of the numerical solution for heat-transfer equations in heterogeneous media. The proposed approach reformulates the mathematical model within Chebyshev space, adjusting the basis functions to accommodate the spatial changes of the diffusion coefficient K(X), enabling a stable and rapidly solvable linear system matrix. Numerical results demonstrated high accuracy in approximating the true solution with distinctive convergence speed compared to traditional methods, in addition to a significant reduction in numerical error and execution time. The results confirm the efficiency of the proposed algorithm in solving problems involving heterogeneous media, making it a promising tool for application in materials engineering, heat transfer in composites, and high-precision physical modeling.