Exploring spectral collocation methods for diffusion models with variable coefficients in heat transfer studies

Abstract

The diffusion equation with variable coefficients is widely applied in heat transfer to model the distribution of temperature in materials with spatially varying thermal properties, allowing for accurate simulation of heat conduction in heterogeneous media, accounting for changes in material composition and thermal conductivity. In this work, an effective method for numerically solving the one-dimensional diffusion equation with variable coefficients is presented. This approach utilizes a hybrid method that combines Lucas and Fibonacci polynomials with finite differences to convert the problem into a time-discrete form using the forward finite difference approach. Then, the derivative of the function is estimated using Fibonacci polynomials. The proposed method is applied to both linear and nonlinear one-dimensional problems, and its efficacy is verified by comparing the obtained results with those of alternative methods. The accuracy and effectiveness of the proposed method are demonstrated through comparison with the exact solution and other methods found in the literature.