Applications of Orthogonal Polynomials in Simulations of Mass Transfer Diffusion Equation Arising in Food Engineering

Abstract

In this paper, Chebyshev polynomials—which are ultraspherical in the first and second kind and hence symmetric, while the third and fourth order are not ultraspherical and are hence non-symmetric—are used for the simulation of two-dimensional mass transfer equation arising during the convective air drying processes of food products subject to Robin and Neumann boundary conditions. These simulations are used to improve the quality of dried food products and for prediction of the moisture distributions. The equation is discretized in both temporal and special variables by using the second order finite difference scheme and spectral method based on Chebyshev polynomial with the help of fast Fourier transform on tensor product grid, respectively. A system of algebraic equations is obtained after applying the proposed numerical scheme, which is then solved by an appropriate iterative method. The error analysis of the proposed scheme is provided. Some numerical examples are presented to confirm the numerical efficiency and theoretical justification of the proposed scheme. Our numerical scheme has an exponential rate of convergence, which means that one can achieve a very accurate solution using a few collocation points, as opposed to the other available techniques which are very slow in terms of convergence and consume a lot of time. In order to further validate the accuracy of our numerical method, a comparison is made with the exact solution using different norms.