Numerical Treatment of the Time Fractional Diffusion Wave Problem Using Chebyshev Polynomials

Abstract

This paper introduces an efficient numerical method based on applying the typical Petrov–Galerkin approach ($\mathrm{PGA}$) to solve the time fractional diffusion wave equation ($\mathcal{TFDWE}$). The method utilises asymmetric polynomials, namely, shifted second-kind Chebyshev polynomials ($\mathrm{SSKCPs}$). New derivative formulas are derived and used for these polynomials to establish the operational matrices of their derivatives. The paper presents rigorous error bounds for the proposed method in Chebyshev-weighted Sobolev space and demonstrates its accuracy and efficiency through several illustrative numerical examples. The results reveal that the method achieves high accuracy with relatively low polynomial degrees.