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

This paper introduces an efficient numerical method based on applying the typical Petrov–Galerkin approach (PGA) to solve the time fractional diffusion wave equation (TFDWE). The method utilises asymmetric polynomials, namely, shifted second-kind Chebyshev polynomials (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.