Approximate Petrov-Galerkin Solution for the Time Fractional Diffusion Wave Equation

This paper discusses the Petrov-Galerkin method's application in solving the time fractional diffusion wave equation (TFDWE). The method is based on using two modified sets of shifted fourth-kind Chebyshev polynomials (FKCPs) as basis functions. The explicit forms of all spectral matrices were reported. These forms are essential to transforming the TFDWE and its underlying homogeneous conditions into a matrix system. An appropriate algorithm can be used to solve this system to obtain the desired approximate solutions. The error analysis of the method was studied in depth. Four numerical examples were provided that included comparisons with other existing methods in the literature.