Numerical Scheme for Modified Anomalous Time-Fractional Sub-Diffusion Equations Using the Shifted Dickson Polynomials of the Second Kind

Abstract

This paper develops a numerical algorithm for treating the modified anomalous time-fractional sub-diffusion problems (MAFSDPs). The proposed numerical algorithm relies on the tau method. The basis functions, namely, shifted Dickson polynomials of the second kind, are employed to obtain the proposed numerical solutions. Many theoretical formulas of the Dickson polynomials of the second kind and their shifted polynomials, such as the linearization formula, derivative relations, and some specific definite integrals, are developed. These formulas will serve as a fundamental basis for designing our proposed numerical algorithm. The approximate solution is expressed as a truncated double expansion in shifted Dickson basis functions. The utilization of the tau method transforms the equation, along with its underlying conditions, into a system of algebraic equations that can be numerically treated. Rigorous convergence of the double-shifted expansion is studied. Numerical examples are included to verify the accuracy and applicability of the proposed algorithm. In addition, comparisons with some existing numerical methods are presented to confirm the superior performance of our algorithm.