Stability analysis of a second-order difference scheme for the time-fractional mixed sub-diffusion and diffusion-wave equation

TL;DR

This paper develops and analyzes a second-order stable numerical scheme for a complex time-fractional mixed sub-diffusion and diffusion-wave equation, providing theoretical guarantees and numerical validation.

Contribution

It introduces a second-order accurate, unconditionally stable difference scheme for the time-fractional mixed SDDWE, including a broader class with multi-term derivatives.

Findings

The scheme is unconditionally stable.

Numerical results confirm second-order accuracy.

The method effectively handles multi-term fractional derivatives.

Abstract

This study investigates a class of initial-boundary value problems pertaining to the time-fractional mixed sub-diffusion and diffusion-wave equation (SDDWE). To facilitate the development of a numerical method and analysis, the original problem is transformed into a new integro-differential model which includes the Caputo derivatives and the Riemann-Liouville fractional integrals with orders belonging to (0,1). By providing an a priori estimate of the solution, we have established the existence and uniqueness of a numerical solution for the problem. We propose a second-order method to approximate the fractional Riemann-Liouville integral and employ an L2 type formula to approximate the Caputo derivative. This results in a method with a temporal accuracy of second-order for approximating the considered model. The proof of the unconditional stability of the proposed difference scheme is established. Moreover, we demonstrate the proposed method's potential to construct and analyze a second-order L2-type numerical scheme for a broader class of the time-fractional mixed SDDWEs with multi-term time-fractional derivatives. Numerical results are presented to assess the accuracy of the method and validate the theoretical findings.