Symmetric fractional order reduction method with $L1$ scheme on graded mesh for time fractional nonlocal diffusion-wave equation of Kirchhoff type

TL;DR

This paper introduces a fully-discrete numerical scheme combining finite element and $L1$ methods to solve a time fractional nonlocal diffusion-wave equation of Kirchhoff type, with proven stability and error estimates.

Contribution

It develops a linearized scheme with an $ extit{a priori}$ bound and error estimate for a complex fractional PDE, verified through numerical experiments.

Findings

The scheme is stable and convergent under certain conditions.

Numerical results confirm theoretical error estimates.

The method effectively handles fractional derivatives of order between 1 and 2.

Abstract

In this article, we propose a linearized fully-discrete scheme for solving a time fractional nonlocal diffusion-wave equation of Kirchhoff type. The scheme is established by using the finite element method in space and the $L1$ scheme in time. We derive the $\alpha$-robust \textit{a priori} bound and \textit{a priori} error estimate for the fully-discrete solution in $L^{\infty}\big(H^1_0(\Omega)\big)$ norm, where $\alpha \in (1,2)$ is the order of time fractional derivative. Finally, we perform some numerical experiments to verify the theoretical results.