Fractional Crank-Nicolson-Galerkin finite element methods for nonlinear time fractional parabolic problems with time delay

TL;DR

This paper introduces a new numerical scheme combining fractional Crank-Nicolson, Galerkin finite elements, and extrapolation to efficiently solve nonlinear time fractional parabolic problems with delays, supported by error analysis and numerical validation.

Contribution

The paper develops a novel linearized numerical scheme with a discrete fractional Grönwall inequality for improved error estimation in nonlinear time fractional PDEs with delay.

Findings

The scheme achieves accurate solutions verified by numerical examples.

Error estimates are rigorously derived using the new inequality.

Numerical results demonstrate the method's effectiveness and stability.

Abstract

A linearized numerical scheme is proposed to solve the nonlinear time fractional parabolic problems with time delay. The scheme is based on the standard Galerkin finite element method in the spatial direction, the fractional Crank-Nicolson method and extrapolation methods in the temporal direction. A novel discrete fractional Gr\"{o}nwall inequality is established. Thanks to the inequality, the error estimate of fully discrete scheme is obtained. Several numerical examples are provided to verify the effectiveness of the fully discrete numerical method.