A new class of semi-implicit methods with linear complexity for nonlinear fractional differential equations

TL;DR

This paper introduces a new class of semi-implicit methods for nonlinear fractional differential equations that are unconditionally stable, computationally efficient, and have proven effectiveness through numerical simulations.

Contribution

The paper develops a novel semi-implicit scheme with linear complexity for nonlinear fractional differential equations, including stability analysis and an efficient fast convolution strategy.

Findings

Unconditionally stable semi-implicit schemes are developed.

The methods have linear computational complexity.

Numerical simulations confirm effectiveness for complex systems.

Abstract

We propose a new class of semi-implicit methods for solving nonlinear fractional differential equations and study their stability. Several versions of our new schemes are proved to be unconditionally stable by choosing suitable parameters. Subsequently, we develop an efficient strategy to calculate the discrete convolution for the approximation of the fractional operator in the semi-implicit method and we derive an error bound of the fast convolution. The memory requirement and computational cost of the present semi-implicit methods with a fast convolution are about $O(N\log n_T)$ and $O(Nn_T\log n_T)$, respectively, where $N$ is a suitable positive integer and $n_T$ is the final number of time steps. Numerical simulations, including the solution of a system of two nonlinear fractional diffusion equations with different fractional orders in two-dimensions, are presented to verify the effectiveness of the semi-implicit methods.