Efficient and fast predictor-corrector method for solving nonlinear fractional differential equations with non-singular kernel

TL;DR

This paper introduces efficient predictor-corrector methods for nonlinear Caputo-Fabrizio fractional differential equations, achieving high accuracy and reduced computational complexity through a recurrence relation-based fast algorithm.

Contribution

The paper presents a novel fast predictor-corrector scheme with improved computational efficiency for nonlinear fractional differential equations with a non-singular kernel.

Findings

Achieves second-order accuracy with linear interpolation.

Achieves third-order accuracy with quadratic interpolation.

Reduces computational complexity to approximately O(N) operations.

Abstract

Efficient and fast predictor-corrector methods are proposed to deal with nonlinear Caputo-Fabrizio fractional differential equations, where Caputo-Fabrizio operator is a new proposed fractional derivative with a smooth kernel. The proposed methods achieve a uniform accuracy order with the second-order scheme for linear interpolation and the third-order scheme for quadratic interpolation. The convergence analysis is proved by using the discrete Gronwall's inequality. Furthermore, applying the recurrence relation of the memory term, it reduces CPU time executed the proposed methods. The proposed fast algorithm requires approximately $O(N)$ arithmetic operations while $O(N^2)$ is required in case of the regular predictor-corrector schemes, where $N$ is the total number of time step. The following numerical examples demonstrate the accuracy of the proposed methods as well as the efficiency; nonlinear fractional differential equations, time-fraction sub-diffusion, and time-fractional advection-diffusion equation. The theoretical convergence rates are also verified by numerical experiments.