A Fast Compact Finite Difference Method for Fractional Cattaneo Equation Based on Caputo–Fabrizio Derivative

Zhengguang Liu,Haili Qiao,Aijie Cheng

doi:10.1155/2020/3842946

Abstract

The Cattaneo equations with Caputo–Fabrizio fractional derivative are investigated. A compact finite difference scheme of Crank–Nicolson type is presented and analyzed, which is proved to have temporal accuracy of second order and spatial accuracy of fourth order. Since this derivative is defined with an integral over the whole passed time, conventional direct solvers generally take computational complexity of OMN2 and require memory of OMN, with M and N the number of space steps and time steps, respectively. We develop a fast evaluation procedure for the Caputo–Fabrizio fractional derivative, by which the computational cost is reduced to OMN operations; meanwhile, only OM memory is required. In the end, several numerical experiments are carried out to verify the theoretical results and show the applicability of the fast compact difference procedure.

Highlights

Fractional diffusion equations have become a strong and forceful tool to describe the phenomenon of anomalous diffusion, and more research works have been obtained in the last decades [1,2,3,4,5,6]
We develop and analyze a fast compact finite difference procedure for the Cattaneo equation equipped with time-fractional derivative without singular kernel. e timefractional derivative is of Caputo–Fabrizio type with the order of α(1 < α < 2)
Compact difference discretization is applied to obtain a high-order approximation for spatial derivatives of integer order in the partial differential equation, and the Caputo–Fabrizio fractional derivative is discretized by means of Crank–Nicolson approximation

Summary

Introduction

Fractional diffusion equations have become a strong and forceful tool to describe the phenomenon of anomalous diffusion, and more research works have been obtained in the last decades [1,2,3,4,5,6]. I.e., storage requirement and computation cost of an algorithm, researchers devote themselves to reduce storage requirement and computational time by analyzing the particular structure of coefficient matrices arising from the discretization system or reutilizing the intermediate data skillfully We call these algorithms fast methods, including fast finite difference methods [23,24,25,26,27,28], fast finite element methods [29], and fast collocation methods [30, 31].

Some Notations and Definitions

Stability Analysis and Optimal Error Estimates

Compact Finite Difference Scheme in Two Dimensions

Stability Analysis

Numerical Experiments

Findings

Conclusion