New Numerical Aspects of Caputo-Fabrizio Fractional Derivative Operator

Sania Qureshi,Dumitru Baleanu,Norodin A Rangaig

doi:10.3390/math7040374

Abstract

In this paper, a new definition for the fractional order operator called the Caputo-Fabrizio (CF) fractional derivative operator without singular kernel has been numerically approximated using the two-point finite forward difference formula for the classical first-order derivative of the function f (t) appearing inside the integral sign of the definition of the CF operator. Thus, a numerical differentiation formula has been proposed in the present study. The obtained numerical approximation was found to be of first-order convergence, having decreasing absolute errors with respect to a decrease in the time step size h used in the approximations. Such absolute errors are computed as the absolute difference between the results obtained through the proposed numerical approximation and the exact solution. With the aim of improved accuracy, the two-point finite forward difference formula has also been utilized for the continuous temporal mesh. Some mathematical models of varying nature, including a diffusion-wave equation, are numerically solved, whereas the first-order accuracy is not only verified by the error analysis but also experimentally tested by decreasing the time-step size by one order of magnitude, whereupon the proposed numerical approximation also shows a one-order decrease in the magnitude of its absolute errors computed at the final mesh point of the integration interval under consideration.

Highlights

Differential equations of arbitrary real order ν > 0 are used to model various physical models arising in many branches of science and engineering
The present paper fundamentally aims to propose a numerical approximation for the Caputo-Fabrizio (CF) operator using a two-point finite difference formula for the f 0 (t) term, as well as offer a discussion of error analysis associated with the proposed approximation
We introduced the numerical approximation of a fractional derivative using the CF operator where two-point approximation is utilized and the truncation error of the two-point finite difference approach has been proven to have an accuracy which is dependent on the fractional order

Summary

Introduction

Differential equations of arbitrary real order ν > 0 are used to model various physical models arising in many branches of science and engineering. Based upon the error analysis discussed, the current work shows that the local truncation error term of the approximation consists of a positive constant that depends upon the fractional operator ν, leading to the expression of the form O(h), which proves the first-order convergence of the proposed numerical approximation. We present some important definitions used in the present study, along with a few of the properties associated with fractional derivative operators that need to be known at this stage

Mathematical Preliminaries

Proposed Numerical Approximation

Temporal Mesh for Two-Point Finite Difference

Results and Discussion

Conclusions