- Research
- Open Access
Numerical approximation of the space-time Caputo-Fabrizio fractional derivative and application to groundwater pollution equation
- Abdon Atangana^{1} and
- Rubayyi T Alqahtani^{2}Email author
https://doi.org/10.1186/s13662-016-0871-x
© Atangana and Alqahtani 2016
- Received: 17 April 2016
- Accepted: 24 May 2016
- Published: 14 June 2016
Abstract
Recently, Caputo and Fabrizio proposed a new derivative with fractional order without singular kernel. The derivative has several interesting properties that are useful for modeling in many branches of sciences. For instance, the derivative is able to describe substance heterogeneities and configurations with different scales. In order to accommodate researchers dealing with numerical analysis, we propose a numerical approximation in time and space. We modified the advection dispersion equation by replacing the time derivative with the new fractional derivative. We solve numerically the modified equation using the proposed numerical approximation. The stability and convergence analysis of the numerical scheme were presented together with some simulations.
Keywords
- Caputo-Fabrizio derivative
- numerical approximation
- advection diffusion equation
- stability analysis
1 Introduction
In the last decade, many physical problems have been modeled using the concept of noninteger-order derivative. These derivatives of fractional order range from Riemann-Liouville via Caputo to Caputo-Fabrizio [1, 2]. We can find in the literature many analytical approaches to deal with differential equations with fractional equations [3–10]. Most of these techniques are dealing with linear fractional differential equations. However, most fractional differential equations describing real-world problems are highly complicated and cannot sometime be handled via analytical methods. In order to solve these problems in many cases, researchers rely on the use of numerical methods because these problems have initial conditions, boundary condition, and source terms that turn hard to find an analytical solution.
Several numerical approaches in connection with derivatives of fractional order describing real-world problems alter essentially in the many in which the derivative of fractional order is tailored [11–24]. Approximation representation of a derivative of fractional order has a highly complicated formula compared to those of integer order because fractional derivatives are nonlocal, and therefore the calculation at a particular point requires knowledge of the function further out of the region close to that point. Accordingly, finite difference approximations of derivatives of fractional order engage a quantity of points that alters according to how faraway we are from the border line [12–14].
The most recent derivative of fractional order was proposed by Caputo and Fabrizio [1], who demonstrated that the new-fangled derivative encompasses extra encouraging properties in comparison with the old version. They demonstrated, for example, that it can depict substance heterogeneities and configurations with different scales, which obviously cannot be overseeing with the prominent local theories and also the well-known fractional derivative. An additional application is in the investigation of the macroscopic behaviors of some materials, associated with nonlocal communications between atoms, which are recognized to be important of the properties of material. We present the definition of the Caputo fractional derivative.
Definition 1
Definition 2
The aim of this paper is to propose a numerical approximation of the space and time Caputo-Fabrizio derivative of fractional order that will be used by researchers in the field of numerical analysis.
2 Caputo-Fabrizio approximations
Theorem 1
Proof
Theorem 2
Proof
3 Application to some well-known equations
In this section, we present a numerical solution of the time fractional advection diffusion equation in heterogeneous medium. The fractional derivative used here is of the Caputo-Fabrizio type.
4 Stability analysis of the numerical scheme
Theorem 3
Proof
5 Convergence analysis of the numerical solution
Theorem 4
6 Numerical simulation for different values of alpha
7 Conclusions
We have proposed in this work the numerical approximation of the newly proposed derivative of fractional order in order to fit this derivative in the scope of numerical investigations. The new derivative is easy to use even numerically and display important characteristics that cannot be observed in the commonly used fractional derivatives. In order to test the possible application of the new numerical approximation of the new Caputo-Fabrizio derivative of fractional order, we presented a model of advection diffusion equation with the time fractional of the new derivative. We solved this equation numerically using the Crank-Nicolson technique. We showed the stability analysis together with some numerical simulations for different values of alpha.
Declarations
Acknowledgements
We acknowledge the Editorial Board and the referees for their efforts and constructive criticism, which have improved the manuscript.
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Authors’ Affiliations
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