Approximation of the Riesz–Caputo Derivative by Cubic Splines

Francesca Pitolli,Chiara Sorgentone,Enza Pellegrino

doi:10.3390/a15020069

Francesca Pitolli, Chiara Sorgentone + Show 1 more

Open Access

https://doi.org/10.3390/a15020069

Copy DOI

Journal: Algorithms	Publication Date: Feb 21, 2022
Citations: 16	License type: CC BY 4.0

Affiliation: Sapienza University of Rome, University of L'Aquila

Abstract

Differential problems with the Riesz derivative in space are widely used to model anomalous diffusion. Although the Riesz–Caputo derivative is more suitable for modeling real phenomena, there are few examples in literature where numerical methods are used to solve such differential problems. In this paper, we propose to approximate the Riesz–Caputo derivative of a given function with a cubic spline. As far as we are aware, this is the first time that cubic splines have been used in the context of the Riesz–Caputo derivative. To show the effectiveness of the proposed numerical method, we present numerical tests in which we compare the analytical solution of several boundary differential problems which have the Riesz–Caputo derivative in space with the numerical solution we obtain by a spline collocation method. The numerical results show that the proposed method is efficient and accurate.

Highlights

Models using fractional derivatives in time and/or in space are commonly used in various fields of sciences, such as biology, physics, mechanics, economics, control theory, just to cite a few
In the field of geophysics, fractional differential equations are widely used for modeling anomalous diffusion in porous media, a phenomenon in which particles have been observed to spread at a rate which is incompatible with classical Brownian motion
The Riesz fractional derivative has been shown to be more attractive compared to the left-sided derivative since it is able to take into account contributions from both sides of the domain [2,10]

Summary

Introduction

In the field of geophysics, fractional differential equations are widely used for modeling anomalous diffusion in porous media, a phenomenon in which particles have been observed to spread at a rate which is incompatible with classical Brownian motion. In this context, the Riesz fractional derivative has been shown to be more attractive compared to the left-sided derivative since it is able to take into account contributions from both sides of the domain [2,10]. In literature the Riesz derivative defined through the left and right Riemann–Liouville derivatives is commonly used to model fractional diffusion, the

Objectives

Methods

Results

Conclusion