A Fractional B-spline Collocation Method for the Numerical Solution of Fractional Predator-Prey Models

Francesca Pitolli

doi:10.3390/fractalfract2010013

Francesca Pitolli

Open Access

https://doi.org/10.3390/fractalfract2010013

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Journal: Fractal and Fractional	Publication Date: Feb 17, 2018
Citations: 27	License type: CC BY 4.0

Affiliation: Sapienza University of Rome

Abstract

We present a collocation method based on fractional B-splines for the solution of fractional differential problems. The key-idea is to use the space generated by the fractional B-splines, i.e., piecewise polynomials of noninteger degree, as approximating space. Then, in the collocation step the fractional derivative of the approximating function is approximated accurately and efficiently by an exact differentiation rule that involves the generalized finite difference operator. To show the effectiveness of the method for the solution of nonlinear dynamical systems of fractional order, we solved the fractional Lotka-Volterra model and a fractional predator-pray model with variable coefficients. The numerical tests show that the method we proposed is accurate while keeping a low computational cost.

Highlights

Fractional Calculus generalizes to real order the well known concept of derivative and integral of integer order
We present a collocation method based on fractional B-splines for the solution of fractional differential problems
In the last decades the interest in fractional differential and integral equations is rapidly growing and models involving fractional derivatives and/or integrals are widely used in several fields, from physics to continuum mechanics, from biophysics to electro-chemistry, from signal processing to control theory

Summary

Introduction

Fractional Calculus generalizes to real order the well known concept of derivative and integral of integer order. Even if the birth of fractional calculus can be set at the beginning of the 18th century, its use for the modeling of real-life problems is very recent. The reason why the fractional derivative is more suitable to model real-world problems is related to its non-local behavior that is able to introduce in a elegant way into the model, either memory effects in time or non-locality in space. In this paper we refer to the traditional Caputo fractional derivative introduced in [10] and extensively used to model a great variety of mechanical and biological phenomena. We refer the interested reader to the literature (see, for instance, [11,12,13,14,15] and references therein)

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