On the Numerical Solution of Fractional Boundary Value Problems by a Spline Quasi-Interpolant Operator

Francesca Pitolli

doi:10.3390/axioms9020061

Abstract

Boundary value problems having fractional derivative in space are used in several fields, like biology, mechanical engineering, control theory, just to cite a few. In this paper we present a new numerical method for the solution of boundary value problems having Caputo derivative in space. We approximate the solution by the Schoenberg-Bernstein operator, which is a spline positive operator having shape-preserving properties. The unknown coefficients of the approximating operator are determined by a collocation method whose collocation matrices can be constructed efficiently by explicit formulas. The numerical experiments we conducted show that the proposed method is efficient and accurate.

Highlights

Boundary value problems having fractional derivative in space are used to describe physical phenomena in which nonlocality effect are peculiar
In this paper we present a collocation method, based on a spline quasi-interpolant operator, for the solution of boundary value problems having Caputo derivative in space
We show the performance of the proposed method by solving some fractional boundary value problems (FBVPs)

Summary

Introduction

Boundary value problems having fractional derivative in space are used to describe physical phenomena in which nonlocality effect are peculiar. They are used to model anomalous diffusion in biological tissues, viscoplastic materials in mechanical engineering or control of dynamical systems (see References [1,2,3,4,5] and references therein). In this paper we are concerned with the Caputo fractional derivative [6]. The theoretical analysis of fractional boundary value problems (FBVPs) having Caputo derivative in space was addressed, for instance, in References [6,7,8,9,10,11]. We want to mention that in recent years other types of fractional derivatives were introduced, like the He’s derivative [14]

Methods

Results

Conclusion