On the General Solutions of Some Non-Homogeneous Div-Curl Systems with Riemann–Liouville and Caputo Fractional Derivatives

Briceyda B. Delgado,Jorge E. Macías-Díaz

doi:10.3390/fractalfract5030117

Briceyda B. Delgado, Jorge E. Macías-Díaz

Open Access

https://doi.org/10.3390/fractalfract5030117

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Abstract

In this work, we investigate analytically the solutions of a nonlinear div-curl system with fractional derivatives of the Riemann–Liouville or Caputo types. To this end, the fractional-order vector operators of divergence, curl and gradient are identified as components of the fractional Dirac operator in quaternionic form. As one of the most important results of this manuscript, we derive general solutions of some non-homogeneous div-curl systems that consider the presence of fractional-order derivatives of the Riemann–Liouville or Caputo types. A fractional analogous to the Teodorescu transform is presented in this work, and we employ some properties of its component operators, developed in this work to establish a generalization of the Helmholtz decomposition theorem in fractional space. Additionally, right inverses of the fractional-order curl, divergence and gradient vector operators are obtained using Riemann–Liouville and Caputo fractional operators. Finally, some consequences of these results are provided as applications at the end of this work.

Highlights

We extend some results from vector calculus to the fractional case—for instance, the space fractional Helmholtz Decomposition Theorem provided by Propositions 6 and 7
The key tools used are the decompositions of the fractional Teodorescu transform in the Riemann–Liouville case (52) and in the Caputo case (69) as well as various properties associated with these fractional operators, which are thoroughly established in this manuscript
As the most important result, we prove an existence theorem for the solutions of a div-curl system, considering fractional differential operators of the Riemann–Liouville and Caputo types

Summary

Introduction

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. We consider fractional derivatives of the Riemann–Liouville and the Caputo types and provide extensions of the definitions of the main differential operators from vector calculus using these fractional operators In such a way, we present fractional forms of the divergence, the rotational and the gradient operators. Some properties among these operators are established, and a useful factorization theorem for the fractional Laplace operators is proven. A theorem providing necessary and sufficient conditions for the existence of Caputo fractional hyper-conjugate pairs is proven in this stage, along with a result of the existence of a right inverse for the fractional gradient This manuscript closes with a section of concluding remarks

Fractional Calculus

Fractional Quaternionic Analysis

Fundamental Solutions

Fractional Vector Calculus

Properties of the Fractional Teodorescu Transform

Riemann–Liouville System

Caputo System

Application

Conclusions