Eigenfunctions and fundamental solutions of the fractional Laplace and Dirac operators using Caputo derivatives

M Ferreira,N Vieira

doi:10.1080/17476933.2016.1250401

Abstract

In this paper, we study eigenfunctions and fundamental solutions for the three parameter fractional Laplace operator where and the fractional derivatives , , are in the Caputo sense. Applying integral transform methods, we describe a complete family of eigenfunctions and fundamental solutions of the operator in classes of functions admitting a summable fractional derivative. The solutions are expressed using the Mittag–Leffler function. From the family of fundamental solutions obtained, we deduce a family of fundamental solutions of the corresponding fractional Dirac operator, which factorizes the fractional Laplace operator introduced in this paper.

Highlights

In the last decades the interest in fractional calculus increased substantially
Among all the subjects there is a considerable interest in the study of ordinary and partial fractional differential equations regarding the mathematical aspects and methods of their solutions and their applications in diverse areas such as physics, chemistry, engineering, optics or quantum mechanics
The aim of this paper is to present a closed formula for the family of eigenfunctions and fundamental solutions of the three-parameter fractional Laplace operator using Caputo derivatives, as well as a family of fundamental solutions of the associated fractional Dirac operator

Summary

Introduction

In the last decades the interest in fractional calculus increased substantially. Among all the subjects there is a considerable interest in the study of ordinary and partial fractional differential equations regarding the mathematical aspects and methods of their solutions and their applications in diverse areas such as physics, chemistry, engineering, optics or quantum mechanics (see, for example [11, 12, 13, 15, 16, 17, 24]). We propose a fractional Laplace operator in 3-dimensional space using Caputo derivatives with different order for each direction. Previous approaches for this type of operator using Riemann-Liouville derivatives in two and three dimensions can be found in [22] and [7], respectively. The aim of this paper is to present a closed formula for the family of eigenfunctions and fundamental solutions of the three-parameter fractional Laplace operator using Caputo derivatives, as well as a family of fundamental solutions of the associated fractional Dirac operator.

Fractional calculus and special functions

Eigenfunctions and fundamental solution of the fractional Laplace operator

Fundamental solution of the fractional Dirac operator