Fundamental Solution for Natural Powers of the Fractional Laplace and Dirac Operators in the Riemann–Liouville Sense

A Di Teodoro,N Vieira,M Ferreira

doi:10.1007/s00006-019-1029-1

A Di Teodoro, N Vieira + Show 1 more

Open Access

https://doi.org/10.1007/s00006-019-1029-1

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Abstract

In this paper, we study the fundamental solution of natural powers of the n-parameter fractional Laplace and Dirac operators defined via Riemann–Liouville fractional derivatives. To do this we use iteration through the fractional Poisson equation starting from the fundamental solutions of the fractional Laplace $$\Delta _{a^+}^\alpha $$ and Dirac $$D_{a^+}^\alpha $$ operators, admitting a summable fractional derivative. The family of fundamental solutions of the corresponding natural powers of fractional Laplace and Dirac operators are expressed in operator form using the Mittag–Leffler function.

Highlights

During the last decades, the study of the so-called fractional Laplace operator has received the attention of several authors
In this paper we consider a n-parameter fractional Laplace operator dened in n-dimensional space and the associated n-parameter fractional Dirac operator over a Cliord algebra, both dened via Riemann-Liouville fractional derivatives with dierent fractional order of dierentiation for each direction
There the authors studied eigenfunctions and fundamental solutions for the three-parameter fractional Laplace operator dened with Caputo and Riemann-Liouville fractional derivatives, and derived fundamental solutions for the corresponding fractional Dirac operator which factorizes the fractional Laplace operator

Summary

Introduction

The study of the so-called fractional Laplace operator has received the attention of several authors (see for example [1, 14] and references therein indicated). There the authors studied eigenfunctions and fundamental solutions for the three-parameter fractional Laplace operator dened with Caputo and Riemann-Liouville fractional derivatives, and derived fundamental solutions for the corresponding fractional Dirac operator which factorizes the fractional Laplace operator. In both cases, the authors applied an operational approach via Laplace transform to construct general families of fundamental solutions.

Fractional Calculus

Cliord analysis

Conclusions and Future Work