A higher dimensional fractional Borel‐Pompeiu formula and a related hypercomplex fractional operator calculus

Milton Ferreira,R Sören Kraußhar,Nelson Vieira,M Manuela Rodrigues

doi:10.1002/mma.5602

Abstract

In this paper, we develop a fractional integro‐differential operator calculus for Clifford algebra‐valued functions. To do that, we introduce fractional analogues of the Teodorescu and Cauchy‐Bitsadze operators, and we investigate some of their mapping properties. As a main result, we prove a fractional Borel‐Pompeiu formula based on a fractional Stokes formula. This tool in hand allows us to present a Hodge‐type decomposition for the fractional Dirac operator. Our results exhibit an amazing duality relation between left and right operators and between Caputo and Riemann‐Liouville fractional derivatives. We round off this paper by presenting a direct application to the resolution of boundary value problems related to Laplace operators of fractional order.

Highlights

Clifford analysis offers a higher dimensional generalization of the classical theory of complex holomorphic functions
As a main result we prove a fractional Borel-Pompeiu formula based on a fractional Stokes formula
A main tool that Clifford holomorphic function theory uses in the treatment of boundary value problems is the Teodorescu operator, which is the right inverse of the Dirac operator

Summary

Introduction

Clifford analysis offers a higher dimensional generalization of the classical theory of complex holomorphic functions. The advantage of fractional models consists in the possibility of using fractional derivatives to describe the memory and hereditary properties of various materials and processes Another field of application consists in addressing differential equations related to flows with permeable boundaries, such as for instance dam-fill problems which provides a further important motivation to develop three-dimensional generalizations of harmonic and Clifford analysis tools for the fractional setting. This decomposition represents a main result in the paper apart from proving the generalizations of the Borel-Pompeiu formulae in the context of Caputo derivatives. We round off this paper by giving an immediate application to the resolution of boundary value problems involving the fractional Laplace operators

Fractional calculus and special functions

Clifford analysis

Fundamental solutions revisited

Fractional Teodorescu and Cauchy-Bitsadze operators

Hodge-type decomposition

Conclusion