A Time-Fractional Borel–Pompeiu Formula and a Related Hypercomplex Operator Calculus

Abstract

In this paper we develop a time-fractional operator calculus in fractional Clifford analysis. Initially we study the Lp-integrability of the fundamental solutions of the multi-dimensional time-fractional diffusion operator and the associated time-fractional parabolic Dirac operator. Then we introduce the time-fractional analogues of the Teodorescu and Cauchy-Bitsadze operators in a cylindrical domain, and we investigate their main mapping properties. As a main result, we prove a time-fractional version of the Borel-Pompeiu formula based on a time-fractional Stokes' formula. This tool in hand allows us to present a Hodge-type decomposition for the forward time-fractional parabolic Dirac operator with left Caputo fractional derivative in the time coordinate. The obtained results exhibit an interesting duality relation between forward and backward parabolic Dirac operators and Caputo and Riemann-Liouville time-fractional derivatives. We round off this paper by giving a direct application of the obtained results for solving time-fractional boundary value problems.

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

“…Throughout the paper, we show that we can always re-obtain the results of the classical function theory for the Dirac operator when switching to the limit case when = (1, … , 1). The analogous of the results presented in this paper for the case of the time-fractional parabolic Dirac operator can be found in Ferreira et al 21 The structure of the paper reads as follows. In the Section 2, we recall some basic definitions from the fractional calculus, special functions, and Clifford analysis.…”

“…We give a direct proof of the theorem in order to confirm that (34) is indeed the solution of (21). The proof uses the fact that C a + 1 1+ 1…”

“…Throughout the paper, we show that we can always re‐obtain the results of the classical function theory for the Dirac operator when switching to the limit case when α = (1,…,1). The analogous of the results presented in this paper for the case of the time‐fractional parabolic Dirac operator can be found in Ferreira et al…”

On the construction of solutions for the parabolic Dirac operator and equivalent systems

“…This implies that ⃗ T 1,Ω is a right inverse of the divergence operator and has vanishing rotational (see proposition 3.2 in Delgado and Porter [11]). In the next result, we will apply the above solution to four specific div-curl systems derived from (20). Moreover, we will construct functions in ker D +…”

A quaternionic perturbed fractional $ψ-$Fueter operator calculus

“…Fractional hyperholomorphic function theory is a very recent topic of research, see [4,6,7,10,11,16,20,28] for more details. In particular, the interest for considering fractional Laplace and Dirac type operators is devoted in [1,2,8,9,21].…”