Abstract
In this paper we develop a time-fractional operator calculus in fractional Clifford analysis. Initially we study the $$L_p$$ -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.
Highlights
Nowadays, one of the most studied fractional partial differential equation is the time-fractional diffusion equation due to its wide range of applications
We have the work of Gurlebeck and Sproßig based on a Borel-Pompeiu formula and on an orthogonal decomposition of the underlying function space where one of the components is the kernel of the corresponding Dirac operator [11]
In order to do that we initially study the Lp-integrability of the fundamental solutions of the time-fractional diffusion and the time-fractional parabolic Dirac operators deduced in [10]
Summary
One of the most studied fractional partial differential equation is the time-fractional diffusion equation due to its wide range of applications (see [8, 17, 22, 23, 25, 30] and the references therein indicated). We have the work of Gurlebeck and Sproßig based on a Borel-Pompeiu formula and on an orthogonal decomposition of the underlying function space where one of the components is the kernel of the corresponding Dirac operator [11] This theory was successfully applied to a large type of equations, e.g., Lame equations, Maxwell equations, and Navier-Stokes equations. The results exhibit an interesting “double duality” between forward and backward time-fractional parabolic Dirac operators, and between Caputo and Riemann-Liouville time-fractional derivatives This double duality appears in a non-trivial generalization of the Stokes’ formula as well as in the time-fractional Borel-Pompeiu formula and in the Hodge-type decomposition. This represents the main result of the paper, apart from the time-fractional Borel-Pompeiu formula. We round off this paper by giving an immediate application to the resolution of time-fractional boundary value problems
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