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A higher dimensional fractional BorelāPompeiu formula and a related hypercomplex fractional operator calculus
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Abstract(s)
In this paper, we develop a fractional integro-differential operator calculus for Clifford-algebra valued functions. To do that we introduce fractional analogs 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.
Description
Acknowledgements:
The work of M. Ferreira, M.M. Rodrigues and N. Vieira was supported by Portuguese funds through CIDMACenter for Research and Development in Mathematics and Applications, and FCTāFundaĆ§Ć£o para a CiĆŖncia e
a Tecnologia, within project UID/MAT/04106/2019. The work of the authors was supported by the project New Function Theoretic Methods in Computational Electrodynamics / Neue funktionentheoretische Methoden fĀØur instationĀØare PDE, funded by Programme for Cooperation in Science between Portugal and Germany (āPrograma de AĀøcĖoes Integradas Luso-AlemĖas/2017ā - AcĆ§Ć£o No. A-19/08 - DAAD-PPP Deutschland-Portugal, Ref: 57340281). N. Vieira was also supported by FCT via the FCT Researcher Program 2014 (Ref: IF/00271/2014).
Keywords
Fractional Clifford analysis Fractional derivatives Stokes's formula Borel-Pompeiu formula Cauchy's integral formula Hodge-type decomposition
Citation
Ferreira, M, KrauĪ²har, RS, Rodrigues, MM, Vieira, N. A higher dimensional fractional BorelāPompeiu formula and a related hypercomplex fractional operator calculus. Math Meth Appl Sci. 2019; 42(10): 3633ā 3653. https://doi.org/10.1002/mma.5602
Publisher
Wiley Online Library