A Time-Fractional Borel–Pompeiu Formula and a Related Hypercomplex Operator Calculus
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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.
KeywordsFractional Clifford analysis Fractional derivatives Time-fractional parabolic Dirac operator Fundamental solution Borel–Pompeiu formula
Mathematics Subject ClassificationPrimary 30G35 Secondary 35R11 26A33 35A08 30E20 45P05
The work of M. Ferreira, M.M. Rodrigues and N. Vieira was supported by Portuguese funds through CIDMA-Center for Research and Development in Mathematics and Applications, and FCT–Fundação para a Ciência e a Tecnologia, within project UID/MAT/04106/2019. The work of the authors was supported by the project New Function Theoretical Methods in Computational Electrodynamics / Neue funktionentheoretische Methoden für instationäre PDE, funded by Programme for Cooperation in Science between Portugal and Germany (“Programa de Ações Integradas Luso-Alemãs 2017” - DAAD-CRUP - Acção No. A-15/17 / DAAD-PPP Deutschland-Portugal, Ref: 57340281). N. Vieira was also supported by FCT via the FCT Researcher Program 2014 (Ref: IF/00271/2014).
- 14.Idczak, D., Walczak, S.: Fractional Sobolev spaces via Riemann-Liouville derivatives. J. Funct. Spaces Appl. (2013). https://doi.org/10.1155/2013/128043
- 19.Luchko, Y.: On some new properties of the fundamental solution to the multi-dimensional space- and time-fractional diffusion-wave equation. Mathematics 5(4), Article ID: 76 (2017)Google Scholar
- 20.Luchko, Y.: Multi-dimensional fractional wave equation and some properties of its fundamental solution. Commun. Appl. Ind. Math., 6(1), Article ID 485 (2014)Google Scholar
- 21.Luchko, Y.: Fractional wave equation and damped waves. J. Math. Phys., 54(3), Article ID: 031505 (2013)Google Scholar