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A Time-Fractional Borel–Pompeiu Formula and a Related Hypercomplex Operator Calculus
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Advisor(s)
Abstract(s)
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 analogs 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.
UID/MAT/04106/2019. A-15/17 / DAAD-PPP IF/00271/2014
UID/MAT/04106/2019. A-15/17 / DAAD-PPP IF/00271/2014
Description
The final version is published in Complex Analysis and Operator Theory, 13-No.6, (2019). Received: 8 May 2018 / Accepted: 24 December 2018 / Published online: 11 January 2019.
Acknowledgment: 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–Funda¸c˜ao para a Ciˆencia 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 f¨ur instation¨are PDE, funded by Programme for Cooperation in Science between Portugal and Germany (“Programa de A¸c˜oes Integradas Luso-Alem˜as 2017” - DAAD-CRUP - Ação 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).
Acknowledgment: 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–Funda¸c˜ao para a Ciˆencia 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 f¨ur instation¨are PDE, funded by Programme for Cooperation in Science between Portugal and Germany (“Programa de A¸c˜oes Integradas Luso-Alem˜as 2017” - DAAD-CRUP - Ação 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).
Keywords
Fractional Clifford analysis Fractional derivatives Time-fractional parabolic Dirac operator Fundamental solution Borel-Pompeiu formula
Citation
Ferreira, M., Rodrigues, M.M. & Vieira, N. A Time-Fractional Borel–Pompeiu Formula and a Related Hypercomplex Operator Calculus. Complex Anal. Oper. Theory 13, 2495–2526 (2019). https://doi.org/10.1007/s11785-018-00887-7
Publisher
Springer Nature