Abstract
In this paper, we study the fundamental solution of natural powers of the n-parameter fractional Laplace and Dirac operators defined via Riemann–Liouville fractional derivatives. To do this we use iteration through the fractional Poisson equation starting from the fundamental solutions of the fractional Laplace \(\Delta _{a^+}^\alpha \) and Dirac \(D_{a^+}^\alpha \) operators, admitting a summable fractional derivative. The family of fundamental solutions of the corresponding natural powers of fractional Laplace and Dirac operators are expressed in operator form using the Mittag–Leffler function.
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Acknowledgements
A. Di Teodoro was supported by Colegio de Ciencias e Ingenierías de la Universidad San Francisco de Quito. The work of M. Ferreira 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. N. Vieira was also supported by FCT via the FCT Researcher Program 2014 (Ref: IF/00271/2014).
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Dedicated to Professor Sirkka-Liisa Eriksson on occasion of her 60th birthday.
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This article is part of the Topical Collection on FTHD 2018, edited by Sirkka-Liisa Eriksson, Yuri M. Grigoriev, Ville Turunen, Franciscus Sommen and Helmut Malonek. Provided Funding information has to be tagged.
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Teodoro, A.D., Ferreira, M. & Vieira, N. Fundamental Solution for Natural Powers of the Fractional Laplace and Dirac Operators in the Riemann–Liouville Sense. Adv. Appl. Clifford Algebras 30, 3 (2020). https://doi.org/10.1007/s00006-019-1029-1
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DOI: https://doi.org/10.1007/s00006-019-1029-1
Keywords
- Fractional Clifford analysis
- Fractional derivatives
- Fundamental solution
- Poisson’s equation
- Laplace transform