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 Eigenfunctions and fundamental solutions of the fractional Laplace and Dirac operators using Caputo derivatives
Please use this identifier to cite or link to this item http://hdl.handle.net/10773/18083

title: Eigenfunctions and fundamental solutions of the fractional Laplace and Dirac operators using Caputo derivatives
authors: Ferreira, M.
Vieira, N.
keywords: Fractional partial differential equations
Fractional Laplace and Dirac operators
Caputo derivative
Fundamental solution
issue date: 1-Jun-2017
publisher: Taylor & Francis
abstract: In this paper we study eigenfunctions and fundamental solutions for the three parameter fractional Laplace operator ${}^C\!\Delta_+^{(\alpha,\beta,\gamma)}:= {}^C\!D_{x_0^+}^{1+\alpha} +{}^C\!D_{y_0^+}^{1+\beta} +{}^C\!D_{z_0^+}^{1+\gamma},$ where $(\alpha, \beta, \gamma) \in \,]0,1]^3$ and the fractional derivatives ${}^C\!D_{x_0^+}^{1+\alpha}$, ${}^C\!D_{y_0^+}^{1+\beta}$, ${}^C\!D_{z_0^+}^{1+\gamma}$ are in the Caputo sense. Applying integral transform methods we describe a complete family of eigenfunctions and fundamental solutions of the operator ${}^C\!\Delta_+^{(\alpha,\beta,\gamma)}$ in classes of functions admitting a summable fractional derivative. The solutions are expressed using the Mittag-Leffler function. From the family of fundamental solutions obtained we deduce a family of fundamental solutions of the corresponding fractional Dirac operator, which factorizes the fractional Laplace operator introduced in this paper.
URI: http://hdl.handle.net/10773/18083
ISSN: 1747-6933
publisher version/DOI: http://dx.doi.org/10.1080/17476933.2016.1250401
source: Complex Variables and Elliptic Equations
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