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Title: Eigenfunctions and fundamental solutions of the fractional Laplace and Dirac operators: the Riemann-Liouville case
Author: Ferreira, Milton
Vieira, Nelson
Keywords: Fractional partial differential equations
Fractional Laplace and Dirac operators
Riemann-Liouville derivatives and integrals of fractional order
Eigenfunctions and fundamental solution
Laplace transform
Mittag-Leffler function
Issue Date: Jun-2016
Publisher: Springer International Publishing
Abstract: In this paper we study eigenfunctions and fundamental solutions for the three parameter fractional Laplace operator $\Delta_+^{(\alpha,\beta,\gamma)}:= D_{x_0^+}^{1+\alpha} +D_{y_0^+}^{1+\beta} +D_{z_0^+}^{1+\gamma},$ where $(\alpha, \beta, \gamma) \in \,]0,1]^3$, and the fractional derivatives $D_{x_0^+}^{1+\alpha}$, $D_{y_0^+}^{1+\beta}$, $D_{z_0^+}^{1+\gamma}$ are in the Riemann-Liouville sense. Applying operational techniques via two-dimensional Laplace transform we describe a complete family of eigenfunctions and fundamental solutions of the operator $\Delta_+^{(\alpha,\beta,\gamma)}$ in classes of functions admitting a summable fractional derivative. Making use of the Mittag-Leffler function, a symbolic operational form of the solutions is presented. From the obtained family of fundamental solutions we deduce a family of fundamental solutions of the fractional Dirac operator, which factorizes the fractional Laplace operator. We apply also the method of separation of variables to obtain eigenfunctions and fundamental solutions.
Peer review: yes
DOI: 10.1007/s11785-015-0529-9
ISSN: 1661-8254
Appears in Collections:CIDMA - Artigos
CHAG - Artigos

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