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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 |
URI: | http://hdl.handle.net/10773/15637 |
DOI: | 10.1007/s11785-015-0529-9 |
ISSN: | 1661-8254 |
Appears in Collections: | CIDMA - Artigos CHAG - Artigos |
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artigo39_VF.pdf | 399.72 kB | Adobe PDF | View/Open |
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