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Eigenfunctions and fundamental solutions of the fractional Laplace and Dirac operators: the Riemann-Liouville case
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Advisor(s)
Abstract(s)
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.
Description
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
Citation
Ferreira M., and Vieira N., Eigenfunctions and fundamental solutions of the fractional Laplace and Dirac operators: the Riemann-Liouville case, Complex Anal. Oper. Theory, 10(5), 2016, 1081-1100
Publisher
Springer Nature [academic journals on nature.com]