Eigenfunctions and fundamental solutions of the fractional Laplace and Dirac operators using Caputo derivatives

Ferreira, Milton; Vieira, Nelson Felipe Loureiro

http://hdl.handle.net/10400.8/3811

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Authors

Ferreira, Milton

Vieira, Nelson Felipe Loureiro

Abstract(s)

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.

Keywords

Fractional partial differential equations Fractional Laplace and Dirac operators Caputo derivative Eigenfunctions Fundamental solution

URI

http://hdl.handle.net/10400.8/3811

Citation

M. Ferreira & N. Vieira (2017) Eigenfunctions and fundamental solutions of the fractional Laplace and Dirac operators using Caputo derivatives, Complex Variables and Elliptic Equations, 62:9, 1237-1253, DOI: 10.1080/17476933.2016.1250401