Ferreira, MiltonVieira, Nelson Felipe Loureiro2019-02-072019-02-072017-06M. 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: 10.1080/17476933.2016.12504011747-69331747-6941http://hdl.handle.net/10400.8/3811In 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.engFractional partial differential equationsFractional Laplace and Dirac operatorsCaputo derivativeEigenfunctionsFundamental solutionjournal article10.1080/17476933.2016.1250401