Eigenfunctions and fundamental solutions of the fractional Laplace and Dirac operators using Caputo derivatives - Ferreira, M. (2018)
In this paper we study eigenfunctions and fundamental solutions for the three parameter fractional Laplace operator ${}^C\!Δ_+^{(α,β,γ)}:= {}^C\!D_{x_0^+}^{1+α} +{}^C\!D_{y_0^+}^{1+β} +{}^C\!D_{z_0^+}^{1+γ},$ where $(α, β, γ) \in \,]0,1]^3$ and the fractional derivatives ${}^C\!D_{x_0^+}^{1+α}$, ${}^C\!D_{y_0^+}^{1+β}$, ${}^C\!D_{z_0^+}^{1+γ}$ are in the Caputo sense. Applying integral transform methods we describe a complete family of eigenfunctions and fundamental solutions of the operator ${}^C\!Δ_+^{(α,β,γ)}$ 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.