Eigenfunctions and fundamental solutions of the fractional Laplace and Dirac operators: the Riemann-Liouville case

• Autor(a) principal: Ferreira, Milton
• Data de Publicação: 2016
• Outros Autores: Vieira, Nelson
• Tipo de documento: Artigo
• Idioma: eng
• Título da fonte: Repositórios Científicos de Acesso Aberto de Portugal (RCAAP)
• Texto Completo: http://hdl.handle.net/10773/15637
• topic: 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

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.