Eigenfunctions and fundamental solutions for the Fractional Laplacian in 3 dimensions

• Autor(a) principal: Ferreira, Milton dos Santos
• Data de Publicação: 2015
• Outros Autores: Vieira, Nelson Felipe Loureiro
• Tipo de documento: Artigo
• Idioma: eng
• Título da fonte: Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos)
• Texto Completo: http://hdl.handle.net/10773/16281
• topic: Fractional Laplace operator; Riemann-Liouville fractional derivatives; Eigenfunctions; Fundamental solution; Mittag-Leffler function

Recently there has been a surge of interest in PDEs involving fractional derivatives in different fields of engineering. In this extended abstract we present some of the results developed in \cite{FV}. We compute the fundamental solution for the three-parameter fractional Laplace operator $\Delta^{(\alpha,\beta,\gamma)}$ with $(\alpha, \beta, \gamma) \in \,]0,1]^3$ by transforming the eigenfunction equation into an integral equation and applying the method of separation of variables. The obtained solutions are expressed in terms of Mittag-Leffer functions. For more details we refer the interested reader to \cite{FV} where it is also presented an operational approach based on the two Laplace transform.