Fundamental solutions of the time fractional diffusion-wave and parabolic Dirac operators

• Main Author: Ferreira, Milton dos Santos
• Publication Date: 2017
• Other Authors: Vieira, Nelson Felipe Loureiro
• Format: Article
• Language: eng
• Source: Repositórios Científicos de Acesso Aberto de Portugal (RCAAP)
• Download full: http://hdl.handle.net/10773/16282
• topic: Fractional Laplace operator; Riemann-Liouville fractional derivatives; Eigenfunctions; Mittag-Leffler function; Time fractional diffusion-wave operator; Time fractional parabolic Dirac operator; Fundamental solutions; Caputo fractional derivative; Fractional moments

In this paper we study the multidimensional time fractional diffusion-wave equation where the time fractional derivative is in the Caputo sense with order $\beta \in ]0,2].$ Applying operational techniques via Fourier and Mellin transforms we obtain an integral representation of the fundamental solution (FS) of the time fractional diffusion-wave operator. Series representations of the FS are explicitly obtained for any dimension. From these we derive the FS for the time fractional parabolic Dirac operator in the form of integral and series representation. Fractional moments of arbitrary order $\gamma>0$ are also computed. To illustrate our results we present and discuss some plots of the FS for some particular values of the dimension and of the fractional parameter.