Eigenfunctions and Fundamental Solutions for the Fractional Laplacian in 3 dimensions ∗ M. Ferreira †,‡ N. Vieira ‡ † School of Technology and Management, Polytechnic Institute of Leiria P-2411-901, Leiria, Portugal. ‡ CIDMA - Center for Research and Development in Mathematics and Applications Department of Mathematics, University of Aveiro Campus Universit´ ario de Santiago, 3810-193 Aveiro, Portugal. E-mails: milton.ferreira@ipleiria.pt,nloureirovieira@gmail.com Abstract 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 [3]. We compute the fundamental solution for the three-parameter fractional Laplace operator Δ (α,β,γ) with (α,β,γ) ∈ ]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 [3] where it is also presented an operational approach based on the two Laplace transform. Keywords: Fractional Laplace operator; Riemann-Liouville fractional derivatives; Eigenfunctions; Fun- damental solution; Mittag-Leffler function. MSC 2010: Primary 35R11; Secondary 30G35 26A33 35P10 35A22 35A08 1 Introduction The problems with the fractional Laplacian attracted in the last years a lot of attention, due especially to their large range of applications. The fractional Laplacian appears in probabilistic framework as well as in mathematical finance as infinitesimal generators of the stable L´ evy processes [1]. One can find problems involving the fractional Laplacian in mechanics and in elastostatics, for example, a Signorini obstacle problem originating from linear elasticity [2]. The aim of this paper is to present an explicit expression for the family of eigenfunctions and fundamental solutions of the three-parameter fractional Laplace. For the sake of simplicity we restrict ourselves to the three dimensional case, however the results can be generalized for an arbitrary dimension. The two dimensional case was already studied in [10]. Connections between fractional calculus and Clifford analysis were considered recently in [5, 9]. * Accepted author’s manuscript (AAM) published in Digital Proceedings, 20 th International Conference on the Applications of Computer Science and Mathematics in Architecture and Civil Engineering, K. G¨ urlebeck and T. Lahmer (eds.), July 20–22 2015, Bauhaus-University Weimar, 30–35. The final publication is available at https://e-pub.uni-weimar.de/opus4/frontdoor/index/index/docId/2451. 1