Department of Pure Mathematics, Faculty of Science, University of Porto, Campo Alegre street, 687, 4169-007 Porto, Portugal
Copyright © 2010 Semyon Yakubovich. This is an open access article distributed under the
Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We deal with the following fractional generalization of the Laplace equation for rectangular domains , which is associated with the Riemann-Liouville fractional derivatives , , where , . Reducing the left-hand side of this equation to
the sum of fractional integrals by and , we then use the operational technique for
the conventional right-sided Laplace transformation and its extension to generalized
functions to describe a complete family of eigenfunctions and fundamental solutions
of the operator in classes of functions represented by the left-sided fractional
integral of a summable function or just admitting a summable fractional derivative.
A symbolic operational form of the solutions in terms of the Mittag-Leffler functions
is exhibited. The case of the separation of variables is also considered. An analog
of the fractional logarithmic solution is presented. Classical particular cases of
solutions are demonstrated.