  The objective of this paper is to derive analytical solutions of fractional
order Laplace, Poisson and Helmholtz equations in two variables derived from
the corresponding standard equations in two dimensions by replacing the integer
order partial derivatives with fractional Riesz-Feller derivative and
generalized Riemann-Liouville fractional derivative recently defined by Hilfer.
The Fourier-Laplace transform method is employed to obtain the solutions in
terms of Mittag-Leffler functions, Fox $H$-function and an integral operator
containing a Mittag-Leffler function in the kernel. Results for fractional wave
equation are presented as well. Some interesting special cases of these
equations are considered. Asymptotic behavior and series representation of
solutions are analyzed in detail. Many previously obtained results can be
derived as special cases of those presented in this paper.
