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The objective of this paper is to derive analytical solutions of fractionalorder Laplace, Poisson and Helmholtz equations in two variables derived fromthe corresponding standard equations in two dimensions by replacing the integerorder partial derivatives with fractional Riesz-Feller derivative andgeneralized Riemann-Liouville fractional derivative recently defined by Hilfer.The Fourier-Laplace transform method is employed to obtain the solutions interms of Mittag-Leffler functions, Fox $H$-function and an integral operatorcontaining a Mittag-Leffler function in the kernel. Results for fractional waveequation are presented as well. Some interesting special cases of theseequations are considered. Asymptotic behavior and series representation ofsolutions are analyzed in detail. Many previously obtained results can bederived as special cases of those presented in this paper.

Fractional Helmholtz and fractional wave equations with Riesz-Feller and generalized Riemann-Liouville fractional derivatives