Fractional Helmholtz and Fractional Wave Equations with Riesz-Feller and Generalized Riemann-Liouville Fractional Derivatives

Abstract

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 replacingthe 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 thesolutions 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 also are presented. 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 the results in this paper.