On a Fractional Master Equation

Anitha Thomas

doi:10.1155/2011/346298

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Home > Journals > Int. J. Differ. Equ. > Volume 2011 > Article

2011 On a Fractional Master Equation

Anitha Thomas

Int. J. Differ. Equ. 2011: 1-13 (2011). DOI: 10.1155/2011/346298

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Abstract

A fractional order time-independent form of the wave equation or diffusion equation in two dimensions is obtained from the standard time-independent form of the wave equation or diffusion equation in two-dimensions by replacing the integer order partial derivatives by fractional Riesz-Feller derivative and Caputo derivative of order $\alpha ,\beta ,1<\Re (\alpha )\le 2$ and $1<\Re (\beta )\le 2$ respectively. In this paper, we derive an analytic solution for the fractional time-independent form of the wave equation or diffusion equation in two dimensions in terms of the Mittag-Leffler function. The solutions to the fractional Poisson and the Laplace equations of the same kind are obtained, again represented by means of the Mittag-Leffler function. In all three cases, the solutions are represented also in terms of Fox's $H$ -function.