Abstract
This article is in continuation of the authors research attempts to derive computational solutions of an unified reaction-diffusion equation of distributed order associated with Caputo derivatives as the time-derivative and Riesz-Feller derivative as space derivative. This article presents computational solutions of distributed order fractional reaction-diffusion equations associated with Riemann-Liouville derivatives of fractional orders as the time-derivatives and Riesz-Feller fractional derivatives as the space derivatives. The method followed in deriving the solution is that of joint Laplace and Fourier transforms. The solution is derived in a closed and computational form in terms of the familiar Mittag-Leffler function. It provides an elegant extension of results available in the literature. The results obtained are presented in the form of two theorems. Some results associated specifically with fractional Riesz derivatives are also derived as special cases of the most general result. It will be seen that in case of distributed order fractional reaction-diffusion, the solution comes in a compact and closed form in terms of a generalization of the Kampé de Fériet hypergeometric series in two variables. The convergence of the double series occurring in the solution is also given.
Highlights
Distributed order sub-diffusion is discussed by Naber [1]
Reaction-diffusion models associated with Riemann-Liouville fractional derivative as the time derivative and Riesz-Feller derivative as the space derivative are recently discussed by Haubold et al [21]
The main object of this article is to investigate the computational solutions of fraction reaction-diffusion Equations (1) and (24)
Summary
Distributed order sub-diffusion is discussed by Naber [1]. Distributed order fractional diffusion systems are studied, among others, by Saxton [2,3,4], Langlands [5], Sokolov et al [6], Sokolov and Klafter [7], Saxena and Pagnini [8] and Nikolova and Boyadijiev [9], and recent monographs on the subject [10,11,12,13]. Chen et al [33] have derived the fundamental and numerical solution of a reaction-diffusion equation associated with the Riesz fractional derivative as the space derivative. Reaction-diffusion models associated with Riemann-Liouville fractional derivative as the time derivative and Riesz-Feller derivative as the space derivative are recently discussed by Haubold et al [21]. Such equations in case of Caputo fractional derivative are solved by Saxena et al [27]. The main object of this article is to investigate the computational solutions of fraction reaction-diffusion Equations (1) and (24). Due to the general character of the derived results, many known results given earlier by Chen et al [33], Haubold et al [22] and Pagnini and Mainardi [34], Saxena et al [27], readily follow as special cases of our derived results
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