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Axioms 2015, 4(2), 120-133; doi:10.3390/axioms4020120

Computational Solutions of Distributed Order Reaction-Diffusion Systems Associated with Riemann-Liouville Derivatives

1
Department of Mathematics and Statistics, Jai Narain Vyas University, Jodhpur 342004, India
2
Centre for Mathematical and Statistical Sciences, Peechi Campus, KFRI, Peechi 680653, Kerala, India
3
Department of Mathematics and Statistics, McGill University, Montreal H3A 2K6, Canada
4
Office for Outer Space Affairs, United Nations, P.O. Box 500, Vienna International Centre, Vienna 1400, Austria
*
Author to whom correspondence should be addressed.
Academic Editor: Ángel Garrido
Received: 16 October 2014 / Revised: 25 November 2014 / Accepted: 12 March 2015 / Published: 2 April 2015
View Full-Text   |   Download PDF [228 KB, uploaded 2 April 2015]

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. View Full-Text
Keywords: Mittag-Leffler function; Riesz-Feller fractional derivative; H-function; Riemann-Liouville fractional derivative; Caputo derivative; Laplace transform; Fourier transform; Riesz derivative Mittag-Leffler function; Riesz-Feller fractional derivative; H-function; Riemann-Liouville fractional derivative; Caputo derivative; Laplace transform; Fourier transform; Riesz derivative
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. (CC BY 4.0).

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MDPI and ACS Style

Saxena, R.K.; Mathai, A.M.; Haubold, H.J. Computational Solutions of Distributed Order Reaction-Diffusion Systems Associated with Riemann-Liouville Derivatives. Axioms 2015, 4, 120-133.

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