Space-Time Fractional Reaction-Diffusion Equations Associated with a Generalized Riemann–Liouville Fractional Derivative
This paper deals with the investigation of the computational solutions of a unified fractional reaction-diffusion equation, which is obtained from the standard diffusion equation by replacing the time derivative of first order by the generalized Riemann–Liouville fractional derivative defined by oth...
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doaj-6f36ea843c3b4a749b9f98c6c8591ed32020-11-24T21:25:48ZengMDPI AGAxioms2075-16802014-08-013332033410.3390/axioms3030320axioms3030320Space-Time Fractional Reaction-Diffusion Equations Associated with a Generalized Riemann–Liouville Fractional DerivativeRam K. Saxena0Arak M. Mathai1Hans J. Haubold2Department of Mathematics and Statistics, Jai Narain Vyas University, Jodhpur 342005, IndiaCentre for Mathematical and Statistical Sciences, Peechi Campus, KFRI, Peechi 680653, Kerala, IndiaCentre for Mathematical and Statistical Sciences, Peechi Campus, KFRI, Peechi 680653, Kerala, IndiaThis paper deals with the investigation of the computational solutions of a unified fractional reaction-diffusion equation, which is obtained from the standard diffusion equation by replacing the time derivative of first order by the generalized Riemann–Liouville fractional derivative defined by others and the space derivative of second order by the Riesz–Feller fractional derivative and adding a function ɸ(x, t). The solution is derived by the application of the Laplace and Fourier transforms in a compact and closed form in terms of Mittag–Leffler functions. The main result obtained in this paper provides an elegant extension of the fundamental solution for the space-time fractional diffusion equation obtained by others and the result very recently given by others. At the end, extensions of the derived results, associated with a finite number of Riesz–Feller space fractional derivatives, are also investigated.http://www.mdpi.com/2075-1680/3/3/320fractional operatorsfractional reaction-diffusionRiemann-Liouville fractional derivativeRiesz-Feller fractional derivativeMittag-Leffler function |
collection |
DOAJ |
language |
English |
format |
Article |
sources |
DOAJ |
author |
Ram K. Saxena Arak M. Mathai Hans J. Haubold |
spellingShingle |
Ram K. Saxena Arak M. Mathai Hans J. Haubold Space-Time Fractional Reaction-Diffusion Equations Associated with a Generalized Riemann–Liouville Fractional Derivative Axioms fractional operators fractional reaction-diffusion Riemann-Liouville fractional derivative Riesz-Feller fractional derivative Mittag-Leffler function |
author_facet |
Ram K. Saxena Arak M. Mathai Hans J. Haubold |
author_sort |
Ram K. Saxena |
title |
Space-Time Fractional Reaction-Diffusion Equations Associated with a Generalized Riemann–Liouville Fractional Derivative |
title_short |
Space-Time Fractional Reaction-Diffusion Equations Associated with a Generalized Riemann–Liouville Fractional Derivative |
title_full |
Space-Time Fractional Reaction-Diffusion Equations Associated with a Generalized Riemann–Liouville Fractional Derivative |
title_fullStr |
Space-Time Fractional Reaction-Diffusion Equations Associated with a Generalized Riemann–Liouville Fractional Derivative |
title_full_unstemmed |
Space-Time Fractional Reaction-Diffusion Equations Associated with a Generalized Riemann–Liouville Fractional Derivative |
title_sort |
space-time fractional reaction-diffusion equations associated with a generalized riemann–liouville fractional derivative |
publisher |
MDPI AG |
series |
Axioms |
issn |
2075-1680 |
publishDate |
2014-08-01 |
description |
This paper deals with the investigation of the computational solutions of a unified fractional reaction-diffusion equation, which is obtained from the standard diffusion equation by replacing the time derivative of first order by the generalized Riemann–Liouville fractional derivative defined by others and the space derivative of second order by the Riesz–Feller fractional derivative and adding a function ɸ(x, t). The solution is derived by the application of the Laplace and Fourier transforms in a compact and closed form in terms of Mittag–Leffler functions. The main result obtained in this paper provides an elegant extension of the fundamental solution for the space-time fractional diffusion equation obtained by others and the result very recently given by others. At the end, extensions of the derived results, associated with a finite number of Riesz–Feller space fractional derivatives, are also investigated. |
topic |
fractional operators fractional reaction-diffusion Riemann-Liouville fractional derivative Riesz-Feller fractional derivative Mittag-Leffler function |
url |
http://www.mdpi.com/2075-1680/3/3/320 |
work_keys_str_mv |
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1725982593081409536 |