Space-Time Fractional Reaction-Diffusion Equations Associated with a Generalized Riemann–Liouville Fractional Derivative

Ram Saxena,Arak Mathai,Hans Haubold

doi:10.3390/axioms3030320

Abstract

This paper deals with the investigation of the computational solutions of an 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 in Hilfer et al. , and the space derivative of second order by the Riesz-Feller fractional derivative, and adding a function $\phi(x,t)$. 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 earlier by Mainardi et al., and the result very recently given by Tomovski et al.. At the end, extensions of the derived results, associated with a finite number of Riesz-Feller space fractional derivatives, are also investigated.

Highlights

Standard reaction-diffusion equations are an important class of partial differential equations to investigate nonlinear behavior
We investigate the solution of a unified model of fractional diffusion system (1)
Consider the unified fractional reaction-diffusion model in Equation (1) were η, t > 0, x ∈ R, α, θ, μ, ν are real parameters with the constraints: 0 < α ≤ 2, |θ| < min(α, 2 − α) and Dtμ,ν is the generalized Riemann–Liouville fractional derivative operator defined by Equation (A9)

Summary

Introduction

Standard reaction-diffusion equations are an important class of partial differential equations to investigate nonlinear behavior. Interest has developed by several authors in the applications of reaction-diffusion models in pattern formation in physical sciences. In this connection, one can refer to Whilhelmsson and Lazzaro [1], Hundsdorfer and Verwer [2] and Sandev et al [3]. For recent and related works on fractional kinetic equations and reaction-diffusion problems, one can refer to papers by [13,14,15,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32]

Unified Fractional Reaction-Diffusion Equation

Special Cases of Theorem 1

Finite Number of Riesz–Feller Space Fractional Derivatives

Special Cases of Theorem 2

Conclusions

Mathematical Preliminaries