Abstract
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.
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
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]
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