Fractional diffusion equation with a generalized Riemann–Liouville time fractional derivative

Abstract

In this paper, the solution of a fractional diffusion equation with a Hilfer-generalized Riemann–Liouville time fractional derivative is obtained in terms of Mittag–Leffler-type functions and Fox's H-function. The considered equation represents a quite general extension of the classical diffusion (heat conduction) equation. The methods of separation of variables, Laplace transform, and analysis of the Sturm–Liouville problem are used to solve the fractional diffusion equation defined in a bounded domain. By using the Fourier–Laplace transform method, it is shown that the fundamental solution of the fractional diffusion equation with a generalized Riemann–Liouville time fractional derivative defined in the infinite domain can be expressed via Fox's H-function. It is shown that the corresponding solutions of the diffusion equations with time fractional derivative in the Caputo and Riemann–Liouville sense are special cases of those diffusion equations with the Hilfer-generalized Riemann–Liouville time fractional derivative. The asymptotic behaviour of the solutions are found for large values of the spatial variable. The fractional moments of the fundamental solution of the fractional diffusion equation are obtained. The obtained results are relevant in the context of glass relaxation and aquifer problems.

Generalized Langevin Equation and the Prabhakar Derivative

“…The case with δ = 0 corresponds to the so-called Hilfer composite fractional derivative of order 0 < µ < 1 and type 0 ≤ν ≤ 1, which is given [24]. This composite fractional derivative has been successfully applied in description of dielectric and viscoelastic phenomena [25,26].…”

“…Here we note that different fractional equations have been used for modeling anomalous diffusion in various systems, including fractional reaction-diffusion equations [27,28] and their application [29], fractional relaxation and diffusion equations [5,6,9,10,[24][25][26], fractional cable equation [30], etc.…”

The Analytical Solution of Parabolic Volterra Integro-Differential Equations in the Infinite Domain

“…It is worth noting that when κ = 1, the three-parameter Mittag-Leffler function can be expressed [13,14]:…”

“…It is worth noting that, when w = 0 and a = 0, integral operator E w;γ,κ a+;α,β ϕ would correspond to the Riemann-Liouville integral operator [13].…”

Operational method for solving multi-term fractional differential equations with the generalized fractional derivatives

“…The concept of GRLFD has appeared in the theoretical modeling of broadband dielectric relaxation spectroscopy for glasses [5]. Some properties and applications of the GRLFD are given in [3,4,18,20,21,22]. Several authors (see [2,18,22] [2,18,22,23].…”