The two forms of fractional relaxation of distributed order

We introduce the linear operators of fractional integration and fractional differentiation in the framework of the Riemann-Liouville fractional calculus. Particular attention is devoted to the technique of Laplace transforms for treating these operators in a way accessible to applied scientists, avoiding unproductive generalities and excessive mathematical rigor. By applying this technique we shall derive the analytical solutions of the most simple linear integral and differe...

This paper deals with the investigation of the solution of an unified fractional reaction-diffusion equation associated with the Caputo derivative as the time-derivative and Riesz-Feller fractional derivative as the space-derivative. The solution is derived by the application of the Laplace and Fourier transforms in closed form in terms of the H-function. The results derived are of general nature and include the results investigated earlier by many authors, notably by Mainard...

We describe a general operational method that can be used in the analysis of fractional initial and boundary value problems with additional analytic conditions. As an example, we derive analytic solutions of some fractional generalisation of differential equations of mathematical physics. Fractionality is obtained by substituting the ordinary integer-order derivative with the Caputo fractional derivative. Furthermore, operational relations between ordinary and fractional diff...

This article is in continuation of our earlier article [37] in which computational solution of an unified reaction-diffusion equation of distributed order associated with Caputo derivatives as the time-derivative and Riesz-Feller derivative as space derivative is derived. In this article, we present computational solutions of distributed order fractional reaction-diffusion equations associated with Riemann-Liouville derivatives of fractional orders as the time-derivatives and...

The fundamental solution of the fractional diffusion equation of distributed order in time (usually adopted for modelling sub-diffusion processes) is obtained based on its Mellin-Barnes integral representation. Such solution is proved to be related via a Laplace-type integral to the Fox-Wright functions. A series expansion is also provided in order to point out the distribution of time-scales related to the distribution of the fractional orders. The results of the time fracti...

The relaxation functions introduced empirically by Debye, Cole-Cole, Cole-Davidson and Havriliak-Negami are, each of them, solutions to their respective kinetic equations. In this work, we propose a generalization of such equations by introducing a fractional differential operator written in terms of the Riemann-Liouville fractional derivative of order $\gamma$, $0 < \gamma \leq 1$. In order to solve the generalized equations, the Laplace transform methodology is introduced a...

When $0<\alpha<1$, the approximation for the Caputo derivative $$y^{(\alpha)}(x) = \frac{1}{\Gamma(2-\alpha)h^\alpha}\sum_{k=0}^n \sigma_k^{(\alpha)} y(x-kh)+O\bigl(h^{2-\alpha}\bigr),$$ where $\sigma_0^{(\alpha)} = 1, \sigma_n^{(\alpha)} = (n-1)^{1-a}-n^{1-a}$ and $$\sigma_k^{(\alpha)} = (k-1)^{1-\alpha}-2k^{1-a}+(k+1)^{1-\alpha},\quad (k=1...,n-1),$$ has accuracy $O\bigl(h^{2-\alpha}\bigr)$. We use the expansion of $\sum_{k=0}^n k^\alpha$ to determine an approximation for t...

This paper focuses on the equivalent expression of fractional integrals/derivatives with an infinite series. A universal framework for fractional Taylor series is developed by expanding an analytic function at the initial instant or the current time. The framework takes into account of the Riemann-Liouville definition, the Caputo definition, the constant order and the variable order. On this basis, some properties of fractional calculus are confirmed conveniently. An intuitiv...

We consider diffusion type equations with a distributed order derivative in the time variable. This derivative is defined as the integral in $\alpha$ of the Caputo-Dzhrbashian fractional derivative of order $\alpha \in (0,1)$ with a certain positive weight function. Such equations are used in physical literature for modeling diffusion with a logarithmic growth of the mean square displacement. In this work we develop a mathematical theory of such equations, study the derivativ...

We introduce two kinds of fractional integral operators; the one is defined via the exponential-integral function $$ E_1(x)=\int_x^\infty \frac{e^{-t}}{t}\,dt,\quad x>0, $$ and the other is defined via the special function $$ \mathcal{S}(x)=e^{-x} \int_0^\infty \frac{x^{s-1}}{\Gamma(s)}\,ds,\quad x>0. $$ We establish different properties of these operators, and we study the relationship between the fractional integrals of first kind and the fractional integrals of second kind...