Exact solutions of triple-order time-fractional differential equations for anomalous relaxation and diffusion I: The accelerating case

Abstract

In recent years the interest around the study of anomalous relaxation and diffusion processes is increased due to their importance in several natural phenomena. Moreover, a further generalization has been developed by introducing time-fractional differentiation of distributed order which ranges between 0 and 1. We refer to accelerating processes when the driving power law has a changing-in-time exponent whose modulus tends from less than 1 to 1, and to decelerating processes when such an exponent modulus decreases in time moving away from the linear behaviour. Accelerating processes are modelled by a time-fractional derivative in the Riemann–Liouville sense, while decelerating processes by a time-fractional derivative in the Caputo sense. Here the focus is on the accelerating case while the decelerating one is considered in the companion paper. After a short reminder about the derivation of the fundamental solution for a general distribution of time-derivative orders, we consider in detail the triple-order case for both accelerating relaxation and accelerating diffusion processes and the exact results are derived in terms of an infinite series of $H$-functions. The method adopted is new and it makes use of certain properties of the generalized Mittag-Leffler function and the $H$-function, moreover it provides an elegant generalization of the method introduced by Langlands (2006) [T.A.M. Langlands, Physica A 367 (2006) 136] to study the double-order case of accelerating diffusion processes.

Introduction

Anomalous dynamics is frequently met in processes through complex and/or disordered media, e.g. dispersion in plasmas [1] or diffusion with obstacles [2] or binding [3], self-diffusion of surfactant molecules [4] or protein movements [5], [6], and light scattering in a cold atomic cloud [7]. A useful mathematical tool for physical investigation and description of such phenomena is fractional calculus, see for example Refs. [8], [9], [10], [11], [12], [13] for anomalous relaxation and Refs. [14], [15], [16], [17], [18], [19], [20] for anomalous diffusion. Recently, the extension of fractional differential equations to distributed-order fractional differential equations has permitted to describe also processes whose scaling law changes in time [21], [22], [23], [12], [24], [25], [26], [27], [28]. However an early idea of the time-fractional derivative of distributed order was proposed in 1969 by Caputo [29], and later re-proposed by Caputo himself [30], [31] and Bagley and Torvik [32], [33]. In particular, when the time-fractional derivative operator is opportunely chosen, it permits to model phenomena whose driving power law can be expressed by a growing-in-time exponent modulus that from less than 1 tends to 1, or an exponent modulus that decreases in time moving away from the linear behaviour, and we refer to them as accelerating and decelerating processes, respectively [34], [35]. The accelerating processes are modelled by a Riemann–Liouville (R–L) time-fractional derivative while the decelerating processes by a Caputo (C) time-fractional derivative. However, when single-order fractional differential equations are considered the two forms (R–L) and (C) are equivalent. The present paper is focused on the accelerating case while the decelerating case is considered in the companion paper [36].

A double-order time-fractional diffusion equation has been exactly solved by Langlands [37], [38]. The main object of the present article is to further include one more fractional time derivative and investigate its solution by the application of the generalized Mittag-Leffler function. Then the triple-order time-fractional differential equations considered in the present paper are:

$$\frac{du}{dt}=-\lambda [P{}_t{D}^{1-\alpha }+Q{}_t{D}^{1-\beta }+T{}_t{D}^{1-\gamma }]u\left(t\right)\text{,}t\in R_0^{+}\text{,}$$$$\frac{\partial u}{\partial t}=[P{}_t{D}^{1-\alpha }+Q{}_t{D}^{1-\beta }+T{}_t{D}^{1-\gamma }]\frac{{\partial }^2}{\partial x^2}u(x,t)\text{,}t\in R_0^{+},x\in R\text{,}$$

with $0<\alpha <\beta <\gamma <1$, where $\lambda$ is a positive constant and ${}_t{D}^{\nu }$ is the Riemann–Liouville time-fractional derivative operator of order $\nu >0$ [39], [40], which for a sufficiently well-behaved function $f\left(t\right)$ is defined as

$${}_t{D}^{\nu }f\left(t\right)=\{\begin{array}{cc}\frac{d^m}{d{t}^m}\left[\frac{1}{\Gamma (m-\nu )}{\int }_0^t\frac{f\left(\tau \right)d\tau }{{(t-\tau )}^{\nu +1-m}}\right]\text{,}\hfill & m-1<\nu <m\text{,}\hfill \\ \frac{d^m}{d{t}^m}f\left(t\right)\text{,}\hfill & \nu =m\text{.}\hfill \end{array}$$

The rest of the paper is organized as follows. In Section 2 the fundamental solution for a general distribution of time-derivative orders is recalled together with general considerations which motivate the name accelerating. In Section 3 the exact solutions for both triple-order time-fractional relaxation and diffusion equations are obtained using a new method based on certain properties of Mittag-Leffler and $H$-functions. Concluding remarks are given in Section 4.