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

Ram K. Saxena and Gianni Pagnini

Physica A: Statistical Mechanics and its Applications, 2011, vol. 390, issue 4, 602-613

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.

Keywords: Anomalous relaxation; Anomalous diffusion; Triple-order time-fractional differential equation; Exact solution; H-function; Generalized Mittag-Leffler function (search for similar items in EconPapers)
Date: 2011
References: View complete reference list from CitEc
Citations: View citations in EconPapers (2)

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Persistent link: https://EconPapers.repec.org/RePEc:eee:phsmap:v:390:y:2011:i:4:p:602-613

DOI: 10.1016/j.physa.2010.10.012

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