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

Author

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  • Saxena, Ram K.
  • Pagnini, Gianni

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.

Suggested Citation

  • Saxena, Ram K. & Pagnini, Gianni, 2011. "Exact solutions of triple-order time-fractional differential equations for anomalous relaxation and diffusion I: The accelerating case," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 390(4), pages 602-613.
  • Handle: RePEc:eee:phsmap:v:390:y:2011:i:4:p:602-613
    DOI: 10.1016/j.physa.2010.10.012
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    Cited by:

    1. Awad, Emad, 2019. "On the time-fractional Cattaneo equation of distributed order," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 518(C), pages 210-233.
    2. Sandev, Trifce & Sokolov, Igor M. & Metzler, Ralf & Chechkin, Aleksei, 2017. "Beyond monofractional kinetics," Chaos, Solitons & Fractals, Elsevier, vol. 102(C), pages 210-217.

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