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On the time-fractional Cattaneo equation of distributed order

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  • Awad, Emad

Abstract

The present work revisits the time-fractional Cattaneo equation (TFCE) by considering a generic class of time-fractional derivatives of distributed order, α∈(0,1]. At first, we shed light on two (main) forms of TFCE; one (natural generalization) uses the Caputo fractional derivative and the other (Compte–Metzler generalization) is defined in the Riemann–Liouville sense, and the equivalence condition between their solutions is provided. Some special cases of these two forms are addressed. Hence, different versions of distributed-order TFCE are proposed. The fundamental solution of the two general forms of distributed-order TFCE are derived using integral transform technique. The non-negativity of solutions of TFCE and its distributed-order versions is examined using the “special-type functions” technique; Gorenflo, Luchko and Stojanović (2013). Two classes of order distribution are considered: discrete-order and continuous-order distributions, and some random choices of non-negative order densities affirm graphically the non-negativity of the general forms of distributed-order TFCE. The double-order (as an example on the discrete-order), and exponentially exp(λα) and sinusoidally sin(πμα) distributed-order (as an example on the continuous-order) are chosen. Finally, connections between some models of distributed-order TFCE and continuous time random walk theory are established, and the corresponding mean-squared displacements are discussed.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:phsmap:v:518:y:2019:i:c:p:210-233
    DOI: 10.1016/j.physa.2018.12.005
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    References listed on IDEAS

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    1. Želi, Velibor & Zorica, Dušan, 2018. "Analytical and numerical treatment of the heat conduction equation obtained via time-fractional distributed-order heat conduction law," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 492(C), pages 2316-2335.
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    5. Qi, Haitao & Jiang, Xiaoyun, 2011. "Solutions of the space-time fractional Cattaneo diffusion equation," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 390(11), pages 1876-1883.
    6. 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.
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