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Solution of a modified fractional diffusion equation

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  • Langlands, T.A.M.

Abstract

Recently, a modified fractional diffusion equation has been proposed [I. Sokolov, J. Klafter, From diffusion to anomalous diffusion: a century after Einstein's brownian motion, Chaos 15 (2005) 026103; A.V. Chechkin, R. Gorenflo, I.M. Sokolov, V.Yu. Gonchar, Distributed order time fractional diffusion equation, Frac. Calc. Appl. Anal. 6 (3) (2003) 259279; I.M. Sokolov, A.V. Checkin, J. Klafter, Distributed-order fractional kinetics, Acta. Phys. Pol. B 35 (2004) 1323.] for describing processes that become less anomalous as time progresses by the inclusion of a second fractional time derivative acting on the diffusion term. In this letter we give the solution of the modified equation on an infinite domain. In contrast to the solution of the traditional fractional diffusion equation, the solution of the modified equation requires an infinite series of Fox functions instead of a single Fox function.

Suggested Citation

  • Langlands, T.A.M., 2006. "Solution of a modified fractional diffusion equation," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 367(C), pages 136-144.
    • Handle: RePEc:eee:phsmap:v:367:y:2006:i:c:p:136-144
    DOI: 10.1016/j.physa.2005.12.012
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    References listed on IDEAS

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    1. Raberto, Marco & Scalas, Enrico & Mainardi, Francesco, 2002. "Waiting-times and returns in high-frequency financial data: an empirical study," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 314(1), pages 749-755.
    2. Scalas, Enrico & Gorenflo, Rudolf & Mainardi, Francesco, 2000. "Fractional calculus and continuous-time finance," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 284(1), pages 376-384.
    3. Mainardi, Francesco & Raberto, Marco & Gorenflo, Rudolf & Scalas, Enrico, 2000. "Fractional calculus and continuous-time finance II: the waiting-time distribution," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 287(3), pages 468-481.
    4. Enrico Scalas & Rudolf Gorenflo & Hugh Luckock & Francesco Mainardi & Maurizio Mantelli & Marco Raberto, 2004. "Anomalous waiting times in high-frequency financial data," Quantitative Finance, Taylor & Francis Journals, vol. 4(6), pages 695-702.
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    Cited by:

    1. Chen, Y. & Chen, Chang-Ming, 2018. "Numerical simulation with the second order compact approximation of first order derivative for the modified fractional diffusion equation," Applied Mathematics and Computation, Elsevier, vol. 320(C), pages 319-330.
    2. 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.
    3. 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.
    4. Awad, Emad & Sandev, Trifce & Metzler, Ralf & Chechkin, Aleksei, 2021. "Closed-form multi-dimensional solutions and asymptotic behaviors for subdiffusive processes with crossovers: I. Retarding case," Chaos, Solitons & Fractals, Elsevier, vol. 152(C).
    5. Tawfik, Ashraf M. & Elkamash, I.S., 2022. "On the correlation between Kappa and Lévy stable distributions," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 601(C).
    6. Guo, Gang & Li, Kun & Wang, Yuhui, 2015. "Exact solutions of a modified fractional diffusion equation in the finite and semi-infinite domains," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 417(C), pages 193-201.
    7. Guo, Gang & Chen, Bin & Zhao, Xinjun & Zhao, Fang & Wang, Quanmin, 2015. "First passage time distribution of a modified fractional diffusion equation in the semi-infinite interval," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 433(C), pages 279-290.

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