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Fractional Diffusion and Wave Propagation

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Handbook of Geomathematics

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Abstract

In this chapter, a short overview of the current research towards applications of the partial differential equations of an arbitrary (not necessarily integer) order for modeling of the anomalous transport processes (diffusion, heat transfer, and wave propagation) in the nonhomogeneous media is presented. On the microscopic level, these processes are described by the continuous time random walk (CTRW) model that is a starting point for derivation of some deterministic equations for the time- and space-averaged quantities that characterize the transport processes on the macroscopic level. In this work, the deterministic models are derived in the form of the partial differential equations of the fractional order. In particular, a generalized time-fractional diffusion equation and a time- and space-fractional wave equation are introduced and analyzed in detail. Finally, some open questions and directions for further work are suggested.

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Author information

Authors and Affiliations

  1. Department of Mathematics, Physics, and Chemistry, Beuth Technical University of Applied Sciences Berlin, Berlin, Germany

    Yuri Luchko

Authors
  1. Yuri Luchko
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Correspondence to Yuri Luchko .

Editor information

Editors and Affiliations

  1. Geomathematics Group, University of Kaiserslautern, Kaiserslautern, Germany

    Willi Freeden

  2. Department of Mathematics, University of Central Florida, Orlando, Florida, USA

    M. Zuhair Nashed

  3. Computational Mathematics, Technische Universität Braunschweig, Braunschweig, Germany

    Thomas Sonar

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Luchko, Y. (2015). Fractional Diffusion and Wave Propagation. In: Freeden, W., Nashed, M., Sonar, T. (eds) Handbook of Geomathematics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-54551-1_60

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