Fractional Diffusion and Wave Propagation

  • Yuri LuchkoEmail author
Reference work entry


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.


Fundamental Solution Fractional Derivative Fractional Differential Equation Pulse Velocity Continuous Time Random Walk 
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© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Department of Mathematics, Physics, and ChemistryBeuth Technical University of Applied Sciences BerlinBerlinGermany

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