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Stochastic solution to a time-fractional attenuated wave equation

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Abstract

The power law wave equation uses two different fractional derivative terms to model wave propagation with power law attenuation. This equation averages complex nonlinear dynamics into a convenient, tractable form with an explicit analytical solution. This paper develops a random walk model to explain the appearance and meaning of the fractional derivative terms in that equation, and discusses an application to medical ultrasound. In the process, a new strictly causal solution to this fractional wave equation is developed.

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Acknowledgements

This research was partially supported by NIH grant R01-EB012079 and NSF grants DMS-1025486 and DMS-0803360.

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Authors and Affiliations

  1. Department of Statistics and Probability, Michigan State University, East Lansing, MI, 48824, USA

    Mark M. Meerschaert, Peter Straka & Yuzhen Zhou

  2. Department of Electrical and Computer Engineering, Michigan State University, East Lansing, MI, 48824, USA

    Robert J. McGough

Authors
  1. Mark M. Meerschaert
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  2. Peter Straka
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  3. Yuzhen Zhou
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  4. Robert J. McGough
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Correspondence to Mark M. Meerschaert.

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Meerschaert, M.M., Straka, P., Zhou, Y. et al. Stochastic solution to a time-fractional attenuated wave equation. Nonlinear Dyn 70, 1273–1281 (2012). https://doi.org/10.1007/s11071-012-0532-x

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