Stochastic solution to a time-fractional attenuated wave equation

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 explai...

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Bibliographic Details
Published in:Nonlinear dynamics 2012-10, Vol.70 (2), p.1273-1281
Main Authors: Meerschaert, Mark M., Straka, Peter, Zhou, Yuzhen, McGough, Robert J.
Format: Article
Language:English
Subjects:
Automotive Engineering
Classical Mechanics
Control
Dynamical Systems
Engineering
Exact solutions
Mechanical Engineering
Nonlinear dynamics
Original Paper
Power law
Random walk
Vibration
Wave attenuation
Wave equations
Wave propagation
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Summary: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.
ISSN:0924-090X
1573-269X
DOI:10.1007/s11071-012-0532-x