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FRACTIONAL WAVE EQUATIONS WITH ATTENUATION.

Peter Straka1, Mark M Meerschaert1, Robert J McGough2

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

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Summary
This summary is machine-generated.

This study introduces stochastic and weak solutions for fractional wave equations with power-law attenuation, crucial for understanding sound wave propagation in inhomogeneous media like medical ultrasound.

Keywords:
Fractional derivativeattenuationcontinuous time random walkdispersionstable lawsubordinationwave equation

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Area of Science:

  • Physics
  • Applied Mathematics
  • Acoustics

Background:

  • Fractional wave equations model power-law frequency attenuation in inhomogeneous media.
  • These models are particularly relevant for medical ultrasound applications.
  • Previous work includes models by Caputo, Szabo, Chen, Holm, and Kelly et al.

Purpose of the Study:

  • To develop stochastic solutions for the power-law wave equation proposed by Kelly et al.
  • To develop weak solutions for the power-law wave equation proposed by Kelly et al.

Main Methods:

  • Development of stochastic solution methodologies.
  • Formulation of weak solution frameworks.
  • Analysis of the power-law wave equation within these frameworks.

Main Results:

  • Successful derivation of stochastic solutions.
  • Successful derivation of weak solutions.
  • Demonstration of applicability to power-law attenuation phenomena.

Conclusions:

  • The developed stochastic and weak solutions provide a robust mathematical framework for analyzing wave propagation with attenuation.
  • These solutions enhance the understanding and modeling capabilities for applications like medical ultrasound.