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STOCHASTIC SOLUTIONS FOR FRACTIONAL WAVE EQUATIONS.

Mark M Meerschaert1, René L Schilling2, Alla Sikorskii3

  • 1Department of Statistics & Probability, Michigan State University, East Lansing MI 48824 USA. URL: http://www.stt.msu.edu/users/mcubed/

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The fractional wave equation governs a stochastic wave propagation model. Deterministic time is replaced by the inverse of a stable subordinator, linking fractional calculus to stochastic processes.

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Area of Science:

  • Physics
  • Applied Mathematics
  • Stochastic Processes

Background:

  • The classical wave equation describes wave propagation using second-order time derivatives.
  • Fractional calculus offers a way to model anomalous diffusion and wave phenomena with memory effects.
  • Stochastic processes are essential for modeling systems with inherent randomness.

Purpose of the Study:

  • To establish a connection between fractional wave equations and stochastic models of wave propagation.
  • To demonstrate that a specific fractional wave equation governs a particular stochastic process.
  • To elucidate the relationship between the order of the fractional derivative and the properties of the stochastic subordinator.

Main Methods:

  • The study employs the Caputo fractional derivative to generalize the wave equation.
  • It introduces a stable subordinator process to model time in a stochastic framework.
  • The paper mathematically derives the governing equation for this stochastic wave propagation model.

Main Results:

  • The fractional wave equation, utilizing a Caputo derivative of order $\alpha$ (1 < $\alpha$ < 2), is shown to govern the stochastic model.
  • Deterministic time is replaced by the inverse of a stable subordinator with index $\beta = \alpha/2$.
  • This establishes a direct link between fractional calculus parameters and stochastic process characteristics.

Conclusions:

  • Fractional wave equations provide a powerful framework for describing stochastic wave propagation.
  • The choice of the fractional derivative order directly dictates the nature of the underlying stochastic process.
  • This work bridges the gap between deterministic fractional models and stochastic phenomena in wave dynamics.