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January 1, 1989Journal of Mathematical Physics

Fractional diffusion and wave equations

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Authors

WSW. R. SchneiderWWWalter Wyss

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Overview

Theoretical analysis reveals closed-form Green’s functions for fractional diffusion and wave equations in arbitrary dimensions, indicating fractional diffusion acts as a probability density.

Key Points

  • Formulate fractional diffusion and wave equations as integrodifferential equations and derive their exact Green's functions across arbitrary spatial dimensions.
  • Replaced standard time derivatives in diffusion and wave equations with convolution integrals using power-law kernels involving the gamma function.
  • Vary the fractional order parameter α over the continuous intervals (0,1) for diffusion and (1,2) for wave propagation in arbitrary space dimensions.
  • Obtained exact closed-form Green’s functions for both fractional diffusion and fractional wave equations expressed through Fox functions.
  • Demonstrated that the fundamental solution (Green’s function) for the fractional diffusion equation satisfies the mathematical properties of a true probability density.

Cite This Study

Schneider et al. (1989) studied this question.

synapsesocial.com/papers/69d8d050d2f7327e70ae4601https://doi.org/10.1063/1.528578
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