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Stochastic solution of space-time fractional diffusion equations.

Mark M Meerschaert¹, David A Benson, Hans-Peter Scheffler

¹Department of Mathematics, University of Nevada, Reno, Nevada 89557-0084, USA. mcubed@unr.edu

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics

|May 15, 2002

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Replacing integer time derivatives with fractional derivatives in diffusion equations leads to inverse stable subordinators and Mittag-Leffler distributions. This provides explicit solutions for space-time fractional diffusion, enhancing understanding of these complex models.

Area of Science:

Physics
Mathematics
Applied Mathematics

Background:

Classical and anomalous diffusion equations commonly utilize integer derivatives.
Pseudodifferential operators, including fractional derivatives, are also employed in spatial dimensions.

Purpose of the Study:

To investigate the implications of replacing integer time derivatives with fractional derivatives in diffusion equations.
To derive explicit solutions for space-time fractional diffusion equations.
To gain deeper insight into the mathematical meaning of these fractional diffusion models.

Main Methods:

The study employs fractional calculus, specifically focusing on the subordination of stochastic solutions.
Analysis involves inverse stable subordinator processes and their associated probability distributions.

Related Experiment Videos

The method is applied to diffusion equations with multiscaling space-fractional derivatives.

Main Results:

Replacing the integer time derivative with a fractional derivative subordinates the stochastic solution to an inverse stable subordinator.
The probability distributions arising from this substitution are of the Mittag-Leffler type.
Explicit solutions are obtained for space-time fractional diffusion equations.

Conclusions:

Fractional time derivatives in diffusion models lead to a specific class of stochastic processes (inverse stable subordinators).
Mittag-Leffler distributions naturally emerge in these fractional diffusion scenarios.
The findings offer a clearer interpretation and explicit solvability for complex fractional diffusion equations.