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Fractional kinetic equations: solutions and applications.
Alexander I. Saichev1, George M. Zaslavsky
1Radiophysics Department, Nizhniy Novgorod State University, 23 Gagarin Str., Nizhniy Novgorod, 603600, Russia.
Chaos (Woodbury, N.Y.)
|June 5, 2003
Summary
This study introduces a fractional diffusion equation to model anomalous kinetics in chaotic systems. It presents a method to decompose solutions into fractal Brownian and Levy-type processes for better understanding complex dynamics.
Area of Science:
- Physics
- Mathematical Physics
- Nonlinear Dynamics
Background:
- Anomalous kinetics in dynamical systems often exhibit complex behaviors not captured by standard diffusion models.
- Fractional calculus provides a framework to describe systems with memory and non-local interactions, relevant for chaotic motion.
Purpose of the Study:
- To introduce and analyze a symmetrized fractional diffusion equation with a source term.
- To develop and apply a method for finding asymptotic solutions to this generalized equation.
- To extend fractional calculus concepts to the Kolmogorov-Feller equation.
Main Methods:
- Utilized a method analogous to separation of variables for solving the fractional diffusion equation.
- Analyzed asymptotic solutions by decomposing them into fractal Brownian motion and Levy-type processes.
- Introduced a fractional generalization of the Kolmogorov-Feller equation.
Main Results:
- Derived different asymptotic solutions for the symmetrized fractional diffusion equation.
- The solution method offers a clear physical interpretation.
- Successfully generalized the Kolmogorov-Feller equation within a fractional calculus framework.
Conclusions:
- The fractional diffusion equation provides a powerful tool for modeling anomalous kinetics in chaotic systems.
- The proposed solution method offers physical insights into complex dynamical processes.
- Fractional generalizations of diffusion and Kolmogorov-Feller equations are valuable for advanced theoretical studies.