Related papers: The Fractional Chapman-Kolmogorov Equation

The notion of fractional dynamics is related to equations of motion with one or a few terms with derivatives of a fractional order. This type of equation appears in the description of chaotic dynamics, wave propagation in fractal media, and…

In this study the general formula for differential and integral operations of fractional calculus via fractal operators by the method of cumulative diminution and cumulative growth is obtained. The under lying mechanism in the success of…

It is pointed out that, for the fractional Fokker-Planck equation for subdiffusion proposed by Metzler, Barkai, and Klafter [Phys. Rev. Lett. 82 (1999) 3563], there are four types of infinitely many exact solutions associated with the newly…

We address the problem of constructing approximations based on orthogonal polynomials that preserve an arbitrary set of moments of a given function without loosing the spectral convergence property. To this aim, we compute the constrained…

The boundary conditions for the Fokker-Planck equations, forward and backward ones are directly derived from the Chapman-Kolmogorov equation for M-dimensional region with boundaries. The boundaries are assumed, in addition, to be able to…

We obtain necessary optimality conditions for variational problems with a Lagrangian depending on a Caputo fractional derivative, a fractional and an indefinite integral. Main results give fractional Euler-Lagrange type equations and…

Based on the continuous time random walk, we derive the Fokker-Planck equations with Caputo-Fabrizio fractional derivative, which can effectively model a variety of physical phenomena, especially, the material heterogeneities and structures…

Two approximations, derived from continuous expansions of Riemann-Liouville fractional derivatives into series involving integer order derivatives, are studied. Using those series, one can formally transform any problem that contains…

We relate the convergence of time-changed processes driven by fractional equations to the convergence of corresponding Dirichlet forms. The fractional equations we dealt with are obtained by considering a general fractional operator in…

We propose a fast and stable method for constructing matrix approximations to fractional integral operators applied to series in the Chebyshev fractional polynomials. This method utilizes a recurrence relation satisfied by the fractional…

In this paper, we introduce a new classical fractional particle model incorporating fractional first derivatives. This model represents a natural extension of the standard classical particle with kinetic energy being quadratic in fractional…

We use the fractional integrals to describe fractal media. We consider the fractal media as special ("fractional") continuous media. We discuss the possible experimental testing of the continuous medium model for fractal media that is…

Fractional derivative can be defined as a fractional power of derivative. The commutator (i/h)[H, ], which is used in the Heisenberg equation, is a derivation on a set of observables. A derivation is a map that satisfies the Leibnitz rule.…

Semi-Markov processes are a generalization of Markov processes since the exponential distribution of time intervals is replaced with an arbitrary distribution. This paper provides an integro-differential form of the Kolmogorov's backward…

Assuming a fractal distribution of matter in the universe, consequences that follow from the General Theory of Relativity and the Copernican Principle for fractal cosmology are examined. The change in perspective necessary to deal with a…

Fractional derivatives are generalization to classical integer-order derivatives. The rules which are true for classical derivative need not hold for the fractional derivatives, for example, we cannot simply add the fractional orders…

We demonstrate that the Fokker-Planck equation can be generalized into a 'Fractional Fokker-Planck' equation, i.e. an equation which includes fractional space differentiations, in order to encompass the wide class of anomalous diffusions…

Starting from a recent result expressing the Lerch zeta function as a fractional derivative, we consider further fractional derivatives of the Lerch zeta function with respect to different variables. We establish a partial differential…

For the particles undergoing the anomalous diffusion with different waiting time distributions for different internal states, we derive the Fokker-Planck and Feymann-Kac equations, respectively, describing positions of the particles and…

A Langevin equation with a special type of additive random source is considered. This random force presents a fractional order derivative of white noise, and leads to a power-law time behavior of the mean square displacement of a particle,…