Related papers: Fractional Action-Like Variational Problems
Fractional kinetic theory plays a vital role in describing anomalous diffusion in terms of complex dynamics generating semi-Markovian processes. Recently, the variational principle and associated Levy Ansatz have been proposed in order to…
We introduce a stochastic fractional calculus. As an application, we present a stochastic fractional calculus of variations, which generalizes the fractional calculus of variations to stochastic processes. A stochastic fractional…
Fractional calculus allows one to generalize the linear, one-dimensional, diffusion equation by replacing either the first time derivative or the second space derivative by a derivative of fractional order. The fundamental solutions of…
Field equations with time and coordinates derivatives of noninteger order are derived from stationary action principle for the cases of power-law memory function and long-range interaction in systems. The method is applied to obtain a…
We develop Cresson's nondifferentiable calculus of variations on the space of H\"{o}lder functions. Several quantum variational problems are considered: with and without constraints, with one and more than one independent variable, of first…
In this paper the author presents the results of the preliminary investigation of fractional dynamical systems based on the results of numerical simulations of fractional maps. Fractional maps are equivalent to fractional differential…
In this paper we present multivariate space-time fractional Poisson processes by considering common random time-changes of a (finite-dimensional) vector of independent classical (non-fractional) Poisson processes. In some cases we also…
In the present work, we formulate a necessary condition for functionals with Lagrangians depending on fractional derivatives of differentiable functions to possess an extremum. The Euler-Lagrange equation we obtained generalizes previously…
Fractional dynamics of relativistic particle is discussed. Derivatives of fractional orders with respect to proper time describe long-term memory effects that correspond to intrinsic dissipative processes. Relativistic particle subjected 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…
The aim of this paper is to bring together two approaches to non-conservative systems -- the generalized variational principle of Herglotz and the fractional calculus of variations. Namely, we consider functionals whose extrema are sought,…
We introduce a discrete-time fractional calculus of variations. First and second order necessary optimality conditions are established. Examples illustrating the use of the new Euler-Lagrange and Legendre type conditions are given. They…
In this paper we provide a definition of fractional gradient operators, related to directional derivatives. We develop a fractional vector calculus, providing a probabilistic interpretation and mathematical tools to treat multidimensional…
Fractional analysis is applied to describe classical dynamical systems. Fractional derivative can be defined as a fractional power of derivative. The infinitesimal generators {H, .} and L=G(q,p) \partial_q+F(q,p) \partial_p, which are used…
We review some techniques from non-linear analysis in order to investigate critical paths for the action functional in the calculus of variations applied to physics. Previous attempts to analyse when these are minima ex- ist, but mainly…
Fractional operators play an important role in modeling nonlocal phenomena and problems involving coarse-grained and fractal spaces. The fractional calculus of variations with functionals depending on derivatives and/or integrals of…
Using the fact that extremum of variation of generalized action can lead to the fractional dynamics in the case of systems with long-range interaction and long-term memory function, we consider two different applications of the action…
We consider the fractional generalization of nonholonomic constraints defined by equations with fractional derivatives and provide some examples. The corresponding equations of motion are derived using variational principle.
We review some applications of fractional calculus developed by the author (partly in collaboration with others) to treat some basic problems in continuum and statistical mechanics. The problems in continuum mechanics concern mathematical…
It is argued that the evolution of complex phenomena ought to be described by fractional, differential, stochastic equations whose solutions have scaling properties and are therefore random, fractal functions. To support this argument we…