The Dual Action of Fractional Multi Time Hamilton Equations

The fractional multi time Lagrangian equations has been derived for dynamical systems within Riemann-Liouville derivatives. The fractional multi time Hamiltonian is introduced as Legendre transformation of multi time Lagrangian. The corresponding fractional Euler-Lagrange and the Hamilton equations are obtained and the fractional multi time constant of motion are discussed.     

Fractional statistical mechanics

The Liouville and first Bogoliubov hierarchy equations with derivatives of noninteger order are derived. The fractional Liouville equation is obtained from the conservation of probability to find a system in a fractional volume element. This equation is used to obtain Bogoliubov hierarchy and fractional kinetic equations with fractional derivatives. Statistical mechanics of fractional generalization of the Hamiltonian systems is discussed. Liouville and Bogoliubov equations with fractional coordinate and momenta derivatives are considered as a basis to derive fractional kinetic equations. The Fokker-Planck-Zaslavsky equation that has fractional phase-space derivatives is obtained from the fractional Bogoliubov equation. The linear fractional kinetic equation for distribution of the charged particles is considered.     

Fractional Newtonian mechanics

In the present paper, we have introduced the generalized Newtonian law and fractional Langevin equation. We have derived potentials corresponding to different kinds of forces involving both the right and the left fractional derivatives. Illustrative examples have worked out to explain the formalism.     

Lower Bounds for an Integral Involving Fractional Laplacians and the Generalized Navier-Stokes Equations in Besov Spaces

When estimating solutions of dissipative partial differential equations in L^p-related spaces, we often need lower bounds for an integral involving the dissipative term. If the dissipative term is given by the usual Laplacian −Δ, lower bounds can be derived through integration by parts and embedding inequalities. However, when the Laplacian is replaced by the fractional Laplacian (−Δ)^α, the approach of integration by parts no longer applies. In this paper, we obtain lower bounds for the integral involving (−Δ)^α by combining pointwise inequalities for (−Δ)^α with Bernstein's inequalities for fractional derivatives. As an application of these lower bounds, we establish the existence and uniqueness of solutions to the generalized Navier-Stokes equations in Besov spaces. The generalized Navier-Stokes equations are the equations resulting from replacing −Δ in the Navier-Stokes equations by (−Δ)^α.     

Langevin Equation Involving Three Fractional Orders

In this paper, the existence and uniqueness of initial value problems for nonlinear Langevin equation involving three fractional orders are discussed. We use a new norm that is convenient for the fractional and singular differential equations. This norm and the contraction mapping principles are the main tools for investigating the existence and uniqueness of the desired issue. The fractional derivatives are described in Caputo sense.

     

Cosmological Models with Fractional Derivatives and FractionalAction Functional

Cosmological models of a scalar field with dynamical equations containing fractional derivatives or derived from the Einstein-Hilbert action of fractional Order, are constructed. A number of exact solutions to those equations of fractional cosmological models in both cases is given.     

Fractional differential kinetics of dispersive transport as the consequence of its self-similarity

Analysis of the experimental data on anomalous transport in disordered semiconductors reveals a universal behavior of the transient photocurrent, which indicates the self-similarity of the transport process. The kinetic equations of the process, which contain a time derivative of a fractional order α ∈ (0, 1), have been derived on the basis of the self-similarity principle. These equations satisfy the “correspondence principle”: in the limit α → 1, they are transformed to the normal-diffusion equations. The solutions of the equations with fractional derivatives are expressed in terms of stable densities and are in good agreement with the experimental data.     

Fractional dynamics of systems with long-range space interaction and temporal memory

Field equations with time and coordinate derivatives of noninteger order are derived from a stationary action principle for the cases of power-law memory function and long-range interaction in systems. The method is applied to obtain a fractional generalization of the Ginzburg-Landau and nonlinear Schrödinger equations. As another example, dynamical equations for particle chains with power-law interaction and memory are considered in the continuous limit. The obtained fractional equations can be applied to complex media with/without random parameters or processes.     

On the Fractional Hamilton and Lagrange Mechanics

The fractional generalization of Hamiltonian mechanics is constructed by using the Lagrangian involving fractional derivatives. In this paper the equation of projectile motion with air friction using fractional Hamiltonian mechanics and equation for current loop involving electric source, a resistor, an inductor and a capacitor has been obtained. Furthermore, fractional optics has been introduced.     

Analytical Approach for Nonlinear Partial Differential Equations of Fractional Order

The purpose of the paper is to present analytical and numerical solutions of a degenerate parabolic equation with time-fractional derivatives arising in the spatial diffusion of biological populations. The homotopy-perturbation method is employed for solving this class of equations, and the time-fractional derivatives are described in the sense of Caputo. Comparisons are made with those derived by Adomian's decomposition method, revealing that the homotopy perturbation method is more accurate and convenient than the Adomian's decomposition method. Furthermore, the results reveal that the approximate solution continuously depends on the time-fractional derivative and the proposed method incorporating the Caputo derivatives is a powerful and efficient technique for solving the fractional differential equations without requiring linearization or restrictive assumptions. The basis ideas presented in the paper can be further applied to solve other similar fractional partial differential equations.     

Similarity solutions for a class of Fractional Reaction-Diffusion equation

This work studies exact solvability of a class of fractional reaction-diffusion equation with the Riemann-Liouville fractional derivatives on the half-line in terms of the similarity solutions. We derived the conditions for the equation to possess scaling symmetry even with the fractional derivatives. Relations among the scaling exponents are determined, and the appropriate similarity variable introduced. With the similarity variable we reduced the differential partial differential equation to a fractional ordinary differential equation. Exactly solvable systems are then identified by matching the resulted ordinary differential equation with the known exactly solvable fractional ones. Several examples involving the three-parameter Mittag-Leffler function (Kilbas-Saigo function) are presented. The models discussed here turn out to correspond to superdiffusive systems.     

Fractional Fokker-Planck equation for fractal media

We consider the fractional generalizations of equation that defines the medium mass. We prove that the fractional integrals can be used to describe the media with noninteger mass dimensions. Using fractional integrals, we derive the fractional generalization of the Chapman-Kolmogorov equation (Smolukhovski equation). In this paper fractional Fokker-Planck equation for fractal media is derived from the fractional Chapman-Kolmogorov equation. Using the Fourier transform, we get the Fokker-Planck-Zaslavsky equations that have fractional coordinate derivatives. The Fokker-Planck equation for the fractal media is an equation with fractional derivatives in the dual space.     

Fractional vector calculus and fractional Maxwell’s equations

The theory of derivatives and integrals of non-integer order goes back to Leibniz, Liouville, Grunwald, Letnikov and Riemann. The history of fractional vector calculus (FVC) has only 10 years. The main approaches to formulate a FVC, which are used in the physics during the past few years, will be briefly described in this paper. We solve some problems of consistent formulations of FVC by using a fractional generalization of the Fundamental Theorem of Calculus. We define the differential and integral vector operations. The fractional Green’s, Stokes’ and Gauss’s theorems are formulated. The proofs of these theorems are realized for simplest regions. A fractional generalization of exterior differential calculus of differential forms is discussed. Fractional nonlocal Maxwell’s equations and the corresponding fractional wave equations are considered.     

On the Time Fractional Generalized Fisher Equation:Group Similarities and Analytical Solutions

In this letter,the Lie point symmetries of the time fractional Fisher(TFF) equation have been derived using a systematic investigation.Using the obtained Lie point symmetries,TFF equation has been transformed into a different nonlinear fractional ordinary differential equations with the Erd′elyi–Kober fractional derivative which depends on the parameter α.After that some invariant solutions of underlying equation are reported.     

Numerical Solutions of Fractional Boussinesq Equation

Based upon the Adomian decomposition method, a scheme is developed to obtain numerical solutions of a fractional Boussinesq equation with initial condition, which is introduced by replacing some order time and space derivatives by fractional derivatives. The fractional derivatives are described in the Caputo sense. So the traditional Adomian decomposition method for differential equations of integer order is directly extended to derive explicit and numerical solutions of the fractional differential equations. The solutions of our model equation are calculated in the form of convergent series with easily computable components.     

Fractional Hamilton’s equations of motion in fractional time

The Hamiltonian formulation for mechanical systems containing Riemman-Liouville fractional derivatives are investigated in fractional time. The fractional Hamilton’s equations are obtained and two examples are investigated in detail.      

Fractional Nambu Mechanics

The fractional generalization of Nambu mechanics is constructed by using the differential forms and exterior derivatives of fractional orders. The generalized Pfaffian equations are obtained and one example is investigated in details. On leave of absence from Institute of Space Sciences, P.O. Box, MG-23, 76900, Magurele-Bucharest, Romania.     

Fractional hamilton formalism within caputo’s derivative

In this paper we develop a fractional Hamiltonian formulation for dynamic systems defined in terms of fractional Caputo derivatives. Expressions for fractional canonical momenta and fractional canonical Hamiltonian are given, and a set of fractional Hamiltonian equations are obtained. Using an example, it is shown that the canonical fractional Hamiltonian and the fractional Euler-Lagrange formulations lead to the same set of equations.     

Fractional motions

Brownian motion is the archetypal model for random transport processes in science and engineering. Brownian motion displays neither wild fluctuations (the “Noah effect”), nor long-range correlations (the “Joseph effect”). The quintessential model for processes displaying the Noah effect is Lévy motion, the quintessential model for processes displaying the Joseph effect is fractional Brownian motion, and the prototypical model for processes displaying both the Noah and Joseph effects is fractional Lévy motion. In this paper we review these four random-motion models–henceforth termed “fractional motions” –via a unified physical setting that is based on Langevin’s equation, the Einstein–Smoluchowski paradigm, and stochastic scaling limits. The unified setting explains the universal macroscopic emergence of fractional motions, and predicts–according to microscopic-level details–which of the four fractional motions will emerge on the macroscopic level. The statistical properties of fractional motions are classified and parametrized by two exponents—a “Noah exponent” governing their fluctuations, and a “Joseph exponent” governing their dispersions and correlations. This self-contained review provides a concise and cohesive introduction to fractional motions.     

Noether Symmetry and Conserved Quantities of Fractional Birkhoffian System in Terms of Herglotz Variational Problem

The aim of this paper is to study the Herglotz variational principle of the fractional Birkhoffian system and its Noether symmetry and conserved quantities. First, the fractional Pfaff-Herglotz action and the fractional PfaffHerglotz principle are presented. Second, based on different definitions of fractional derivatives, four kinds of fractional Birkhoff's equations in terms of the Herglotz variational principle are established. Further, the definition and criterion of Noether symmetry of the fractional Birkhoffian system in terms of the Herglotz variational problem are given. According to the relationship between the symmetry and the conserved quantities, the Noether's theorems within four different fractional derivatives are derived, which can reduce to the Noether's theorem of the Birkhoffian system in terms of the Herglotz variational principle under the classical conditions. As applications of the Noether's t heorems of the fractional Birkhoffian system in terms of the Herglotz variational principle, an example is given at the end of this paper.