Fractional dynamics of systems with long-range interaction

We consider one-dimensional chain of coupled linear and nonlinear oscillators with long-range power wise interaction defined by a term proportional to 1/|n-m|^{\alpha+1}. Continuous medium equation for this system can be obtained in the so-called infrared limit when the wave number tends to zero. We construct a transform operator that maps the system of large number of ordinary differential equations of motion of the particles into a partial differential equation with the Riesz fractional derivative of order \alpha, when 0<\alpha<2. Few models of coupled oscillators are considered and their synchronized states and localized structures are discussed in details. Particularly, we discuss some solutions of time-dependent fractional Ginzburg-Landau (or nonlinear Schrodinger) equation.Comment: arXiv admin note: substantial overlap with arXiv:nlin/051201

Oscillation of a time fractional partial differential equation

We consider a time fractional partial differential equation subject to the Neumann boundary condition. Several sufficient conditions are established for oscillation of solutions of such equation by using the integral averaging method and a generalized Riccati technique. The main results are illustrated by examples

FORCED OSCILLATION FOR A CLASS OF FRACTIONAL PARABOLIC PARTIAL DIFFERENTIAL EQUATIONS

We investigate the oscillation of class of time fractional partial dierential equationof the formfor (x; t) 2 R+ = G; R+ = [0;1); where is a bounded domain in RN with a piecewisesmooth boundary @ ; 2 (0; 1) is a constant, D +;t is the Riemann-Liouville fractional derivativeof order of u with respect to t and is the Laplacian operator in the Euclidean N- space RNsubject to the Neumann boundary conditionWe will obtain sucient conditions for the oscillation of class of fractional partial dierentialequations by utilizing generalized Riccatti transformation technique and the integral averagingmethod. We illustrate the main results through examples

Hamiltonian formalism of fractional systems

In this paper we consider a generalized classical mechanics with fractional derivatives. The generalization is based on the time-clock randomization of momenta and coordinates taken from the conventional phase space. The fractional equations of motion are derived using the Hamiltonian formalism. The approach is illustrated with a simple-fractional oscillator in a free state and under an external force. Besides the behavior of the coupled fractional oscillators is analyzed. The natural extension of this approach to continuous systems is stated. The interpretation of the mechanics is discussed.Comment: 16 pages, 5 figure

Periodically Forced Nonlinear Oscillators With Hysteretic Damping

We perform a detailed study of the dynamics of a nonlinear, one-dimensional oscillator driven by a periodic force under hysteretic damping, whose linear version was originally proposed and analyzed by Bishop in [1]. We first add a small quadratic stiffness term in the constitutive equation and construct the periodic solution of the problem by a systematic perturbation method, neglecting transient terms as $t\rightarrow \infty$. We then repeat the analysis replacing the quadratic by a cubic term, which does not allow the solutions to escape to infinity. In both cases, we examine the dependence of the amplitude of the periodic solution on the different parameters of the model and discuss the differences with the linear model. We point out certain undesirable features of the solutions, which have also been alluded to in the literature for the linear Bishop's model, but persist in the nonlinear case as well. Finally, we discuss an alternative hysteretic damping oscillator model first proposed by Reid [2], which appears to be free from these difficulties and exhibits remarkably rich dynamical properties when extended in the nonlinear regime.Comment: Accepted for publication in the Journal of Computational and Nonlinear Dynamic

Numerical Investigation of the Fractional Oscillation Equations under the Context of Variable Order Caputo Fractional Derivative via Fractional Order Bernstein Wavelets

This article describes an approximation technique based on fractional order Bernstein wavelets for the numerical simulations of fractional oscillation equations under variable order, and the fractional order Bernstein wavelets are derived by means of fractional Bernstein polynomials. The oscillation equation describes electrical circuits and exhibits a wide range of nonlinear dynamical behaviors. The proposed variable order model is of current interest in a lot of application areas in engineering and applied sciences. The purpose of this study is to analyze the behavior of the fractional force-free and forced oscillation equations under the variable-order fractional operator. The basic idea behind using the approximation technique is that it converts the proposed model into non-linear algebraic equations with the help of collocation nodes for easy computation. Different cases of the proposed model are examined under the selected variable order parameters for the first time in order to show the precision and performance of the mentioned scheme. The dynamic behavior and results are presented via tables and graphs to ensure the validity of the mentioned scheme. Further, the behavior of the obtained solutions for the variable order is also depicted. From the calculated results, it is observed that the mentioned scheme is extremely simple and efficient for examining the behavior of nonlinear random (constant or variable) order fractional models occurring in engineering and science.Comment: This is a preprint of a paper whose final and definite form is published Open Access in 'Mathematics' at [http://dx.doi.org/10.3390/math11112503

Differential/Difference Equations

The study of oscillatory phenomena is an important part of the theory of differential equations. Oscillations naturally occur in virtually every area of applied science including, e.g., mechanics, electrical, radio engineering, and vibrotechnics. This Special Issue includes 19 high-quality papers with original research results in theoretical research, and recent progress in the study of applied problems in science and technology. This Special Issue brought together mathematicians with physicists, engineers, as well as other scientists. Topics covered in this issue: Oscillation theory; Differential/difference equations; Partial differential equations; Dynamical systems; Fractional calculus; Delays; Mathematical modeling and oscillations

Combustion instabilities: mating dance of chemical, combustion, and combustor dynamics

Combustion instabilities exist as consequences of interactions among three classes of phenomena: chemistry and chemical dynamics; combustion dynamics; and combustor dynamics. These dynamical processes take place simultaneously in widely different spatial scales characterized by lengths roughly in the ratios (10^(-3) - 10^(-6)):1:(10^3-10^6). However, due to the wide differences in the associated characteristic velocities, the corresponding time scales are all close. The instabilities in question are observed as oscillations having a time scale in the range of natural acoustic oscillations. The apparent dominance of that single macroscopic time scale must not be permitted to obscure the fact that the relevant physical processes occur on three disparate length scales. Hence, understanding combustion instabilities at the practical level of design and successful operation is ultimately based on understanding three distinct sorts of dynamics