nonlinear fractional dynamics on a lattice with long range interactions abstract a unified approach has been developed to study nonlinear dynamics of a 1d lattice of particles with long range power law interaction a classical case is treated in the framework of the generalization of the well known frenkel kontorova chain model for the non nearest interactions quantum dynamics is considered following davydov s approach for molecular excitons in the continuum limit the problem is reduced to dynamical equations with fractional derivatives resulting from the fractional power of the long range interaction fractional generalizations of the sine gordon nonlinear schrodinger and hilbert schrodinger equations have been found there exists a critical value of the power s of the long range potential below the critical value s 3 s 1 2 we obtain equations with fractional derivatives while for s 3 we have the well known nonlinear dynamical equations with space derivatives of integer order long range interaction impact on the quantum lattice propagator has been studied we have shown that the quantum exciton propagator exhibits transition from the well known gaussian like behavior to a power law decay due to the long range interaction a link between 1d quantum lattice dynamics in the imaginary time domain and a random walk model has been discussed

spectral properties of the transition operator associated to a multivariate refinement equation given a finitely supported sequence a on z s and an s s dilation matrix m the transition operator ta is the linear operator defined by tavoau a p b2z s aomabuvobu where a2 z s and v lies in 0oz s u the linear space of all finitely supported sequences on z s in this paper we investigate the spectral properties of the transition operator ta and apply these properties to the study of the approximation and smoothness properties of the normalized solution of the refinement equation a p a2z s aoau omau 1999

fractional variations for dynamical systems hamilton and lagrange approaches fractional generalization of an exterior derivative for calculus of variations is defined the hamilton and lagrange approaches are considered fractional hamilton and euler lagrange equations are derived fractional equations are obtained by fractional variation of lagrangian and hamiltonian that have only integer derivatives

continuous limit of discrete systems with long range interaction discrete systems with long range interactions are considered continuous medium models as continuous limit of discrete chain system are defined long range interactions of chain elements that give the fractional equations for the medium model are discussed the chain equations of motion with long range interaction are mapped into the continuum equation with the riesz fractional derivative we formulate the consistent definition of continuous limit for the systems with long range interactions in this paper we consider a wide class of long range interactions that give fractional medium equations in the continuous limit the power law interaction is a special case of this class

temporal monotonicity of the solutions of some semilinear parabolic equations with fractional diffusion abstract suppose that the functions g and are nonnegative and satisfy suitable regularity conditions then we prove in this work that the parabolic semilinear problem t u t x u t x g x f u t x x t 0 x r n u 0 x x x r n has a unique positive and time monotone solution here is the fractional laplacian with 0 2 and the source term f is a convex function with f 0 0 moreover using the temporal monotonicity we show that the elliptic equation v x g x f v x x x r d with boundary condition lim x v x 0 has a positive solution we provide also sufficient conditions for the integrability of both solutions

multifractality of one electron eigen states in 1d disordered long range models abstract one investigates the spatial structure of one electron eigenstates for the one dimensional anderson model with long range off diagonal disorder h n m 1 1 n m where a b means a uniform random distribution between a and b two cases are considered according to the choices for the on site energy e n namely e n 0 and e n 0 1 for 1 all states are critical and the multifractal spectra f is computed for the states near the center of the band the level spacing distribution function is also obtained and its behavior for both small and large separations are studied

a structure preserving method for a class of nonlinear dissipative wave equations with riesz space fractional derivatives abstract in this manuscript we consider an initial boundary value problem governed by a 1 1 dimensional hyperbolic partial differential equation with constant damping that generalizes many nonlinear wave equations from mathematical physics the model considers the presence of a spatial laplacian of fractional order which is defined in terms of riesz fractional derivatives as well as the inclusion of a generic continuously differentiable potential it is known that the undamped regime has an associated positive energy functional and we show here that it is preserved throughout time under suitable boundary conditions to approximate the solutions of this model we propose a finite difference discretization based on fractional centered differences some discrete quantities are proposed in this work to estimate the energy functional and we show that the numerical method is capable of conserving the discrete energy under the same boundary conditions for which the continuous model is conservative moreover we establish suitable computational constraints under which the discrete energy of the system is positive the method is consistent of second order and is both stable and convergent the numerical simulations shown here illustrate the most important features of our numerical methodology

a bounded and efficient scheme for multidimensional problems with anomalous convection and diffusion abstract this work is motivated by a generalization of the well known burgers fisher and burgers huxley equations in multiple dimensions considering riesz fractional diffusion and convection initial boundary conditions which are positive and bounded are imposed on a closed and bounded rectangular domain in this manuscript we propose a finite difference method to approximate the positive and bounded solutions of the fractional model the methodology is a linear three steps crank nicolson technique which is based on the use of fractional centered differences the properties of fractional centered differences are employed to establish the existence and the uniqueness of solutions of the finite difference method as well as the capability of the technique to preserve the positivity and the boundedness of the approximations we show in this work that the method is capable of preserving some of the constant solutions of the continuous model additionally we prove that our technique is a second order consistent stable and quadratically convergent scheme suitable bounds for the numerical solutions are also derived in this work finally some illustrative simulations show that the method is able to preserve the positivity and the boundedness of the numerical approximations in agreement with the analytic results proved in this work numerical comparisons provided in this work confirm the rate of convergence of the numerical technique

lattice with long range interaction of power law type for fractional non local elasticity abstract lattice models with long range interactions of power law type are suggested as a new type of microscopic model for fractional non local elasticity using the transform operation we map the lattice equations into continuum equation with riesz derivatives of non integer orders the continuum equations that are obtained from the lattice model describe fractional generalization of non local elasticity models particular solutions and correspondent asymptotic of the fractional differential equations for displacement fields are suggested for the static case

transition to chaos in discrete nonlinear schrodinger equation with long range interaction discrete nonlinear schrodinger dnls equation describes a chain of oscillators with nearest neighbor interactions and a specific nonlinear term we consider its modification with long range interaction through a potential proportional to 1 l1 with fractional 2 and l as a distance between oscillators this model is called dnls it exhibits competition between the nonlinearity and a level of correlation between interacting far distanced oscillators that is defined by the value of we consider transition to chaos in this system as a function of and nonlinearity it is shown that decreasing of with respect to nonlinearity stabilize the system connection of the model to the fractional generalization of the nls called fnls in the long wave approximation is also discussed and some of the results obtained for dnls can be correspondingly extended to the fnls

design and analysis of a dissipative scheme to solve a generalized multi dimensional higgs boson equation in the de sitter space time abstract in this work we design a numerically efficient finite difference technique for the solution of a fractional extension of the higgs boson equation in the de sitter space time the model under investigation is a multidimensional equation with riesz fractional derivatives of orders in 0 1 1 2 which considers a generalized potential and a time dependent diffusion factor an energy integral for the mathematical model is readily available and we propose an explicit and consistent numerical technique based on fractional order centered differences with similar hamiltonian properties as the continuous model a fractional energy approach is used then to prove the properties of stability and convergence of the technique for simulation purposes we consider both the classical and the fractional higgs real valued scalar fields in the 3 1 dimensional de sitter space time and find results qualitatively similar to those available in the literature the present work is the first paper to report on a hamiltonian discretization of the higgs boson equation both fractional and non fractional in the de sitter space time and its numerical analysis more precisely the present manuscript is the first paper of the literature in which a dissipation preserving scheme to solve the multi dimensional fractional higgs boson equation in the de sitter space time is proposed and thoroughly analyzed indeed it is worth pointing out that previous efforts used techniques based on the runge kutta method or discretizations that did not preserve the dissipation nor were rigorously analyzed

fractional vector calculus for fractional advection dispersion abstract we develop the basic tools of fractional vector calculus including a fractional derivative version of the gradient divergence and curl and a fractional divergence theorem and stokes theorem these basic tools are then applied to provide a physical explanation for the fractional advection dispersion equation for flow in heterogeneous porous media

a numerically efficient hamiltonian method for fractional wave equations abstract in this work we consider a partial differential equation that extends the well known wave equation the model under consideration is a multidimensional equation which includes the presence of both a damping term and a fractional laplacian of the riesz type homogeneous dirichlet boundary conditions on a closed and bounded spatial interval are considered in this work the mathematical model has a fractional hamiltonian which is conserved when the damping coefficient is equal to zero and dissipated otherwise motivated by these facts we propose a finite difference method to approximate the solutions of the continuous model the method is an implicit scheme which is based on the use of fractional centered differences to approximate the spatial fractional derivatives of the model a discretized form of the hamiltoninan is also proposed in this work and we prove analytically that the method is capable of preserving dissipating the discrete energy when the continuous model preserves dissipates the energy we establish rigorously the properties of consistency stability and convergence of the method and provide some a priori bounds for the numerical solutions moreover we prove the existence and the uniqueness of the numerical solutions as well as the unconditional stability of the method in the linear regime some computer simulations that assess the capability of the method to preserve dissipate the energy are carried out for illustration purposes

compactlike discrete breathers in systems with nonlinear and nonlocal dispersive terms discrete breathers with purely anharmonic short range interaction potentials localize superexponentially becoming compactlike we analyze their spatial localization properties and their dynamical stability several branches of solutions are identified one of them connects to the well known page and sievers takeno lattice modes another one connects with the compacton solutions of rosenau the absence of linear dispersion allows for extremely long lived time quasiperiodic localized excitations adding long range anharmonic interactions leads to an extreme case of competition between length scales defining the spatial breather localization we show that short and long range interaction terms competition results in the appearance of several characteristic crossover lengths and essentially breaks the concept of compactness of the corresponding discrete breathers

global κ exponential asymptotic stabilization of underactuated surface vessels in this work a time invariant switching control law is proposed to make the desired set point of an underactuated surface vessel globally asymptotically stable with exponential convergence rate exponential stable the construction of the controller relies on discontinuous coordinate input transformations and a lyapunov like switching function the result is the first stabilizing law able to achieve global exponential stability of the underactuated surface vessel the effectiveness of the proposed control law is verified by simulation

lattice fractional calculus integration and differentiation of non integer orders for n dimensional physical lattices with long range particle interactions are suggested the proposed lattice fractional derivatives and integrals are represented by kernels of lattice long range interactions such that their fourier series transformations have a power law form with respect to components of wave vector continuous limits for these lattice fractional derivatives and integrals give the continuum derivatives and integrals of non integer orders with respect to coordinates lattice analogs of fractional differential equations that include suggested lattice differential and integral operators can serve as an important element of microscopic approach to nonlocal continuum models in mechanics and physics

large deviation techniques applied to systems with long range interactions we discuss a method to solve models with long range interactions in the microcanonical and canonical ensemble the method closely follows the one introduced by r s ellis physica d 133 106 1999 which uses large deviation techniques we show how it can be adapted to obtain the solution of a large class of simple models which can show ensemble inequivalence the model hamiltonian can have both discrete ising potts and continuous hmf free electron laser state variables this latter extension gives access to the comparison with dynamics and to the study of non equilibrium effects we treat both infinite range and slowly decreasing interactions and in particular we present the solution of the ising model in one dimension with 0 1

fractional ginzburg landau equation for fractal media abstract we derive the fractional generalization of the ginzburg landau equation from the variational euler lagrange equation for fractal media to describe fractal media we use the fractional integrals considered as approximations of integrals on fractals some simple solutions of the ginzburg landau equation for fractal media are considered and different forms of the fractional ginzburg landau equation or nonlinear schrodinger equation with fractional derivatives are presented the agrawal variational principle and its generalization have been applied

fractional equations of kicked systems and discrete maps starting from kicked equations of motion with derivatives of non integer orders we obtain fractional discrete maps these maps are generalizations of well known universal standard dissipative kicked damped rotator maps the main property of the suggested fractional maps is a long term memory the memory effects in the fractional discrete maps mean that their present state evolution depends on all past states with special forms of weights these forms are represented by combinations of power law functions

dynamics with low level fractionality abstract 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 field theory for the fractional linear oscillator the physical meaning of the derivative of order 2 is dissipation in systems with many spacially coupled elements oscillators the fractional derivative along the space coordinate corresponds to a long range interaction we discuss a method of constructing a solution using an expansion in n with small and positive integer n the method is applied to the fractional linear and nonlinear oscillators and to fractional ginzburg landau or parabolic equations

fractional generalization of gradient and hamiltonian systems we consider a fractional generalization of hamiltonian and gradient systems we use differential forms and exterior derivatives of fractional orders we derive fractional generalization of helmholtz conditions for phase space examples of fractional gradient and hamiltonian systems are considered the stationary states for these systems are derived

algorithm for some anomalously diffusive hyperbolic systems in molecular dynamics theoretical analysis and pattern formation abstract departing from a two dimensional hyperbolic system that describes the interaction between some activator and inhibitor substances in chemical reactions we investigate a general form of that model using a finite difference approach the model under investigation is a nonlinear system consisting of two coupled partial differential equations with generalized reaction terms the presence of two dimensional diffusive terms consisting of fractional operators of the riesz type is considered here using spatial differentiation orders in the set 0 1 1 2 we impose initial conditions on a closed and bounded rectangle and a finite difference methodology based on the use of fractional centered differences is proposed among the most important results of this work we prove the existence and the uniqueness of the solutions of the numerical method and establish analytically the second order consistency of our scheme moreover the discrete energy method is employed to prove the stability and the quadratic convergence of the technique some numerical simulations obtained through our method show the presence of turing patterns and wave instabilities in agreement with some reports found in the literature on superdiffusive hyperbolic activator inhibitor systems we show numerically that the presence of turing patterns is independent of the size of the spatial domain as a new application we show that turing patterns are also present in subdiffusive scenarios

on the solution of a generalized higgs boson equation in the de sitter space time through an efficient and hamiltonian scheme abstract the present work is the first paper in the literature to report on a hamiltonian discretization of the fractional higgs boson equation in the de sitter space time and its theoretical analysis more precisely we design herein a numerically efficient finite difference hamiltonian technique for the solution of a fractional extension of the higgs boson equation in the de sitter space time the model under investigation is a multidimensional equation with generalized potential and riesz space fractional derivatives of orders in 1 2 an energy integral for the model is readily available and we propose a nonlinear implicit and consistent numerical technique based on fractional order centered differences with similar hamiltonian properties in the discrete scenario a fractional energy approach is used then to prove the properties of stability and convergence of the technique for simulation purposes we consider both the classical and the fractional higgs real valued scalar fields in the de sitter space time and find results qualitatively similar to those available in the literature for the sake of convenience we provide the matlab code of an alternative linear discretization of the method presented in this work this linear implicit approach is thoroughly analyzed also

fractal burgers equations abstract the paper studies local and global in time solutions to a class of multidimensional generalized burgers type equations with a fractional power of the laplacian in the principal part and with general algebraic nonlinearity such equations naturally appear in continuum mechanics our results include existence uniqueness regularity and asymptotic behavior of solutions to the cauchy problem as well as a construction of self similar solutions the role of critical exponents is also explained

psi series solution of fractional ginzburg landau equation one dimensional ginzburg landau equations with derivatives of noninteger order are considered using psi series with fractional powers the solution of the fractional ginzburg landau fgl equation is derived the leading order behaviours of solutions about an arbitrary singularity as well as their resonance structures have been obtained it was proved that fractional equations of order with polynomial nonlinearity of order s have the noninteger power like behaviour of order 1 s near the singularity

breathers on lattices with long range interaction we analyze the properties of breathers time periodic spatially localized solutions on chains in the presence of algebraically decaying interactions 1 r s we find that the spatial decay of a breather shows a crossover from exponential short distances to algebraic large distances decay we calculate the crossover distance as a function of s and the energy of the breather next we show that the results on energy thresholds obtained for short range interactions remain valid for s 3 and that for s 3 fractal dispersion at the band edge nonzero thresholds occur for cases where the short range interaction system would yield zero threshold values

on the solution of hyperbolic two dimensional fractional systems via discrete variational schemes of high order of accuracy abstract in this work we consider a general class of damped wave equations in two spatial dimensions the model considers the presence of weyl space fractional derivatives as well as a generic nonlinear potential the system has an associated positive energy functional when damping is not present in which case the model is capable of preserving the energy throughout time meanwhile the energy of the system is dissipated in the damped scenario in this work the weyl space fractional derivatives are approximated through second order accurate fractional centered differences a high order compact difference scheme with fourth order accuracy in space and second order in time is proposed some associated discrete quantities are introduced to estimate the energy functional we prove that the numerical method is capable of conserving the discrete variational structure under the same conditions for which the continuous model is conservative the positivity of the discrete energy of the system is also discussed the properties of consistency solvability stability and convergence of the proposed method are rigorously proved we provide some numerical simulations that illustrate the agreement between the physical properties of the continuous and the discrete models

toward lattice fractional vector calculus an analog of fractional vector calculus for physical lattice models is suggested we use an approach based on the models of three dimensional lattices with long range inter particle interactions the lattice analogs of fractional partial derivatives are represented by kernels of lattice long range interactions where the fourier series transformations of these kernels have a power law form with respect to wave vector components in the continuum limit these lattice partial derivatives give derivatives of non integer order with respect to coordinates in the three dimensional description of the non local continuum the fractional differential operators have the form of fractional partial derivatives of the riesz type as examples of the applications of the suggested lattice fractional vector calculus we give lattice models with long range interactions for the fractional maxwell equations of non local continuous media and for the fractional generalization of the mindlin and aifantis continuum models of gradient elasticity

on the continuum limit for discrete nls with long range lattice interactions we consider a general class of discrete nonlinear schrodinger equations dnls on the lattice h mathbb z with mesh size h 0 in the continuum limit when h 0 we prove that the limiting dynamics are given by a nonlinear schrodinger equation nls on mathbb r with the fractional laplacian as dispersive symbol in particular we obtain that fractional powers frac 1 2 alpha 1 arise from long range lattice interactions when passing to the continuum limit whereas the nls with the usual laplacian describes the dispersion in the continuum limit for short range or quick decaying interactions e g nearest neighbor interactions

the self similar field and its application to a diffusion problem we introduce a continuum approach which accounts for self similarity as a symmetry property of an infinite medium a self similar laplacian operator is introduced which is the source of self similar continuous fields in this way self similar symmetry appears in an analogous manner as transverse isotropy or cubic symmetry of a medium as a consequence of the self similarity the laplacian is a non local fractional operator obtained as the continuum limit of the discrete self similar laplacian introduced recently by michelitsch et al 2009 phys rev e 80 011135 the dispersion relation of the laplacian and its green s function is deduced in closed forms as a physical application of the approach we analyze a self similar diffusion problem the statistical distributions which constitute the solutions of this problem turn out to be levi stable distributions with infinite variances characterizing the statistics of one dimensional levi flights the self similar continuum approach introduced in this paper has the potential to be applied on a variety of scale invariant and fractal problems in physics such as in continuum mechanics electrodynamics and in other fields

fractional dynamics of systems with long range space interaction and temporal memory abstract 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 schrodinger 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

fractional liouville equation on lattice phase space abstract in this paper we propose a lattice analog of phase space fractional liouville equation the liouville equation for phase space lattice with long range jumps of power law types is suggested we prove that the continuum limit transforms this lattice equation into liouville equation with conjugate riesz fractional derivatives of non integer orders with respect to coordinates of continuum phase space an application of the fractional liouville equation with these riesz fractional derivatives to describe properties of plasma like nonlocal media is considered

the restaurant at the end of the random walk recent developments in the description of anomalous transport by fractional dynamics fractional dynamics has experienced a firm upswing during the past few years having been forged into a mature framework in the theory of stochastic processes a large number of research papers developing fractional dynamics further or applying it to various systems have appeared since our first review article on the fractional fokker planck equation metzler r and klafter j 2000a phys rep 339 1 77 it therefore appears timely to put these new works in a cohesive perspective in this review we cover both the theoretical modelling of sub and superdiffusive processes placing emphasis on superdiffusion and the discussion of applications such as the correct formulation of boundary value problems to obtain the first passage time density function we also discuss extensively the occurrence of anomalous dynamics in various fields ranging from nanoscale over biological to geophysical and environmental systems

fractional order difference equations for physical lattices and some applications fractional order operators for physical lattice models based on the grunwald letnikov fractional differences are suggested we use an approach based on the models of lattices with long range particle interactions the fractional order operators of differentiation and integration on physical lattices are represented by kernels of lattice long range interactions in continuum limit these discrete operators of non integer orders give the fractional order derivatives and integrals with respect to coordinates of the grunwald letnikov types as examples of the fractional order difference equations for physical lattices we give difference analogs of the fractional nonlocal navier stokes equations and the fractional nonlocal maxwell equations for lattices with long range interactions continuum limits of these fractional order difference equations are also suggested

fractional systems and fractional bogoliubov hierarchy equations we consider the fractional generalizations of the phase volume volume element and poisson brackets these generalizations lead us to the fractional analog of the phase space we consider systems on this fractional phase space and fractional analogs of the hamilton equations the fractional generalization of the average value is suggested the fractional analogs of the bogoliubov hierarchy equations are derived from the fractional liouville equation we define the fractional reduced distribution functions the fractional analogs of the vlasov equation and the debye radius are considered

the nature of electronic states in a disordered chain with long ranged hopping amplitudes abstract in this work we investigate the nature of the electronic states in a one dimensional tight binding model with on site energetic disorder and long ranged hopping amplitudes these are assumed to follow a power law decay behavior with distance in the form h r 1 r with 1 direct numerical diagonalization finite size scaling and a spectral statistics analysis show that as 1 the localization length exhibits an exponential growth as the upper band edge is approached strong finite size corrections are present in this regime near the upper band edge with the one electron states displaying a strong level repulsion even at large although finite chains