a reaction diffusion equations modelling the effect of fluctuating water levels on prey predator interactions prey predator interactions are influenced by many ecological factors in this paper a prey predator model with spatial diffusion under homogeneous neumann boundary conditions is investigated to study the impact of water level on persistence of two fish populations living in an artificial lake we obtain some important qualitative properties including the existence of the global positive solution the dissipation and permanence of this system the existence of a unique globally stable periodic solution is also presented some numerical examples are presented to verify our results

two energy conserving numerical schemes for the sine gordon equation two explicit conservative numerical schemes for the sine gordon equation are proposed their stability and convergence are proved numerical simulation of a sine gordon soliton shows that the new schemes are very accurate and fast

an averaging theorem for hamiltonian dynamical systems in the thermodynamic limit it is shown how to perform some steps of perturbation theory if one assumes a measure theoretic point of view i e if one renounces to control the evolution of the single trajectories and the attention is restricted to controlling the evolution of the measure of some meaningful subsets of phase space for a system of coupled rotators estimates uniform in n for finite specific energy can be obtained in quite a direct way this is achieved by making reference not to the sup norm but rather following koopman and von neumann to the much weaker l2 norm

periodic duffing delay equations with state dependent impulses conditions are obtained for the duffing equation with delay and state dependent impulses to admit a periodic solution

an almost periodic competitive system subject to impulsive perturbations in this paper an almost periodic system of impulsive differential equations used in modeling two species in competition is considered by the comparison between the solutions of impulsive system and the corresponding non impulsive system and constructing lyapunov function sufficient conditions ensuring the uniformly asymptotic stability of a unique positive almost periodic solution of the system are derived finally an example and its corresponding numerical simulations are presented to explain our theoretical results

symplectic methods for the nonlinear schrodinger equation abstract in this paper we show that the spatial discretization of the nonlinear schrodinger equation leads to a hamiltonian system which can be simulated with symplectic numerical schemes in particular we apply two symplectic integrators to the nonlinear schrodinger equation and we demonstrate that they are able to produce accurate results and to preserve very well the invariants of the original system such as the energy and charge

design and numerical analysis of a logarithmic scheme for nonlinear fractional diffusion reaction equations abstract in this work we consider a parabolic partial differential equation with fractional diffusion that generalizes the well known fisher s and hodgkin huxley equations the spatial fractional derivatives are understood in the sense of riesz and initial boundary conditions on a closed and bounded interval are considered here the mathematical model is presented in an equivalent form and a finite difference discretization based on fractional order centered differences is proposed the scheme is the first explicit logarithmic model proposed in the literature to solve fractional diffusion reaction equations we establish rigorously the capability of the technique to preserve the positivity and the boundedness of the methodology moreover we propose conditions under which the monotonicity of the numerical model is also preserved the consistency the stability and the convergence of the scheme are also proved mathematically and some a priori bounds for the numerical solutions are proposed we provide some numerical simulations in order to confirm that the method is capable of preserving the positivity and the boundedness of the approximations and a numerical study of the convergence of the technique is carried out confirming thus the analytical results

numerical solution of the space fractional fokker planck equation the traditional second order fokker planck equation may not adequately describe the movement of solute in an aquifer because of large deviation from the dynamics of brownian motion densities of stable type have been used to describe the probability distribution of these motions the resulting governing equation of these motions is similar to the traditional fokker planck equation except that the order of the highest derivative is fractional in this paper a space fractional fokker planck equation sffpe with instantaneous source is considered a numerical scheme for solving sffpe is presented using the riemann liouville and grunwald letnikov definitions of fractional derivatives the sffpe is transformed into a system of ordinary differential equations ode then the ode system is solved by a method of lines numerical results for sffpe with a constant diffusion coefficient are evaluated for comparison with the known analytical solution the numerical approximation of sffpe with a time dependent diffusion coefficient is also used to simulate levy motion with stable densities we will show that the numerical method of sffpe is able to more accurately model these heavy tailed motions

preserving energy resp dissipation in numerical pdes using the average vector field method we give a systematic method for discretizing hamiltonian partial differential equations pdes with constant symplectic structure while preserving their energy exactly the same method applied to pdes with constant dissipative structure also preserves the correct monotonic decrease of energy the method is illustrated by many examples in the hamiltonian case these include the sine gordon korteweg de vries nonlinear schrodinger linear time dependent schrodinger and maxwell equations in the dissipative case the examples are the allen cahn cahn hilliard ginzburg landau and heat equations

an energy conservative difference scheme for the nonlinear fractional schrodinger equations abstract in this paper an energy conservative crank nicolson difference scheme for nonlinear riesz space fractional schrodinger equations is studied we give a rigorous analysis of the conservation properties including mass conservation and energy conservation in the discrete sense based on brouwer fixed point theorem the existence of the difference solution is proved by virtue of the energy method the difference solution is shown to be unique and convergent at the order of o 2 h 2 in the l 2 norm with time step and mesh size h finally a linearized iterative algorithm is presented and numerical experiments are given to confirm the theoretical results

a deterministic model for the distribution of the stopping time in a stochastic equation and its numerical solution in this work we consider a stochastic differential equation that generalizes the well known paris equation from fracture of materials the model describes the propagation of cracks on solids and it includes a deterministic summand and a stochastic component in terms of a brownian motion the use of ito s stochastic integral gives an equivalent stochastic integral equation that is further generalized here we note that the probability distribution of the stopping time of the general model satisfies a deterministic diffusion advection partial differential equation for which the solution is known only in a reduced number of particular cases motivated by these analytical results we develop a fast finite difference method to approximate the distribution of the stopping time the method is an explicit exponential like technique that preserves the main features of a probability distribution namely the non negativity the boundedness from above by 1 as well as the spatial monotonicity moreover the method is a monotone technique that is also capable of preserving the temporal monotonicity of the approximations these properties of the proposed methodology are thoroughly established in the present manuscript a continuity condition of the numerical solutions in terms of the initial conditions and the temporal computational parameter is established also together with a limiting property of the methodology when the free parameter tends to infinity for comparison purposes we are providing an implicit and stable discretization of the mathematical model which has a second order of convergence but for which conditions that guarantee the positivity the boundedness and the monotonicity of approximations are not available the numerical simulations obtained with implementations of our techniques show that the explicit method is an efficient scheme that preserves the characteristics of interest non negativity boundedness from above by 1 and monotonicity and that the numerical approximations are in good agreement with the known exact solutions

time scales to equipartition in the fermi pasta ulam problem finite size effects and thermodynamic limit we investigate numerically the common and the pure fpu models as well as some higher order generalizations we consider initial conditions in which only low frequency normal modes are excited and perform a very accurate systematic study of the equilibrium time as a function of the number n of particles the specific energy e and the parameters and while at any fixed n the equilibrium time is found to be a stretched exponential in 1 e in the thermodynamic limit i e for n at fixed e we observe a crossover to a power law concerning the usually disregarded dependence of teq on and we find it is nontrivial and propose and test a general law a central role is played by the comparison of the fpu models with the toda model

dynamic behaviors of the impulsive periodic multi species predator prey system the dynamic behaviors of an impulsive periodic predator prey model with n preys and m predators are studied in this paper by constructing a suitable lyapunov function and using the comparison theorem of impulsive differential equation sufficient conditions which ensure the permanence and global attractivity of the system are obtained at the same time a set of criteria which guarantee that some species in the system are permanent and globally attractive while the remaining species are driven to extinction is obtained our results show that for the multi species predator prey community impulsivity is one of the important reasons that can change the long time behaviors of species

optimal impulsive harvesting on non autonomous beverton holt difference equations many recent advances in the theory of the optimal economic exploitation of renewable fish resources have been gained by applying optimal control theory however despite these successes much less is known about how seasonal environments affect the maximum sustainable yield msy or population persistence and any effects of relations between intensity and frequency of harvesting assuming that fish populations follow beverton holt equations we investigated impulsive harvesting in seasonal environments focusing on both economic aspects and resource sustainability we first investigated the existence and stability of a periodic solution and its analytic formula and then showed that the population persistence depends on the intensity and frequency of harvesting with the msy as a management objective we investigated optimal impulsive harvesting policies the optimal harvesting effort that maximizes the sustainable yield the corresponding optimal population level and the msy are obtained by using discrete euler lagrange equations and product formulae and their explicit expressions were obtained in terms of the intrinsic growth rate the carrying capacity and the impulsive moments these results imply that harvest timing is of crucial importance to the msy since impulsive differential equations incorporate elements of continuous and discrete systems we can apply all results obtained for beverton holt equations with impulsive effects to periodic logistic equations with impulsive harvesting

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

two boundedness and monotonicity preserving methods for a generalized fisher kpp equation two finite difference methods are proposed for a generalized fisher kpp equation these methods are capable of preserving the skew symmetry of the equation these methods preserve the positivity boundedness and monotonicity of solutions these methods are solvable these methods are convergent a semi explicit finite difference method and an implicit finite difference method are proposed for a generalized fisher kpp equation with two space variables it is proved that these two methods could preserve the skew symmetry of this equation moreover under a condition on the step sizes the semi explicit method is capable of preserving the positivity the boundedness and the spatial and temporal monotonicity of initial approximations and the implicit method is able to preserve these properties with no restriction on the step sizes the stability and convergence of these two methods are also analyzed respectively finally some numerical simulations are provided to verify the validity of our analytical results

the fermi pasta ulam problem and its underlying integrable dynamics this paper is devoted to a numerical study of the familiar fpu model precisely we here discuss revisit and combine together two main ideas on the subject i in the system at small specific energy e e n two well separated time scales are present in the former one a kind of metastable state is produced while in the second much larger one such an intermediate state evolves and reaches statistical equilibrium ii fpu should be interpreted as a perturbed toda model rather than as is typical as a linear model perturbed by nonlinear terms in the view we here present and support the former time scale is the one in which fpu is essentially integrable its dynamics being almost indistinguishable from the toda dynamics the toda actions stay constant for fpu too while the usual linear normal modes do not the angles fill their almost invariant torus and nothing else happens the second time scale is instead the one in which the toda actions significantly evolve and statistical equilibrium is possible we study both fpu like initial states in which only a few degrees of freedom are excited and generic initial states extracted randomly from an approximated microcanonical distribution the study is based on a close comparison between the behavior of fpu and toda in various situations the main technical novelty is the study of the correlation functions of the toda constants of motion in the fpu dynamics such a study allows us to provide a good definition of the equilibrium time i e of the second time scale for generic initial data our investigation shows that is stable in the thermodynamic limit i e the limit of large n at fixed e and that by reducing e ideally the temperature approximately grows following a power law e a with a 5 2

permanence and global attractivity of an impulsive ratio dependent predator prey system in a patchy environment this paper studies an impulsive nonautonomous two species predator prey lotka volterra type dispersal system in a patchy environment in which the prey species can disperse among n different patches but the predator species is confined to one patch and cannot disperse on the basis of inequality estimation techniques and the comparison theory of impulsive differential equations sufficient conditions of integrable form for the permanence and extinction are obtained by constructing appropriate lyapunov function conditions for the global attractivity of the system are established

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

a structure preserving scheme for the kolmogorov fokker planck equation in this paper we introduce a numerical scheme which preserves the behavior of solutions to the kolmogorov equation as time tends to infinity the method presented is based on a self similar change of variables technique to transform the kolmogorov equation into a new form such that the problem of designing structure preserving schemes for the original equation amounts to building a standard scheme for the transformed equation this transformation also has the added benefit of allowing for an exact operator splitting scheme whereas in the original form a standard operator splitting was only second order finally we verify the preservation of long time behavior through numerical simulations

a series expansion for the time autocorrelation of dynamical variables we present here a general iterative formula which gives a formal series expansion for the time autocorrelation of smooth dynamical variables for all hamiltonian systems endowed with an invariant measure we add some criteria theoretical in nature which enable one to decide whether the decay of the correlations is exponentially fast or not one of these criteria is implemented numerically for the case of the fermi pasta ulam system and we find indications which might suggest a sub exponential decay of the time autocorrelation of a relevant dynamical variable

on a stochastic logistic equation with impulsive perturbations a stochastic logistic model with impulsive perturbations is proposed and investigated first we give a new definition of a solution of an impulsive stochastic differential equation isde which is more convenient for use than the existing one using this definition we show that our model has a global and positive solution and obtain its explicit expression then we establish the sufficient conditions for extinction non persistence in the mean weak persistence persistence in the mean and stochastic permanence of the solution the critical value between weak persistence and extinction is obtained in addition the limit of the average in time of the sample path of the solution is estimated by two constants afterwards the lower growth rate and the upper growth rate of the solution are estimated finally sufficient conditions for global attractivity are established

global dynamics of the periodic logistic system with periodic impulsive perturbations this paper studies the global behaviors of the periodic logistic system with periodic impulsive perturbations the results of d d bainov and p s simeonov 1993 are extended and dynamics different from the corresponding continuous system are found it is shown that the system may have a unique positive periodic solution which is globally asymptotically stable or go extinct when the two periods are rational dependent when they are rational independent the system has no periodic solutions however still has a global attractor or go extinct under some conditions

nonlinear normal modes for the toda chain abstract the toda chain is a nonlinear mass spring chain which can in principle be integrated analytically by the action angle theory of classical mechanics we show that the transformation from physical variables to action variables can be implemented very simply on a computer we study the correspondence between action variables and the motions of the chain and find the role of the action variables to be very similar to the role of normal mode amplitudes in the harmonic chain this similarity leads to a partly rigorous partly heuristic normal mode analysis for the toda chain new results are obtained when this normal mode analysis is applied to certain perturbations of the toda chain such as the fermi pasta ulam chain or the toda chain with a mass impurity

finite difference schemes for nonlinear wave equation that inherit energy conservation property abstract we propose two general finite difference schemes that inherit energy conservation property from nonlinear wave equations such as the nonlinear klein gordon equation nlkge one of proposed schemes is implicit and another is explicit many studies exist on fdss that inherit energy conservation property from nlkge and we can derive all of their schemes from the proposed general schemes in this paper the most important feature of our procedure is a rigorous discretization of variational derivatives using summation by parts which implies that the inherited properties are satisfied exactly because of this the derived schemes are expected to be numerically stable and yield solutions converging to pde solutions we make new fdss for fermi pasta ulam equation string vibration equation shimoji kawai equation ske and ebihara equation and verify numerically the inheritance of the energy conservation property for nlkge and ske

extensive adiabatic invariants for nonlinear chains we look for extensive adiabatic invariants in nonlinear chains in the thermodynamic limit considering the quadratic part of the klein gordon hamiltonian by a linear change of variables we transform it into a sum of two parts in involution at variance with the usual method of introducing normal modes our constructive procedure allows us to exploit the complete resonance while keeping the extensive nature of the system next we construct a nonlinear approximation of an extensive adiabatic invariant for a perturbation of the discrete nonlinear schrodinger model the fluctuations of this quantity are controlled via gibbs measure estimates independent of the system size for a large set of initial data at low specific energy finally by numerical calculations we show that our adiabatic invariant is well conserved for times much longer than predicted by our first order theory with fluctuation much smaller than expected according to standard statistical estimates

existence of periodic solution of order one of planar impulsive autonomous system based on the situation that all the recent research about state dependent impulsive differential equations is focused on systems which have explicit solutions this paper try to consider those systems with state dependent impulsion which have not explicit solutions we get an existence theorem of periodic solution of order one for a general planar autonomous impulsive system and by applying it to a special state dependent impulsive differential equations we get the concrete conditions of existence of one order periodic solution of that special system

efficient energy preserving integrators for oscillatory hamiltonian systems in this paper we focus our attention on deriving and analyzing an efficient energy preserving formula for the system of nonlinear oscillatory or highly oscillatory second order differential equations q t mq t f q t where m is a symmetric positive semi definite matrix with m 1 and f q q u q is the negative gradient of a real valued function u q this system is a hamiltonian system with the hamiltonian h p q 1 2 p t p 1 2 q t mq u q the energy preserving formula exactly preserves the hamiltonian we analyze in detail the properties of the energy preserving formula and propose new efficient energy preserving integrators in the sense of numerical implementation the convergence analysis of the fixed point iteration is presented for the implicit integrators proposed in this paper it is shown that the convergence of implicit average vector field methods is dependent on m whereas the convergence of the new energy preserving integrators is independent of m the fermi pasta ulam problem and the sine gordon equation are carried out numerically to show the competence and efficiency of the novel integrators in comparison with the well known average vector field methods in the scientific literature

energy dissipation preserving birkhoffian multi symplectic methods for maxwell s equations with dissipation terms in this paper we propose two new energy dissipation preserving birkhoffian multi symplectic methods birkhoffian and birkhoffian box for maxwell s equations with dissipation terms after investigating the non autonomous and autonomous birkhoffian formalism for maxwell s equations with dissipation terms we first apply a novel generating functional theory to the non autonomous birkhoffian formalism to propose our birkhoffian scheme and then implement a central box method to the autonomous birkhoffian formalism to derive the birkhoffian box scheme we have obtained four formal local conservation laws and three formal energy global conservation laws we have also proved that both of our derived schemes preserve the discrete version of the global local conservation laws furthermore the stability dissipation and dispersion relations are also investigated for the schemes theoretical analysis shows that the schemes are unconditionally stable dissipation preserving for maxwell s equations in a perfectly matched layer pml medium and have second order accuracy in both time and space numerical experiments for problems with exact theoretical results are given to demonstrate that the birkhoffian multi symplectic schemes are much more accurate in preserving energy than both the exponential finite difference time domain fdtd method and traditional hamiltonian scheme we also solve the electromagnetic pulse emp propagation problem and the numerical results show that the birkhoffian scheme recovers the magnitude of the current source and reaction history very well even after long time propagation

crank nicolson method for the fractional diffusion equation with the riesz fractional derivative we examine a numerical method to approximate to a fractional diffusion equation with the riesz fractional derivative in a finite domain which has second order accuracy in time and space level in order to approximate the riesz fractional derivative we use the fractional centered derivative approach we determine the error of the riesz fractional derivative to the fractional centered difference we apply the crank nicolson method to a fractional diffusion equation which has the riesz fractional derivative and obtain that the method is unconditionally stable and convergent numerical results are given to demonstrate the accuracy of the crank nicolson method for the fractional diffusion equation with using fractional centered difference approach

permanence and global stability for nonautonomous n species lotka volterra competitive system with impulses and infinite delays abstract in this paper we consider a class of nonautonomous n species lotka volterra competitive systems with impulses and infinite delays by developing the methods given in teng 2002 25 we give sufficient conditions for permanence and global attractivity of the system

periodicity in a generalized semi ratio dependent predator prey system with time delays and impulses abstract with the help of a continuation theorem based on coincidence degree theory we establish necessary and sufficient conditions for the existence of positive periodic solutions in a generalized semi ratio dependent predator prey system with time delays and impulses which covers many models appeared in the literature when the results reduce to the semi ratio dependent predator prey system without impulses they generalize and improve some known ones

some new structure preserving algorithms for general multi symplectic formulations of hamiltonian pdes many partial differential equations pdes can be written as a multi symplectic hamiltonian system which has three local conservation laws namely multi symplectic conservation law local energy conservation law and local momentum conservation law in this paper we give several systematic methods for discretizing general multi symplectic formulations of hamiltonian pdes including a local energy preserving algorithm a class of global energy preserving methods and a local momentum preserving algorithm the methods are illustrated by the nonlinear schrodinger equation and the korteweg de vries equation numerical experiments are presented to demonstrate the conservative properties of the proposed numerical methods

on metastability in fpu we present an analytical study of the fermi pasta ulam fpu model with periodic boundary conditions we analyze the dynamics corresponding to initial data with one low frequency fourier mode excited we show that correspondingly a pair of kdv equations constitute the resonant normal form of the system we also use such a normal form in order to prove the existence of a metastability phenomenon more precisely we show that the time average of the modal energy spectrum rapidly attains a well defined distribution corresponding to a packet of low frequencies modes subsequently the distribution remains unchanged up to the time scales of validity of our approximation the phenomenon is controlled by the specific energy

periodic lienard type delay equations with state dependent impulses abstract conditions are obtained for lienard type equations with delay and state dependent impulses to admit an absolutely continuous periodic solution with first derivative of bounded variation and consequently with lebesgue integrable second derivative the results are applied to josephson s equation and the nonconservative forced pendulum equation

global existence results for impulsive functional differential equations abstract in this paper we investigate the existence of solutions of impulsive delay differential equation x t f t x t t 0 t k k n x t i k x t t k k n new global existence theorems are established without assuming the global existence of solutions of the corresponding continuous equation

energy conserving schemes for the simulation of musical instrument contact dynamics abstract collisions are an innate part of the function of many musical instruments due to the nonlinear nature of contact forces special care has to be taken in the construction of numerical schemes for simulation and sound synthesis finite difference schemes and other time stepping algorithms used for musical instrument modelling purposes are normally arrived at by discretising a newtonian description of the system however because impact forces are non analytic functions of the phase space variables algorithm stability can rarely be established this way this paper presents a systematic approach to deriving energy conserving schemes for frictionless impact modelling the proposed numerical formulations follow from discretising hamilton s equations of motion generally leading to an implicit system of nonlinear equations that can be solved with newton s method the approach is first outlined for point mass collisions and then extended to distributed settings such as vibrating strings and beams colliding with rigid obstacles stability and other relevant properties of the proposed approach are discussed and further demonstrated with simulation examples the methodology is exemplified through a case study on tanpura string vibration with the results confirming the main findings of previous studies on the role of the bridge in sound generation with this type of string instrument

birkhoff coordinates for the toda lattice in the limit of infinitely many particles with an application to fpu abstract in this paper we study the birkhoff coordinates cartesian action angle coordinates of the toda lattice with periodic boundary condition in the limit where the number n of the particles tends to infinity we prove that the transformation introducing such coordinates maps analytically a complex ball of radius r n in discrete sobolev analytic norms into a ball of radius r n with r r 0 independent of n if and only if 2 then we consider the problem of equipartition of energy in the spirit of fermi pasta ulam we deduce that corresponding to initial data of size r n 2 0 r 1 and with only the first fourier mode excited the energy remains forever in a packet of fourier modes exponentially decreasing with the wave number finally we consider the original fpu model and prove that energy remains localized in a similar packet of fourier modes for times one order of magnitude longer than those covered by previous results which is the time of formation of the packet the proof of the theorem on birkhoff coordinates is based on a new quantitative version of a vey type theorem by kuksin and perelman which could be interesting in itself

fractional schrodinger equation some properties of the fractional schrodinger equation are studied we prove the hermiticity of the fractional hamilton operator and establish the parity conservation law for fractional quantum mechanics as physical applications of the fractional schrodinger equation we find the energy spectra of a hydrogenlike atom fractional bohr atom and of a fractional oscillator in the semiclassical approximation an equation for the fractional probability current density is developed and discussed we also discuss the relationships between the fractional and standard schrodinger equations

dissipation preserving galerkin legendre spectral methods for two dimensional fractional nonlinear wave equations abstract in this paper we consider implicit and linearized dissipation preserving galerkin legendre spectral methods to solve space fractional nonlinear damped wave equation in two dimensions the full discrete schemes preserve dissipation of energy in damped case and conservation of energy in undamped case moreover the stability and convergence analysis of the full discrete schemes are rigorously given furthermore we get that two methods are convergent with second order accuracy in time and optimal error estimates in space in order to reduce the computational cost we adopt matrix diagonalization approach to solve the resulting algebraic systems in numerical implementation numerical results are presented to validate the theoretical analysis

on the numerical integration of fpu like systems abstract this paper concerns the numerical integration of systems of harmonic oscillators coupled by nonlinear terms like the common fpu models we show that the most used integration algorithm namely leap frog behaves very gently with such models preserving in a beautiful way some peculiar features which are known to be very important in the dynamics in particular the selection rules which regulate the interaction among normal modes this explains why leap frog in spite of being a low order algorithm behaves so well as numerical experimentalists always observed at the same time we show how the algorithm can be improved by introducing at a low cost a counterterm which eliminates the dominant numerical error

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

an averaging theorem for fpu in the thermodynamic limit consider an fpu chain composed of n gg 1 particles and endow the phase space with the gibbs measure corresponding to a small temperature beta 1 given a fixed k we construct k packets of normal modes whose energies are adiabatic invariants i e are approximately constant for times of order beta 1 a a 0 for initial data in a set of large measure furthermore the time autocorrelation function of the energy of each packet does not decay significantly for times of order beta the restrictions on the shape of the packets are very mild all estimates are uniform in the number n of particles and thus hold in the thermodynamic limit n rightarrow infty beta 0

local structure preserving algorithms for the good boussinesq equation in this paper we derive a series of local structure preserving algorithms for the good boussinesq equation including multisymplectic geometric structure preserving algorithms local energy preserving algorithms and local momentum preserving algorithms the outstanding advantage of the proposed algorithms is that they conserve these local structures in any time space region exactly for example the proposed local energy preserving algorithms preserve the local energy conservation law in any local domain therefore the local structure preserving algorithms overcome the shortage of global structure preserving algorithms on the boundary conditions especially with suitable boundary conditions such as periodic or homogeneous boundary conditions the local structure preserving algorithms will be global structure preserving algorithms numerical results verify the theoretical analysis

asymptotic stability of lattice solitons in the energy space orbital and asymptotic stability for 1 soliton solutions of the toda lattice equations as well as for small solitary waves of the fpu lattice equations are established in the energy space unlike analogous hamiltonian pdes the lattice equations do not conserve the adjoint momentum in fact the toda lattice equation is a bidirectional model that does not fit in with the existing theory for the hamiltonian systems by grillakis shatah and strauss to prove stability of 1 soliton solutions we split a solution around a 1 soliton into a small solution that moves more slowly than the main solitary wave and an exponentially localized part we apply a decay estimate for solutions to a linearized toda equation which has been recently proved by mizumachi and pego to estimate the localized part we improve the asymptotic stability results for fpu lattices in a weighted space obtained by friesecke and pego

dissipative or conservative finite difference schemes for complex valued nonlinear partial differential equations abstract we propose a new procedure for designing finite difference schemes that inherit energy conservation or dissipation property from complex valued nonlinear partial differential equations pdes such as the nonlinear schrodinger equation the ginzburg landau equation and the newell whitehead equation the procedure is a complex version of the procedure that furihata has recently presented for real valued nonlinear pdes furthermore we show that the proposed procedure can be modified for designing linearly implicit finite difference schemes that inherit energy conservation or dissipation property

differential quadrature method application to initial boundary value problems two non linear dynamical systems have been considered in both cases the governing equation of motion is reduced to two second order non linear non autonomous ordinary differential equations using the differential quadrature method with a careful distribution of sampling points to check the numerical results a comparison with those obtained using the galerkin approach is proposed

positivity preserving dg and central dg methods for ideal mhd equations ideal mhd equations arise in many applications such as astrophysical plasmas and space physics and they consist of a system of nonlinear hyperbolic conservation laws the exact density r and pressure p should be non negative numerically such positivity property is not always satisfied by approximated solutions one can encounter this when simulating problems with low density high mach number or much large magnetic energy compared with internal energy when this occurs numerical instability may develop and the simulation can break down in this paper we propose positivity preserving discontinuous galerkin and central discontinuous galerkin methods for solving ideal mhd equations by following x zhang c w shu journal of computational physics 229 2010 8918 8934 in one dimension the positivity preserving property is established for both methods under a reasonable assumption the performance of the proposed methods in terms of accuracy stability and positivity preserving property is demonstrated through a set of one and two dimensional numerical experiments the proposed methods formally can be of any order of accuracy

a new class of energy preserving numerical integration methods the first ever energy preserving b series numerical integration method for ordinary differential equations is presented and applied to several hamiltonian systems related novel lie algebraic results are also discussed

asymptotic localization of energy in nondisordered oscillator chains we study two popular one dimensional chains of classical anharmonic oscillators the rotor chain and a version of the discrete nonlinear schrodinger chain we assume that the interaction between neighboring oscillators controlled by the parameter 0 is small we rigorously establish that the thermal conductivity of the chains has a nonperturbative origin with respect to the coupling constant and we provide strong evidence that it decays faster than any power law in as 0 the weak coupling regime also translates into a high temperature regime suggesting that the conductivity vanishes faster than any power of the inverse temperature to our knowledge it is the first time that a clear connection has been established between kam like phenomena and thermal conductivity 2015 wiley periodicals inc

numerical methods for the variable order fractional advection diffusion equation with a nonlinear source term in this paper we consider a variable order fractional advection diffusion equation with a nonlinear source term on a finite domain explicit and implicit euler approximations for the equation are proposed stability and convergence of the methods are discussed moveover we also present a fractional method of lines a matrix transfer technique and an extrapolation method for the equation some numerical examples are given and the results demonstrate the effectiveness of theoretical analysis

development of a wavenumber preserving scheme for solving maxwell s equations in curvilinear non staggered grids abstract in this paper a compact finite difference solver for solving the maxwell s equations in curvilinear coordinates is presented the scheme formulated in time domain can theoretically preserve zero divergence condition and scaled wavenumber characteristics in non staggered grids the inherent local conservation laws are also retained discretely all the time the space and time derivative terms are approximated to yield an equal fourth order spatial and temporal accuracy in irregular physical domain maxwell s equations are recast in terms of the contravariant and covariant field variables so that the developed dual preserving solver can be directly implemented in addition in curvilinear coordinates the four components in the metric tensor have been calculated under the guideline that the determinant of the transformation matrix is computed exactly through the computational exercises it is demonstrated that the newly proposed solver with a fairly small numerical scaled wavenumber error in curvilinear coordinates is computationally efficient for use to get the long time accurate maxwell s solutions in irregular physical domain

asymptotic stability of n solitary waves of the fpu lattices we study stability of n solitary wave solutions of the fermi pasta ulam fpu lattice equation solitary wave solutions of the fpu lattice equation cannot be characterized as critical points of conservation laws due to the lack of infinitesimal invariance in the spatial variable in place of standard variational arguments for hamiltonian systems we use an exponential stability property of the linearized fpu equation in a weighted space which is biased in the direction of motion the dispersion of the linearized fpu equation balances the potential term for low frequencies whereas the dispersion is superior for high frequencies we approximate the low frequency part of a solution of the linearized fpu equation by a solution to the linearized korteweg de vries kdv equation around an n soliton solution we prove an exponential stability property of the linearized kdv equation around n solitons by using the linearized backlund transformation and use the result to analyze the linearized fpu equation

an explicit fourth order energy preserving scheme for riesz space fractional nonlinear wave equations abstract in this paper a new explicit fourth order scheme for solving riesz space fractional nonlinear wave equations is developed the scheme is designed by using a novel riesz space fractional difference operator for spatial discretization and a multidimensional extended runge kutta nystrom method for time integration the conservation law of the semi discrete energy stability and convergence of the semi discrete system are investigated numerical experiments show the efficiency and energy conservation of the present scheme

on a positivity preserving numerical model for a linearized hyperbolic fisher kolmogorov petrovski piscounov equation abstract in this work we introduce a nonstandard finite difference scheme to approximate positive solutions of a modified hyperbolic fisher kolmogorov petrovski piscounov equation appearing in the investigation of the dynamics of some populations the technique has a consistency of second order and it provides nonnegative approximations for nonnegative initial profiles the stability analysis of the method is carried out in detail and the scheme is validated against known analytical solutions for some initial boundary value problems

parallel iterative regularization methods for solving systems of ill posed equations in this note two parallel iterative regularization methods for finding a minimal norm solution to a system of ill posed equations involving the so called strongly inverse monotone operators have been investigated some applications of the proposed methods are considered and numerical experiments are discussed

periodicity and global dynamics of an impulsive delay lasota wazewska model abstract the periodicity of an impulsive delay lasota wazewska model is discussed sufficient and necessary condition for the existence of a positive periodic solution is established sufficient conditions for its global attractivity are also obtained via the methods of lyapunov function and comparison

on a nonautonomous competitive system subject to stochastic and impulsive perturbations a nonautonomous impulsive stochastic competitive system is considered sufficient conditions for stochastic permanence and extinction are established the influences of stochastic and impulsive perturbations are shown this paper presents a nonautonomous impulsive stochastic differential model of competition between species which contains the classic two species competitive lotka volterra model as a special case with the theory of impulsive stochastic differential equations a good understanding of stochastic permanence and extinction of system is obtained two specific numerical examples are presented to support the analytical results conclusions are also given

positivity and boundedness preserving schemes for space time fractional predator prey reaction diffusion model the semi implicit schemes for the nonlinear predator prey reaction diffusion model with the space time fractional derivatives are discussed where the space fractional derivative is discretized by the fractional centered difference and wsgd scheme the stability and convergence of the semi implicit schemes are analyzed in the l norm we theoretically prove that the numerical schemes are stable and convergent without the restriction on the ratio of space and time stepsizes and numerically further confirm that the schemes have first order convergence in time and second order convergence in space then we discuss the positivity and boundedness properties of the analytical solutions of the discussed model and show that the numerical solutions preserve the positivity and boundedness the presented numerical examples confirm the theoretical results and convergence orders

numerical solution of the sine gordon equation two new difference schemes are proposed for the sine gordon equation they have the advantage that there is a discrete energy which is conserved their stability and convergence is proved also we show some numerical computations using one of these schemes

a compact fourth order in space energy preserving method for riesz space fractional nonlinear wave equations in this work we investigate numerically a nonlinear hyperbolic partial differential equation with space fractional derivatives of the riesz type the model under consideration generalizes various nonlinear wave equations including the sine gordon and the nonlinear klein gordon models the system considered in this work is conservative when homogeneous dirichlet boundary conditions are imposed motivated by this fact we propose a finite difference method based on fractional centered differences that is capable of preserving the discrete energy of the system the method under consideration is a nonlinear implicit scheme which has various numerical properties among the most interesting numerical features we show that the methodology is consistent of second order in time and fourth order in space moreover we show that the technique is stable and convergent some numerical simulations show that the method is capable of preserving the energy of the discrete system this characteristic of the technique is in obvious agreement with the properties of its continuous counterpart