dispersion of low energy waves for the generalized benjamin bona mahony equation on considere les equations suivantes u t u x f u x u xxx 0 1 et u t u x f u x u xxt 0 2 avec f r r est c on donne le comportement des solutions ondes a basse energie pour certains f

nonlinear stability of θ methods for neutral differential equations in banach space abstract this paper is concerned with the stability properties of theoretical solution and numerical solution to nonlinear neutral differential equations with two delays in banach space some sufficient conditions for the stability and asymptotic stability of the systems are given the numerical stability of methods when applied to the systems is also considered a series of new stability results of methods are obtained

mean square stability of two classes of theta method for neutral stochastic differential delay equations in this paper a stochastic linear theta slt method is introduced and analyzed for neutral stochastic differential delay equations nsddes we give some conditions on neutral item drift and diffusion coefficients which admit that the diffusion coefficient can be highly nonlinear and does not necessarily satisfy a linear growth or global lipschitz condition it is proved that for all positive stepsizes the slt method with 1 2 1 is asymptotically mean stable and so is 0 1 2 under a stronger assumption furthermore we consider the split step theta sst method and obtain a similar but better result that is the sst method with 1 2 1 is exponentially mean stable and so is 0 1 2 finally two numerical examples are given to show the efficiency of the obtained results

stability analysis of numerical schemes for stochastic differential equations stochastic differential equations sdes represent physical phenomena dominated by stochastic processes as for deterministic ordinary differential equations odes various numerical schemes are proposed for sdes in this paper we study the stability of numerical schemes for scalar sdes with respect to the mean square norm which we call ms stability we will show some figures of the ms stability domain or regions for some numerical schemes and present numerical results which confirm it this notion is an extension of absolute stability in numerical methods for odes

resonance and web structure in discrete soliton systems the two dimensional toda lattice and its fully discrete and ultra discrete analogues we present a class of solutions of the two dimensional toda lattice equation its fully discrete analogue and its ultra discrete limit these solutions demonstrate the existence of soliton resonance and web like structure in discrete integrable systems such as differential difference equations difference equations and cellular automata ultra discrete equations

convergence and stability of the backward euler method for jump diffusion sdes with super linearly growing diffusion and jump coefficients abstract this paper firstly investigates convergence of the backward euler method for stochastic differential equations sdes driven by brownian motion and compound poisson process different from existing works we do not impose a globally lipschitz condition on the jump coefficient but formulate a more relaxed condition to allow for its super linear growth it is shown that the mean square convergence order of this method can be arbitrarily close to 1 2 under mild assumptions imposed on sdes allowing for possibly super linearly growing drift diffusion and jump coefficients an exact order 1 2 is recovered when further differentiability assumption is put on the coefficients furthermore the considered method is able to inherit the mean square stability of a wider class of levy noise driven sdes for all stepsizes these results are finally supported by some numerical experiments

mean square stability of two classes of theta methods for numerical computation and simulation of delayed stochastic hopfield neural networks abstract recently the investigation on the stability of the numerical solutions to delayed stochastic differential equations has received an increasing attention but there has been little work on the stability analysis of the numerical solutions to delayed stochastic hopfield neural networks dshnns so far the aim in this paper is to study the mean square exponential stability of the split step theta sst method and the stochastic linear theta slt method for the underlying model it is proved that for any 0 1 2 there exists a constant 0 depending on such that the numerical schemes produced by the sst method and the slt method are mean square exponentially stable for 0 under the same assumptions as those to guarantee the mean square exponential stability of the underlying continuous model for the case 1 2 1 we show the same stability conclusion for all 0 to carry out the required conclusion a novel technique for the stability analysis of discrete numerical schemes with multi time delays namely the weighted sum lyapunov functional method is proposed finally a numerical example is given to illustrate the application of the suggested methods and to verify the stability conclusions obtained

mean square and asymptotic stability of the stochastic theta method stability analysis of numerical methods for ordinary differential equations odes is motivated by the question for what choices of stepsize does the numerical method reproduce the characteristics of the test equation we study a linear test equation with a multiplicative noise term and consider mean square and asymptotic stability of a stochastic version of the theta method we extend some mean square stability results in saito and mitsui siam j numer anal 33 1996 pp 2254 2267 in particular we show that an extension of the deterministic a stability property holds we also plot mean square stability regions for the case where the test equation has real parameters for asymptotic stability we show that the issue reduces to finding the expected value of a parametrized random variable we combine analytical and numerical techniques to get insights into the stability properties for a variant of the method that has been proposed in the literature we obtain precise analytic expressions for the asymptotic stability region this allows us to prove a number of results the technique introduced is widely applicable and we use it to show that a fully implicit method suggested by kloeden and platen numerical solution of stochastic differential equations springer verlag 1992 has an asymptotic stability extension of the deterministic a stability property we also use the approach to explain some numerical results reported in milstein platen and schurz siam j numer anal 35 1998 pp 1010 1019

almost sure exponential stability of numerical solutions for stochastic delay differential equations using techniques based on the continuous and discrete semimartingale convergence theorems this paper investigates if numerical methods may reproduce the almost sure exponential stability of the exact solutions to stochastic delay differential equations sddes the important feature of this technique is that it enables us to study the almost sure exponential stability of numerical solutions of sddes directly this is significantly different from most traditional methods by which the almost sure exponential stability is derived from the moment stability by the chebyshev inequality and the borel cantelli lemma

decay of solutions of some nonlinear wave equations abstract we study the large time behaviour of solutions to the initial value problem for the korteweg de vries equation and for the regularized long wave equation with a dissipative term appended using energy estimates a maximum principle and a transformation of cole hopf type sharp rates of temporal decay of certain norms of the solution are obtained

continuum approach to discreteness we study analytically and numerically continuum models derived on the basis of pade approximations and their effectiveness in modeling spatially discrete systems we not only analyze features of the temporal dynamics that can be captured through these continuum approaches e g shape oscillations radiation effects and trapping but also point out ones that cannot be captured such as peierls nabarro barriers and bloch oscillations we analyze the role of such methods in providing an effective homogenization of spatially discrete as well as of heterogeneous continuum equations finally we develop numerical methods for solving such equations and use them to establish the range of validity of these continuum approximations as well as to compare them with other semicontinuum approximations

exponential mean square stability of numerical methods for systems of stochastic differential equations this paper is concerned with exponential mean square stability of the classical stochastic theta method and the so called split step theta method for stochastic systems first we consider linear autonomous systems under a sufficient and necessary condition for exponential mean square stability of the exact solution it is proved that the two classes of theta methods with q 0 5 are exponentially mean square stable for all positive step sizes and the methods with q 0 5 still unconditionally preserves the exponential mean square stability of the underlying systems but the stochastic theta method does not have this property finally we consider stochastic differential equations with jumps some similar results are derived

convergence and stability of two classes of theta milstein schemes for stochastic differential equations abstract this paper examines convergence and stability of the two classes of theta milstein schemes for stochastic differential equations sdes with non global lipschitz continuous coefficients the split step theta milstein sstm scheme and the stochastic theta milstein stm scheme for 1 2 1 this paper concludes that the two classes of theta milstein schemes converge strongly to the exact solution with the order 1 for 0 1 2 under the additional linear growth condition for the drift coefficient these two classes of the theta milstein schemes are also strongly convergent with the standard order this paper also investigates exponential mean square stability of these two classes of the theta milstein schemes for 1 2 1 these two theta milstein schemes can share the exponential mean square stability of the exact solution for 0 1 2 similar to the convergence under the additional linear growth condition these two theta milstein schemes can also reproduce the exponential mean square stability of the exact solution

almost sure and moment exponential stability in the numerical simulation of stochastic differential equations relatively little is known about the ability of numerical methods for stochastic differential equations sdes to reproduce almost sure and small moment stability here we focus on these stability properties in the limit as the timestep tends to zero our analysis is motivated by an example of an exponentially almost surely stable nonlinear sde for which the euler maruyama em method fails to reproduce this behavior for any nonzero timestep we begin by showing that em correctly reproduces almost sure and small moment exponential stability for sufficiently small timesteps on scalar linear sdes we then generalize our results to multidimensional nonlinear sdes we show that when the sde obeys a linear growth condition em recovers almost surely exponential stability very well under the less restrictive condition that the drift coefficient of the sde obeys a one sided lipschitz condition where em may break down we show that the backward euler method maintains almost surely exponential stability

numerical methods for nonlinear stochastic differential equations with jumps we present and analyse two implicit methods for ito stochastic differential equations sdes with poisson driven jumps the first method ssbe is a split step extension of the backward euler method the second method cssbe arises from the introduction of a compensated martingale form of the poisson process we show that both methods are amenable to rigorous analysis when a one sided lipschitz condition rather than a more restrictive global lipschitz condition holds for the drift our analysis covers strong convergence and nonlinear stability we prove that both methods give strong convergence when the drift coefficient is one sided lipschitz and the diffusion and jump coefficients are globally lipschitz on the way to proving these results we show that a compensated form of the euler maruyama method converges strongly when the sde coefficients satisfy a local lipschitz condition and the pth moment of the exact and numerical solution are bounded for some p 2 under our assumptions both ssbe and cssbe give well defined unique solutions for sufficiently small stepsizes and ssbe has the advantage that the restriction is independent of the jump intensity we also study the ability of the methods to reproduce exponential mean square stability in the case where the drift has a negative one sided lipschitz constant this work extends the deterministic nonlinear stability theory in numerical analysis we find that ssbe preserves stability under a stepsize constraint that is independent of the initial data cssbe satisfies an even stronger condition and gives a generalization of b stability finally we specialize to a linear test problem and show that cssbe has a natural extension of deterministic a stability the difference in stability properties of the ssbe and cssbe methods emphasizes that the addition of a jump term has a significant effect that cannot be deduced directly from the non jump literature

choice of θ and mean square exponential stability in the stochastic theta method of stochastic differential equations this paper examines the relationship of choice of and mean square exponential stability in the stochastic theta method stm of stochastic differential equations sdes and mainly includes the following three results i under the linear growth condition for the drift term when 0 1 2 the stm may preserve the mean square exponential stability of the exact solution but the counterexample shows that the stm cannot reproduce this stability without this linear growth condition ii when 1 2 1 without the linear growth condition for the drift term the stm may reproduce the mean square exponential stability of the exact solution but the bound of the lyapunov exponent cannot be preserved iii when 1 this stm is called as the backward euler maruyama bem method the stm can reproduce not only the mean square exponential stability but also the bound of the lyapunov exponent this paper also gives the sufficient and necessary conditions of the mean square exponential stability of the stm for the linear sde when 0 1 2 and 1 2 1 respectively and the simulations also illustrate these theoretical results

exponential mean square stability of the theta approximations for neutral stochastic differential delay equations in this paper a split step theta sst method is introduced and analyzed for neutral stochastic differential delay equations nsddes it is proved that the sst method with 0 1 2 can recover the exponential mean square stability of the exact solution with some restrictive conditions on stepsize and the drift coefficient but for 1 2 1 the sst can reproduce the exponential mean square stability unconditionally then based on the stability results of sst scheme we examine the exponential mean square stability of the stochastic linear theta slt approximation for nsddes and obtain the similar stability results to that of the sst method moreover for sufficiently small stepsize we show that the decay rate as measured by the lyapunov exponent can be reproduced arbitrarily accurately these results show that different values of theta will drastically affect the exponential mean square stability of the two classes of theta approximations for nsddes

convergence and stability of the semi tamed euler scheme for stochastic differential equations with non lipschitz continuous coefficients recently explicit tamed schemes were proposed to approximate the sdes with the non lipschitz continuous coefficients this work proposes a semi tamed euler scheme which is also explicit to solve the sdes with the drift coefficient equipped with the lipschitz continuous part and non lipschitz continuous part it is shown that the semi tamed euler converges strongly with the standard order one half to the exact solution of the sde we also investigate the stability inheritance of the semi tamed euler schemes and reveal that this scheme does have advantage in reproducing the exponential mean square stability of the exact solution numerical experiments confirm the theoretical analysis

small amplitude solutions of the generalized imbq equation in this paper the global existence of small amplitude solution for the cauchy problem of the multidimensional generalized imbq equation is proved moreover we obtain a nonlinear scattering result of the cauchy problem of the imbq equation for small initial data

global existence of solution of cauchy problem for nonlinear pseudo parabolic equation abstract in this paper we prove that the cauchy problem for the nonlinear pseudo parabolic equation v t v x x t v x x v x f v x v x x g v g v x x admits a unique global generalized solution in c 1 0 h s r a unique global classical solution and asymptotic behavior of the solution we also prove that the cauchy problem for the equation v t v x x t v x x g v g v x x has a unique global generalized solution in c 1 0 w m p r l r a unique global classical solution and asymptotic behavior of the solution

towards a systematic linear stability analysis of numerical methods for systems of stochastic differential equations we develop two classes of test equations for the linear stability analysis of numerical methods applied to systems of stochastic ordinary differential equations of ito type sodes motivated by the theory of stochastic stabilization and destabilization these test equations capture certain fundamental effects of stochastic perturbation in systems of sodes while remaining amenable to analysis before and after discretization we then carry out a linear stability analysis of the theta maruyama method applied to these test equations investigating mean square and almost sure asymptotic stability of the test equilibria we discuss the implications of our work for the notion of a stability of the theta maruyama method and use numerical simulation to suggest extensions of our results to test systems with nonnormal drift coefficients

asymptotic exponential stability of modified truncated em method for neutral stochastic differential delay equations abstract the aim of this paper is to investigate the asymptotic mean square and almost sure exponential stability of modified truncated euler maruyama method for neutral stochastic differential delay equations under non linear conditions it is proved that the so called modified truncated euler maruyama method is mean square almost surely exponentially stable under suitable local lipschitz and khasminskii type conditions an example is provided to support our conclusions

convergence and stability of the split step theta method for stochastic differential equations with piecewise continuous arguments abstract in this paper we consider the stochastic differential equations with piecewise continuous arguments sdepcas in which both the drift and the diffusion coefficients do not satisfy the global lipschitz and linear growth conditions especially the diffusion coefficients are highly non linear growing it is proved that the split step theta sst method with 1 2 1 is strongly convergent to sdepcas under the local lipschitz monotone and one sided lipschitz conditions it is also obtained that the sst method with 1 2 1 preserves the exponential mean square stability of sdepcas under the monotone condition and some condition on the step size without any restriction on the step size there exists 1 2 1 such that the sst method with 1 is exponentially stable in mean square moreover for sufficiently small step size the rate constant can be reproduced some numerical simulations are presented to illustrate the analytical theory

exponential stability of the exact solutions and θ em approximations to neutral sddes with markov switching exponential stability of the exact solutions and em 1 2 1 approximations to neutral stochastic differential delay equations with markov switching are investigated in this paper sufficient conditions are obtained to ensure the p th moment p 1 and almost sure exponential stability of the exact solutions as well as em approximations p 2 an example is provided to support our conclusions

exponential stability in p th mean of solutions and of convergent euler type solutions of stochastic delay differential equations one concept of the stability of a solution of an evolutionary equation relates to the sensitivity of the solution to perturbations in the initial data there are other stability concepts notably those concerned with persistent perturbations results are presented on the stability in p th mean of solutions of stochastic delay differential equations with multiplicative noise and of stochastic delay difference equations the difference equations are of a type found in numerical analysis and we employ our results to obtain mean square stability criteria for the solution of the euler maruyama discretization of stochastic delay differential equations the analysis proceeds as follows we show that an inequality of halanay type derivable via comparison theory can be employed to derive conditions for p th mean stability of a solution we then produce a discrete analogue of the halanay type theory that permits us to develop a p th mean stability analysis of analogous stochastic difference equations the application of the theoretical results is illustrated by deriving mean square stability conditions for solutions and numerical solutions of a constant coefficient linear test equation