transient behavior analysis of vibrationally controlled nonlinear parabolic systems with neumann boundary conditions the transient behavior of vibrationally controlled distributed parameter systems governed by parabolic partial differential equations pdes with neumann boundary conditions is analyzed the analysis uses a certain mapping and a time invariant pde whose trajectories under this mapping yield the approximate moving averages along the trajectories of the vibrationally controlled system this approach considerably enhances the understanding of the behavior of vibrationally controlled distributed parameter systems and facilitates a selection of the parameters of stabilizing vibrations that ensure transient behavior with more desirable properties

almost sure exponential stability of numerical solutions to stochastic delay hopfield neural networks stability of numerical solutions to stochastic delay differential equations have received an increasing attention but there has been so far little work on the stability analysis of numerical solutions to stochastic delay hopfield neural networks the aim of this paper is to study the almost sure exponential stability of numerical solutions to stochastic delay hopfield neural networks by using two approaches the euler method and the backward euler method under some reasonable conditions both the euler scheme and the backward euler scheme are proved to be almost sure exponential stability in particular the euler method and the backward euler method are mainly based on the semimartingale convergence theorem

stability analysis of split step θ milstein method for a class of n dimensional stochastic differential equations abstract in this paper we introduce a split step theta milstein sstm method for n dimensional stochastic delay differential equations sddes the exponential mean square stability of the numerical solutions is analyzed and in accordance with previous findings we prove that the method is exponentially mean square stable if the employed time step is smaller than a given and easily computable upper bound in particular according to our investigation larger time steps can be used in the case 1 2 1 than in the case 0 1 2 numerical results are presented which reveal that the sstm method is conditionally mean square stable and that in the case 1 2 1 the interval of time steps for which the sstm method is theoretically shown to be mean square stable is significantly larger than in the case 0 1 2 it is worth mentioning that the sstm method has never been employed or analyzed for the numerical approximation of sddes at least to the very best of our knowledge

stability analysis for neutral stochastic delay systems with markovian switching abstract this paper considers the exponential stability in p th p 1 moment and the almost sure exponential stability for neutral stochastic delay systems with markovian switching by using the generalized integral inequality and the nonnegative local martingale convergence theorem the lyapunov stability results on the exponential stability in p th p 1 moment and the almost sure exponential stability for such systems are given when the time varying delay is a bounded measurable function and the coefficients in the monotonicity condition are time varying the derived results are illustrated by two examples

the stability with a general decay of stochastic delay differential equations with markovian switching abstract this paper considers the problems on the existence and uniqueness the pth p 1 moment and the almost sure stability with a general decay for the global solution of stochastic delay differential equations with markovian switching when the drift term and the diffusion term satisfy the locally lipschitz condition and the monotonicity condition by using the lyapunov function approach the barbalat lemma and the nonnegative semi martingale convergence theorem some sufficient conditions are proposed to guarantee the existence and uniqueness as well as the stability with a general decay for the global solution of such equations it is mentioned that in this paper the time varying delay is a bounded measurable function the derived stability results are more general which not only include the exponential stability but also the polynomial stability as well as the logarithmic one at last two examples are given to show the effectiveness of the theoretical results obtained

necessary conditions controllability and the value function for differential difference inclusions on deduit des conditions necessaires pour la commande optimale d un systeme contraint par des inclusions differentielles aux differences

asymptotic behavior of a nonlinear kirchhoff type equation with spring boundary conditions in this paper we consider the initial boundary value problem for a nonlinear kirchhoff type equation with the damping term and spring boundary conditions we establish the global existence and uniqueness of solutions to this problem in time and give an example and simulation to illustrate our results for the proof we use the faedo galerkin approximate method finally we study the asymptotic behavior of solutions and some of its simulation results results of this paper are able to apply industrial parts such as a typical model widely used to represent threads wires magnetic tapes belts band saws and so on

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

evaluation of four irrational definite sine integrals using residue theory this paper evaluates four irrational definite sine integrals using complex variable residue theory a solution using this method does not require the choosing of complicated contour integration paths around the singularities and branch points of the integrand the solutions produced are in the form of infinite series which converge rapidly the integrals in this paper can be shown to be complete elliptic integrals of the first and second kind in each case the laurent series was found by re indexing the series to find the constant co efficient of the 1 z term

almost sure exponential stability of backward euler maruyama discretizations for hybrid stochastic differential equations this is a continuation of the first author s earlier paper 1 jointly with pang and deng in which the authors established some sufficient conditions under which the euler maruyama em method can reproduce the almost sure exponential stability of the test hybrid sdes the key condition imposed in 1 is the global lipschitz condition however we will show in this paper that without this global lipschitz condition the em method may not preserve the almost sure exponential stability we will then show that the backward em method can capture the almost sure exponential stability for a certain class of highly nonlinear hybrid sdes

development and application of halanay type theory evolutionary differential and difference equations with time lag we consider extensions developments and modifications of a result due to halanay and the application of halanay type inequalities in the analysis and numerics of retarded functional differential equations difference equations and retarded functional difference equations our emphasis is on the variety structure and development and future development of halanay type results and their applications we classify and present novel results of halanay type linear and non linear discrete semi discrete and continuous and establish their relevance to delay differential equations discretized analogues we consider q methods and difference equations a role for such results in stability and contractivity analysis is made apparent

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

applications of complex variable residue theory to the evaluation of irrational definite integrals iii in this paper four irrational definite integrals were evaluated using complex variable residue theory two of these integrals were proven to be zero the approach to evaluating these integrals is nonconventional but is very useful in spite of this however the method continues to produce correct and valid results all cases were verified using numerical integration techniques

sharp energy decay estimates for the wave equation with a local degenerate dissipation we derive a fast decay estimate for the wave equation with a local degenerate dissipation of the type a x u t in a bounded domain w the dissipative coefficient a x is a nonnegative function only on a neighborhood of some part of the boundary w and may vanish somewhere in w the results obtained extend and improve on earlier results of nakao as well as those of tcheugoue tebou the method of proof is based on multipliers technique and some modified interpolation and difference inequalities

a new razumikhin theorem for delay difference equations abstract in this paper we established a new razumikhin type theorem on stability for delay difference equations

delay dependent stability analysis of numerical methods for stochastic delay differential equations this paper is concerned with the numerical solution of stochastic delay differential equations the focus is on the delay dependent stability of numerical methods for a linear scalar test equation with real coefficients by using the so called root locus technique the full asymptotic stability region in mean square of stochastic theta methods is obtained which is characterized by a sufficient and necessary condition in terms of the drift and diffusion coefficients as well as time stepsize and method parameter theta then this condition is compared with the analytical stability condition it is proved that the backward euler method completely preserves the asymptotic mean square stability of the underlying system and the euler maruyama method preserves the instability of the system our investigation also shows that not all theta methods with q 12 preserve this delay dependent stability some numerical examples are presented to confirm the theoretical results

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

noise induced changes to the behaviour of semi implicit euler methods for stochastic delay differential equations undergoing bifurcation we are concerned with estimating parameter values at which bifurcations occur in stochastic delay differential equations after a brief review of bifurcation we employ a numerical approach and consider how bifurcation values are influenced by the choice of numerical scheme and the step length and by the level of white noise present in the equation in this paper we provide a formulaic relationship between the estimated bifurcation value the level of noise the choice of numerical scheme and the step length we are able to show that in the presence of noise there may be some loss of order in the accuracy of the approximation to the true bifurcation value compared to the use of the same approach in the absence of noise

preserving exponential mean square stability in the simulation of hybrid stochastic differential equations positive results are derived concerning the long time dynamics of fixed step size numerical simulations of stochastic differential equation systems with markovian switching euler maruyama and implicit theta method discretisations are shown to capture exponential mean square stability for all sufficiently small time steps under appropriate conditions moreover the decay rate as measured by the second moment lyapunov exponent can be reproduced arbitrarily accurately new finite time convergence results are derived as an intermediate step in this analysis we also show however that the mean square a stability of the theta method does not carry through to this switching scenario the proof techniques are quite general and hence have the potential to be applied to other numerical methods

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

local hadamard well posedness for nonlinear wave equations with supercritical sources and damping abstract we consider the wave equation with supercritical interior and boundary sources and damping terms the main result of the paper is local hadamard well posedness of finite energy weak solutions the results obtained 1 extend the existence results previously obtained in the literature by allowing more singular sources 2 show that the corresponding solutions satisfy hadamard well posedness conditions during the time of existence this result provides a positive answer to an open question in the area and it allows for the construction of a strongly continuous semigroup representing the dynamics governed by the wave equation with supercritical sources and damping

local soltutions for a nonlinear degenerate hyperbolic equation on etudie l existence et l unicite des solutions locales pour le probleme mixte associe a l equation hyperbolique degenere du type 2 u 2 m u x t 2 dx u 0 ou est un ouvert borne de r n a frontiere lisse et x est un vecteur de r n

exponential stability of euler maruyama solutions for impulsive stochastic differential equations with delay this paper establishes a method to study the exponential stability of euler maruyama em method for impulsive stochastic differential equations with delay by using the properties of m matrix and stochastic analysis technique some conditions under which the em solution is exponentially mean square stable are obtained some examples are provided to illustrate the results

on the nonlinear wave equation utt b t ux 2 uxx f x t u ux ut associated with the mixed homogeneous conditions abstract in this paper we consider the following nonlinear wave equation 1 u tt b t u x 2 u xx f x t u u x u t x 0 1 0 2 u x 0 t h 0 u 0 t 0 u x 1 t h 1 u 1 t 0 3 u x 0 u 0 x u t x 0 u 1 x where h0 h1 are given nonnegative constants and b f u 0 u 1 are given functions in eq 1 the coefficient b t ux 2 containing an integral u x 2 0 1 u x x t 2 dx in this paper we associate with problem 1 3 a linear recursive scheme for which the existence of a local and unique solution is proved by using standard compactness argument in case of b c 2 r 2 b b0 0 b 1 c 1 r 2 b1 0 f c 2 0 r 3 and f 1 c 1 0 r 3 we obtain from the following equation u tt b t u x 2 eb 1 t u x 2 u xx f x t u u x u t ef 1 x t u u x u t associated to 2 3 a weak solution ue x t having an asymptotic expansion of order 2 in e for e sufficiently small

discrete time approximations of stochastic delay equations the milstein scheme in this paper we develop a strong milstein approximation scheme for solving stochastic delay differential equations sddes the scheme has convergence order 1 in order to establish the scheme we prove an infinite dimensional ito formula for tame functions acting on the segment process of the solution of an sdde it is interesting to note that the presence of the memory in the sdde requires the use of the malliavin calculus and the anticipating stochastic analysis of nualart and pardoux given the nonanticipating nature of the sdde the use of anticipating calculus methods in the context of strong approximation schemes appears to be novel

a note on exponential stability in p th mean of solutions of stochastic delay differential equations in a very recent paper baker and buckwar exponential stability in pth mean of solutions and of convergent euler type solutions of stochastic delay differential equations j comput appl math 184 2005 404 427 investigated the exponential stability in pth mean of solutions of stochastic delay differential equations with multiplicative noise and of stochastic difference equations which are euler maruyama discretization of stochastic delay differential equations the dini derivative of the expectation of v t x t along x t is taken in their stability analysis unfortunately the main results derived by them are somewhat restrictive to be applied for practical purposes in this note we shall improve the corresponding results an example is given to illustrate our theory

mean square stability and dissipativity of two classes of theta methods for systems of stochastic delay differential equations in this paper we first study the mean square stability of numerical methods for stochastic delay differential equations under a coupled condition on the drift and diffusion coefficients this condition admits that the diffusion coefficient can be highly nonlinear i e it does not necessarily satisfy a linear growth or global lipschitz condition it is proved that for all positive stepsizes the classical stochastic theta method with q 0 5 is asymptotically mean square stable and the split step theta method with q 0 5 is exponentially mean square stable conditional stability results for the methods with q 0 5 and prove that the method possesses a bounded absorbing set in mean square independent of initial data

mean square stability and almost sure exponential stability of two step maruyama methods of stochastic delay hopfield neural networks abstract in this paper the two step maruyama methods of stochastic delay hopfield neural networks are studied we have found that under what choices of step size the two step maruyama methods of stochastic delay hopfield networks maintain the stability of exact solutions the mean square stability of two step maruyama methods of stochastic delay hopfield neural networks is investigated under suitable conditions also the almost sure exponential stability of two step maruyama methods of stochastic delay hopfield networks is proved using the semi martingale convergence theorem further the comparisons of stability conditions to the previous results in liu and zhu 2015 rathinasamy 2012 and ronghua et al 2010 are given numerical experiments are provided to illustrate our theoretical results

applications of complex variable residue theory to the evaluation solution of irrational definite integrals ii abstract eight irrational definite integrals are proven to be zero using complex analysis as in previous papers the integrals are first expressed as contour integrals by expanding binomially and collecting the contributions to the 1 z term the residue is found in all eight cases the residues are zero and hence the integrals are zero it was not expected that all of the integrals would be zero the result further illustrates the usefulness of this method to find exact values of extremely complex integrals it is again concluded that this technique is widely applicable to whole classes of integrals not yet evaluated

robust exponential stability of uncertain delayed neural networks with stochastic perturbation and impulse effects this paper focuses on the hybrid effects of parameter uncertainty stochastic perturbation and impulses on global stability of delayed neural networks by using the ito formula lyapunov function and halanay inequality we established several mean square stability criteria from which we can estimate the feasible bounds of impulses provided that parameter uncertainty and stochastic perturbations are well constrained moreover the present method can also be applied to general differential systems with stochastic perturbation and impulses

almost sure exponential stability of the backward euler maruyama scheme for stochastic delay differential equations with monotone type condition this paper is a continuation of our previous paper in which the second author with mao and szpruch examined the almost sure stability of the euler maruyama em and the backward euler maruyama bem methods for stochastic delay differential equations sddes in the previous results although the drift coefficient may defy the linear growth condition the diffusion coefficient is required to satisfy the linear growth condition in this paper we want to further relax the condition under monotone type condition this paper will give the almost sure stability of the bem for sddes whose both drift and diffusion coefficients may defy the linear condition this improves the existing results considerably

ms stability of the euler maruyama method for stochastic differential delay equations the aim of this paper is to investigate the stability of the euler maruyama method for the stochastic differential equations with time delay the definition of ms stability of numerical methods is established the condition of the mean square stability of analytical solution is obtained for a linear scalar system with multiplicative noise and the ms stability of the numerical scheme is proved

asymptotic mean square stability of explicit runge kutta maruyama methods for stochastic delay differential equations as relatively little is known about runge kutta type method applied to stochastic delay differential equations we present an explicit runge kutta maruyama rkm method for solving them the mean square stability properties of the numerical solutions generated by the rkm method are investigated and a sufficient condition for stability is obtained and applied to the s rock type methods for stochastic delay differential equations numerical examples are provided to confirm theoretical results

existence and asymptotic stability for the semilinear wave equation with boundary damping and source term in this paper we consider the semilinear wave equation with boundary conditions this work is devoted to prove the existence of solutions and uniform decay rates of the wave equation with boundary damping and source term

exponential stability of θ method for stochastic differential equations in the g framework abstract exponential stability of the stochastic method for stochastic differential equations with g brownian motion called g sdes for brevity is investigated it is proved that under the global lipschitz condition a g sde is p th p 0 1 moment exponentially stable if and only if the stochastic method with a sufficiently small step size is also p th moment exponentially stable the stochastic method for a g sde is p th p 0 1 moment exponentially stable then it is also quasi surely exponentially stable and furthermore the g sde is quasi surely exponentially stable numerical examples are demonstrated to illustrate the obtained theoretical results

the w transform in stability analysis for stochastic linear functional difference equations abstract we apply the method of integral transforms to study stability properties of rather general linear functional difference equations of ito type this approach called the w method is based on integral transforms coming from auxiliary equations that already possess the desired asymptotic properties some sufficient conditions for different kinds of stability are found in this way a general framework is offered as well that can be used to study asymptotic properties of other classes of stochastic difference equations

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

a note on the lasalle type theorems for stochastic differential delay equations the main aim of this note is to improve some results obtained in the author s earlier paper 1999 j math anal appl 236 350 369 from the improved result follow some useful criteria on the stochastic asymptotic stability and boundedness

almost sure exponential stability in the numerical simulation of stochastic differential equations this paper is mainly concerned with whether the almost sure exponential stability of stochastic differential equations sdes is shared with that of a numerical method under the global lipschitz condition we first show that the sde is p th moment exponentially stable for p in 0 1 if and only if the stochastic theta method is p th moment exponentially stable for a sufficiently small step size we then show that the p th moment exponential stability of the sde or the stochastic theta method implies the almost sure exponential stability of the sde or the stochastic theta method respectively hence our new theory enables us to study the almost sure exponential stability of the sdes using the stochastic theta method instead of the method of the lyapunov functions that is we can now carry out careful numerical simulations using the stochastic theta method with a sufficiently small step size delta t if the stochastic theta method is p th moment exponentially stable for a sufficiently small

global existence for the wave equation with nonlinear boundary damping and source terms abstract the paper deals with local and global existence for the solutions of the wave equation in bounded domains with nonlinear boundary damping and source terms the typical problem studied is u tt u 0 in 0 u 0 on 0 0 u u t m 2 u t u p 2 u on 0 1 u 0 x u 0 x u t 0 x u 1 x on where r n n 1 is a regular and bounded domain 0 1 m 1 2 p r where r 2 n 1 n 2 when n 3 r when n 1 2 and the initial data are in the energy space we prove local existence of the solutions in the energy space when m r r 1 p or n 1 2 and global existence when p m or the initial data are inside the potential well associated to the stationary problem

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

exact series solution to the epstein hubbell generalized elliptic type integral using complex variable residue theory in this paper complex variable residue theory is used to arrive at an exact infinite series solution to the epstein hubbell integral this method of solution is different from those used to derive previous solutions to the epstein hubbell integral but produces exactly equivalent results several sets of numerical data are included that verify the results additional information on the improvement of convergence will be forthcoming at a later date

mean square exponential stability for stochastic control systems with discrete time state feedbacks and their numerical schemes in simulation aiming at the qualitative numerical simulation this article concerns the mean square exponential stability mses of the stochastic systems with discrete time state feedbacks and their numerical schemes a time delay dependent stability criterion is obtained for the underlying system the euler maruyama and backward euler maruyama schemes are established and their mses are proved under the same criterion which means that the numerical schemes preserve the stability of the continuous model the illustrative example at the end of this letter shows that the theoretical upper bound for the sampling period limited in the stability criterion is much greater than that obtained by the stability criteria from the literature

stability of analytical and numerical solutions of nonlinear stochastic delay differential equations this paper concerns the stability of analytical and numerical solutions of nonlinear stochastic delay differential equations sddes we derive sufficient conditions for the stability contractivity and asymptotic contractivity in mean square of the solutions for nonlinear sddes the results provide a unified theoretical treatment for sddes with constant delay and variable delay including bounded and unbounded variable delays then the stability contractivity and asymptotic contractivity in mean square are investigated for the backward euler method it is shown that the backward euler method preserves the properties of the underlying sddes the main results obtained in this work are different from those of razumikhin type theorems indeed our results hold without the necessity of constructing or finding an appropriate lyapunov functional

on exponential mean square stability of two step maruyama methods for stochastic delay differential equations we are concerned with the exponential mean square stability of two step maruyama methods for stochastic differential equations with time delay we propose a family of schemes and prove that it can maintain the exponential mean square stability of the linear stochastic delay differential equation for every step size of integral fraction of the delay in the equation numerical results for linear and nonlinear equations show that this family of two step maruyama methods exhibits better stability than previous two step maruyama methods

on mean square stability of two step maruyama methods for nonlinear neutral stochastic delay differential equations the asymptotic mean square stability of two step maruyama methods is considered for nonlinear neutral stochastic differential equations with constant time delay nsddes under the one sided lipschitz condition and the linear growth condition it is proved that a family of implicit two step maruyama methods can preserve the stability of the analytic solution in mean square sense numerical results for both a nonlinear nsdde and a system show that the family of two step maruyama methods have better stability than previous two step maruyama methods

theoretical and numerical analysis for volterra integro differential equations with ito integral under polynomially growth conditions abstract in this paper we theoretically and numerically deal with nonlinear volterra integro differential equations with ito integral under a one sided lipschitz condition and polynomially growth conditions it is proved that both the exact solutions and vector fields are bounded and satisfy a holder condition in the pth moment sense analogously the boundedness and holder condition in the pth moment sense are preserved by the semi implicit euler method for sufficiently small step size moreover by the local truncated errors we prove the strong convergence order 1 finally numerical simulations on stochastic control models and stochastic ginzburg landau equation illustrate our results

stability of the drift implicit and double implicit milstein schemes for nonlinear sdes abstract this paper examines the stability of numerical solutions of nonlinear stochastic differential equations sdes with non global lipschitz continuous coefficients two implicit milstein schemes called drift implicit milstein scheme and double implicit milstein scheme are considered to simulate the underlying sdes it is proved that the schemes can preserve the stability and contractivity in mean square of the underlying systems

asymptotically almost periodic solutions of stochastic functional differential equations abstract in this paper we investigate a class of stochastic functional differential equations of the form dx t ax t f t x t x t dt g t x t x t dw t our main results concern the existence and exponential stability of quadratic mean asymptotically almost periodic mild solutions an example is given to illustrate our results

stabilization for the kirchhoff type equation from an axially moving heterogeneous string modeling with boundary feedback control abstract the first objective of this paper is to make the mathematical model for vibration suppression of an axially moving heterogeneous string in order to describe the geometrical nonlinearity due to finite transverse deformation the exact expression of the strain is used the mathematical modeling is derived first by using hamilton s principle and variational lemma and the derived nonlinear pde system is the kirchhoff type equation with boundary feedback control next we show the existence and uniqueness of strong solutions of the pde system via techniques of functional analysis mainly a theorem of compactness for the analysis of the approximation of the faedo galerkin method and estimate a decay rate for the energy the theoretical results are assured by numerical results of the solution s shape and asymptotic behavior for the system

strong convergence and stability of implicit numerical methods for stochastic differential equations with non globally lipschitz continuous coefficients we are interested in the strong convergence and almost sure stability of euler maruyama em type approximations to the solutions of stochastic differential equations sdes with non linear and non lipschitzian coefficients motivation comes from finance and biology where many widely applied models do not satisfy the standard assumptions required for the strong convergence in addition we examine the globally almost surely asymptotic stability in this non linear setting for em type schemes in particular we present a stochastic counterpart of the discrete lasalle principle from which we deduce stability properties for numerical methods

on local solutions of a mildly degenerate hyperbolic equation in this work we prove the existence of a unique nonlocal solution for the vibrations of a nonhomogeneous stretched string in sobolev spaces and without any smallness conditions on the size of the initial data

convergence and stability of balanced methods for stochastic delay integro differential equations abstract this paper deals with a family of balanced implicit methods for the stochastic delay integro differential equations it is shown that the balanced methods which own the implicit iterative scheme in the diffusion term give strong convergence rate of at least 1 2 it proves that the mean square stability for the stochastic delay integro differential equations is inherited by the strong balanced methods and the weak balanced methods with sufficiently small stepsizes several numerical experiments are given for illustration and show that the fully implicit methods are superior to those of the explicit methods in terms of mean square stabilities

split step theta method for stochastic delay integro differential equations with mean square exponential stability abstract in this paper we propose the split step theta method for stochastic delay integro differential equations by the lagrange interpolation technique and investigate the mean square exponential stability of the proposed scheme it is shown that the split step theta method can inherit the mean square exponential stability of the continuous model under the linear growth condition and the proposed stability condition by the delayed differential and difference inequalities established in the paper a numerical example is given at the end of the paper to illustrate the method and conclusion of the paper in addition the convergence of the split step theta method is proved in the appendix

some nonlinear degenerate wave equations on etudie la resolubilite locale du probleme aux valeurs limites et initiales suivant 2 u t 2 m gradu x t 2 dx u 0 dans 0 u 0 sur w 0 u 0 0 u t 0 dans ou m r est une fonction c 1 non negative sur 0 un domaine borne de r n a frontiere lisse w

a study on the epstein hubbell generalized elliptic type integral using residue theory abstract in this study a new solution method for epstein hubbell generalized elliptic type integral is developed and complex variable residue theory is used to arrive at the self truncating series associated with this integral by means of this method we determined that only a closed term c j 2 2n 2n n 2n n 2 n 0 1 2 contributes to the obtained series

asymptotic behavior analysis of a coupled time varying system application to adaptive systems asymptotic behavior of a partial state of a coupled ordinary and or partial differential equation is investigated it is specifically shown that if a signal x t is a solution to a dynamic system existing for all t spl ges 0 in an abstract banach space and pth p spl ges 1 power integrable then x t spl rarr 0 as t spl rarr spl infin the system is allowed to be nonautonomous and assumes the existence of a lyapunov function since the derivative of the lyapunov function is negative semidefinite stability or uniform stability in the sense of lyapunov would be concluded however this paper further asserts that the partial state which remains in the time derivative of the lyapunov function converges to zero asymptotically

exponential stability of non linear stochastic delay differential system with generalized delay dependent impulsive points abstract this paper is concerned with a non linear stochastic delay differential system with delay dependent impulsive perturbations in this work the size of the jump is defined as a general non linear delay dependent state variable and the solution of the impulsive stochastic delay differential system corresponding to the system without impulsive perturbations is given this work is based on the relation between the solution of the equivalent model of stochastic delay differential system without impulses corresponding to the solution of the system with impulses then the conditions of the exponential stability of the proposed impulsive system are obtained by deriving stability criteria of the corresponding system without impulses the numerical approximation for the stochastic delay system without impulses is developed using the runge kutta maruyama method and it is suitably applied for the corresponding impulsive system finally the obtained theoretical results are illustrated graphically for a stochastic delay system with impulses

almost sure and moment exponential stability of euler maruyama discretizations for hybrid stochastic differential equations positive results are derived concerning the long time dynamics of numerical simulations of stochastic differential equation systems with markovian switching euler maruyama discretizations are shown to capture almost sure and moment exponential stability for all sufficiently small timesteps under appropriate conditions

the split step backward euler method for linear stochastic delay differential equations in this paper the numerical approximation of solutions of linear stochastic delay differential equations sddes in the ito sense is considered we construct split step backward euler ssbe method for solving linear sddes and develop the fundamental numerical analysis concerning its strong convergence and mean square stability it is proved that the ssbe method is convergent with strong order c 12 in the mean square sense the conditions under which the ssbe method is mean square stable ms stable and general mean square stable gms stable are obtained some illustrative numerical examples are presented to demonstrate the order of strong convergence and the mean square stability of the ssbe method

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

retarded differential equations retarded differential equations rdes are differential equations having retarded arguments they arise in many realistic models of problems in science engineering and medicine where there is a time lag or after effect numerical techniques for such problems may be regarded as extensions of dense output methods for ordinary differential equations odes but scalar rdes are inherently infinite dimensional with a richer structure than their ode counterparts we give background material develop a theoretical foundation for the basic numerics and give some results not previously published