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The exponential stability and the polynomial stability have been well studied for the exact and numerical solutions of the stochastic pantograph differential equations (SPDEs), while the result on the general decay stabilities is ...
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The exponential stability and the polynomial stability have been well studied for the exact and numerical solutions of the stochastic pantograph differential equations (SPDEs), while the result on the general decay stabilities is insufficient for SPDEs. In this paper, the sufficient conditions of the almost sure stability with general decay rate of the exact and numerical solutions for SPDEs have been considered. The stability of two different theta numerical methods, namely the split-step theta method and the stochastic linear theta method, have been discussed respectively. From the conditions, we established for the two theta approximations to reproduce the stability of the exact solution, and we see that the conditions for ? . The bound of the Lyapunov exponent of the split-step theta method is little bigger than that of the stochastic linear theta method for sufficiently small step size. To illustrate the theoretical results, we give two examples to examine the almost sure exponential stability.
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Tau-leaping is a family of algorithms for the approximate simulation of the discrete state continuous time Markov chains. A motivation for the development of such methods can be found, for instance, in the fields of chemical kinet...
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Tau-leaping is a family of algorithms for the approximate simulation of the discrete state continuous time Markov chains. A motivation for the development of such methods can be found, for instance, in the fields of chemical kinetics and systems biology. It is known that the dynamical behavior of biochemical systems is often intrinsically stiff representing a serious challenge for their numerical approximation. The naive extension of stiff deterministic solvers to stochastic integration often yields numerical solutions with either impractically large relaxation times or incorrectly resolved covariance. In this paper, we propose a splitting heuristic which helps to resolve some of these issues. The proposed integrator contains a number of unknown parameters which are estimated for each particular problem from the moment equations of the corresponding linearized system. We show that this method is able to reproduce the exact mean and variance of the linear scalar test equation and demonstrates a good accuracy for the arbitrarily stiff systems at least in the linear case. The numerical examples for both linear and nonlinear systems are also provided, and the obtained results confirm the efficiency of the considered splitting approach. (C) 2019 Elsevier B.V. All rights reserved.
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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 ...
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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 theta is an element of [1/2, 1] is asymptotically mean stable and so is theta is an element of [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 theta is an element of [1/2, 1] is exponentially mean stable and so is theta is an element of [0, 1/2). Finally, two numerical examples are given to show the efficiency of the obtained results. (C) 2016 Elsevier B.V. All rights reserved.
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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 so...
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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 theta is an element of [0, 1/2), there exists a constant Delta* > 0 depending on theta such that the numerical schemes produced by the SST method and the SLT method are mean square exponentially stable for Delta is an element of(0, Delta*), under the same assumptions as those to guarantee the mean square exponential stability of the underlying continuous model. For the case theta is an element of [1/2, 1], we show the same stability conclusion for all Delta > 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. (C) 2018 Elsevier B.V. All rights reserved.
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A fractional step theta-method for the approximation of time-dependent viscoelastic fluid flow equations is described and analyzed in this article. The algorithm uses substeps within a time step to sequentially update velocity, pr...
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A fractional step theta-method for the approximation of time-dependent viscoelastic fluid flow equations is described and analyzed in this article. The algorithm uses substeps within a time step to sequentially update velocity, pressure, and stress. This lagged approach to temporal integration requires a resolution of smaller systems than a fully implicit approach while achieving a second order temporal accuracy. We establish a priori error estimates for and provide numerical computations to support the theoretical results and our scheme, demonstrate the capability of this method.
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In this paper, a new split-step theta (SS theta) method for stochastic age-dependent population equations with Poisson jumps is constructed. The main aim of this paper is to investigate the convergence of the SS theta method for s...
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In this paper, a new split-step theta (SS theta) method for stochastic age-dependent population equations with Poisson jumps is constructed. The main aim of this paper is to investigate the convergence of the SS theta method for stochastic age-dependent population equations with Poisson jumps. It is proved that the proposed method is convergent with strong order 1/2 under given conditions. Finally, an example is simulated to verify the results obtained from theory, the results show that the SSh method has better accuracy compared to the Euler method. (C) 2015 Elsevier Inc. All rights reserved.
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In this paper, we investigate the strong convergence rate of the split-step theta (SST) method for a kind of stochastic differential equations with piecewise continuous arguments (SDEPCAs) under some polynomially growing condition...
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In this paper, we investigate the strong convergence rate of the split-step theta (SST) method for a kind of stochastic differential equations with piecewise continuous arguments (SDEPCAs) under some polynomially growing conditions. It is shown that the SST method with theta is an element of [1/2, 1] is strongly convergent with order 1/2 in pth(p >= 2) moment if both drift and diffusion coefficients are polynomially growing with regard to the delay terms, while the diffusion coefficients are globally Lipschitz continuous in non-delay arguments. The exponential mean square stability of the improved split-step theta (ISST) method is also studied without the linear growth condition. With some relaxed restrictions on the step-size, it is proved that the ISST method with theta is an element of (1/2, 1] is exponentially mean square stable under the monotone condition. Without any restriction on the step-size, there exists theta* is an element of (1/2, 1] such that the ISST method with theta is an element of (theta*, 1] is exponentially stable in mean square. Some numerical simulations are presented to illustrate the analytical theory.
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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 accordan...
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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 theta is an element of (1/2, 1] than in the case theta is an element of [0, 1/2]. Numerical results are presented which reveal that the SSTM method is conditionally mean-square stable and that in the case theta is an element of(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 theta is an element of [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. (C) 2018 Elsevier Inc. All rights reserved.
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Neutral stochastic delay differential equations often appear in various fields of science and engineering. The aim of this article is to investigate the strong convergence of the split-step theta (SST) method for the neutral stoch...
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Neutral stochastic delay differential equations often appear in various fields of science and engineering. The aim of this article is to investigate the strong convergence of the split-step theta (SST) method for the neutral stochastic delay differential equations, where the corresponding coefficients may be highly nonlinear with respect to the delay variables. In particular, we reveal that the SST method with θ ε [0,1] strongly converges to the exact solution with the order 1/2. Some numerical results are presented to confirm the obtained results.
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In this paper, we are interested in numerical methods with variable stepsize for stochastic pantograph differential equations (SPDEs). SPDEs are very special stochastic delay differential equations (SDDEs) with unbounded memory. T...
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In this paper, we are interested in numerical methods with variable stepsize for stochastic pantograph differential equations (SPDEs). SPDEs are very special stochastic delay differential equations (SDDEs) with unbounded memory. The problem of computer memory hold, when the numerical methods with constant step-size are applied to the SPDEs. In this work, we construct split-step theta (SS.) methods with variable step-size for SPDEs. The boundedness and strong convergence of the numerical methods are investigated under a local Lipschitz condition and a coupled condition on the drift and diffusion coefficients. It is proved that, the SS. methods with variable step-size for.. [12, 1] converge strongly to the exact solution. In addition, the strong order 0.5 is given under mild assumptions. The mean-square stability (MS-Stability) of the numerical methods with.. (1 2, 1] is given. Finally, some illustrative numerical examples are presented to show the efficiency of the methods, and how MS-Stability of SS. methods depends on the parameter theta for both linear and nonlinear models.
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