This paper deals with almost sure and moment exponential stability of a class of predictor- corrector methods applied to the stochastic differential equations of Ito-type. Stability criteria for this type of methods a...This paper deals with almost sure and moment exponential stability of a class of predictor- corrector methods applied to the stochastic differential equations of Ito-type. Stability criteria for this type of methods are derived. The methods are shown to maintain almost sure and moment exponential stability for all sufficiently small timesteps under appropriate conditions. A numerical experiment further testifies these theoretical results.展开更多
The stability of the first-order and second-order solution moments for a Harrison-type predator-prey model with parametric Gaussian white noise is analyzed in this paper. The moment equations of the system solution ar...The stability of the first-order and second-order solution moments for a Harrison-type predator-prey model with parametric Gaussian white noise is analyzed in this paper. The moment equations of the system solution are obtained under Ito interpretations. The delay-independent stable condition of the first-order moment is identical to that of the deterministic delayed system, and the delay-independent stable condition of the second-order moment depends on the noise intensities. The corresponding critical time delays are determined once the stabilities of moments lose. Further, when the time delays are greater than the critical time delays, the system solution becomes unstable with the increase of noise intensities. Finally, some numerical simulations are given to verify the theoretical results.展开更多
In this paper we will see that, under certain conditions, the techniques of generalized moment problem will apply to numerically solve an Volterra integral equation of first kind or second kind. Volterra integral equa...In this paper we will see that, under certain conditions, the techniques of generalized moment problem will apply to numerically solve an Volterra integral equation of first kind or second kind. Volterra integral equation is transformed into a one-dimensional generalized moment problem, and shall apply the moment problem techniques to find a numerical approximation of the solution. Specifically you will see that solving the Volterra integral equation of first kind f（t） = {a^t K（t, s）x（s）ds a ≤ t ≤ b or solve the Volterra integral equation of the second kind x（t） =f（t）＋{a^t K（t,s）x（s）ds a ≤ t ≤ b is equivalent to solving a generalized moment problem of the form un = {a^b gn（s）x（s）ds n = 0,1,2… This shall apply for to find the solution of an integrodifferential equation of the form x＇（t） = f（t） ＋ {a^t K（t,s）x（s）ds for a ≤ t ≤ b and x（a） = a0 Also considering the nonlinear integral equation： f（x）= {fa^x y（x-t）y（t）dt This integral equation is transformed a two-dimensional generalized moment problem. In all cases, we will find an approximated solution and bounds for the error of the estimated solution using the techniques ofgeneralized moment problem.展开更多
It will be shown that finding solutions from the Poisson and Klein-Gordon equations under Neumann conditions are equivalent to solving an integral equation, which can be treated as a generalized two-dimensional moment...It will be shown that finding solutions from the Poisson and Klein-Gordon equations under Neumann conditions are equivalent to solving an integral equation, which can be treated as a generalized two-dimensional moment problem over a domain that is considered rectangular. The method consists to solve the integral equation numerically using the two-dimensional inverse moments problem techniques. We illustrate the different cases with examples.展开更多
We considerer parabolic partial differential equations: under the conditions , on a region . We will see that an approximate solution can be found using the techniques of generalized inverse moments problem and also b...We considerer parabolic partial differential equations: under the conditions , on a region . We will see that an approximate solution can be found using the techniques of generalized inverse moments problem and also bounds for the error of estimated solution. First we transform the parabolic partial differential equation to the integral equation . Using the inverse moments problem techniques we obtain an approximate solution of . Then we find a numerical approximation of when solving the integral equation , because solving the previous integral equation is equivalent to solving the equation .展开更多
Several available methods, known in literatures, are available for solving nth order differential equations and their complexities differ based on the accuracy of the solution. A successful method, known to researcher...Several available methods, known in literatures, are available for solving nth order differential equations and their complexities differ based on the accuracy of the solution. A successful method, known to researcher in the area of computational electromagnetic and called the Method of Moment (MoM) is found to have its way in this domain and can be used in solving boundary value problems where differential equations are resulting. A simplified version of this method is adopted in this paper to address this problem, and two differential equations examples are considered to clarify the approach and present the simplicity of the method. As illustrated in this paper, this approach can be introduced along with other methods, and can be considered as an attractive way to solve differential equations and other boundary value problems.展开更多
The derivation of moment equations for the theoretical description of electrons is of interest for modelling of gas discharge plasmas and semiconductor devices. Usually, certain artificial closure assumptions are appl...The derivation of moment equations for the theoretical description of electrons is of interest for modelling of gas discharge plasmas and semiconductor devices. Usually, certain artificial closure assumptions are applied in order to derive a closed system of moment equations from the electron Boltzmann equation. Here, a novel four-moment model for the description of electrons in nonthermal plasmas is derived by an expansion of the electron velocity distribution function in Legendre polynomials. The proposed system of partial differential equations is consistently closed by definition of transport coefficients that are determined by solving the electron Boltzmann equation and are then used in the fluid calculations as function of the mean electron energy. It is shown that the four-moment model can be simplified to a new drift-diffusion approximation for electrons without loss of accuracy, if the characteristic frequency of the electric field alteration in the discharge is small in comparison with the momentum dissipation frequency of the electrons. Results obtained by the proposed fluid models are compared to those of a conventional drift-diffusion approximation as well as to kinetic results using the example of low pressure argon plasmas. It is shown that the results provided by the new approaches are in good agreement with kinetic results and strongly improve the accuracy of fluid descriptions of gas discharges.展开更多
In this paper, with the Kronecker's product and Kronecker's sum of matrices, the 2nd order moment equations of linear Ito stochastic systems are dervided. Based on the moment equations obtained, a necessary an...In this paper, with the Kronecker's product and Kronecker's sum of matrices, the 2nd order moment equations of linear Ito stochastic systems are dervided. Based on the moment equations obtained, a necessary and sufficient condition for the mean-square asymptotic stability of linear Ito stochastic systems is obtained.For the time-invariant stochastic systems,the necessary and sufficient condition is just the same as the Hurwitz property of certain matrices related to the coefficient matrices of the systems. An algorithm STILSS is given for testing the mean-square asymptotic stability of time-invariant linear Ito stochastic systems.展开更多
In this paper, the asymptotical p-moment stabifity of stochastic impulsive differential equations is studied, and a comparison theory to ensure the asymptotieal p-moment stability for trivial solution of this system i...In this paper, the asymptotical p-moment stabifity of stochastic impulsive differential equations is studied, and a comparison theory to ensure the asymptotieal p-moment stability for trivial solution of this system is established, from which we can find out whether a stochastic impulsive differential system is stable just from a deterministic comparison system. As an application of this theory, we control the chaos of stochastic Chen system using impulsive method, and a stable region is deduced too. Finally, numerical simulations verify the feasibility of our method.展开更多
Morales-Casique et al.(Adv.Water Res.,29(2006),pp.1238-1255)developed exact first and second nonlocal moment equations for advective-dispersive transport in finite,randomly heterogeneous geologic media.The velocity an...Morales-Casique et al.(Adv.Water Res.,29(2006),pp.1238-1255)developed exact first and second nonlocal moment equations for advective-dispersive transport in finite,randomly heterogeneous geologic media.The velocity and concentration in these equations are generally nonstationary due to trends in heterogeneity,conditioning on site data and the influence of forcing terms.Morales-Casique et al.(Adv.Water Res.,29(2006),pp.1399-1418)solved the Laplace transformed versions of these equations recursively to second order in the standard deviationσY of(natural)log hydraulic conductivity,and iteratively to higher-order,by finite elements followed by numerical inversion of the Laplace transform.They did the same for a space-localized version of the mean transport equation.Here we recount briefly their theory and algorithms;compare the numerical performance of the Laplace-transform finite element scheme with that of a high-accuracy ULTIMATE-QUICKEST algorithm coupled with an alternating split operator approach;and review some computational results due to Morales-Casique et al.(Adv.Water Res.,29(2006),pp.1399-1418)to shed light on the accuracy and computational efficiency of their recursive and iterative solutions in comparison to conditional Monte Carlo simulations in two spatial dimensions.展开更多
In this article, we consider the long time behavior of the solutions to stochastic wave equations driven by a non-Gaussian Lévy process. We shall prove that under some appropriate conditions, the exponential stab...In this article, we consider the long time behavior of the solutions to stochastic wave equations driven by a non-Gaussian Lévy process. We shall prove that under some appropriate conditions, the exponential stability of the solutions holds. Finally, we give two examples to illustrate our results.展开更多
基金supported by NSFC under Grant Nos.11171125 and 91130003NSFH under Grant No. 2011CDB289the Freedom Explore Program of Central South University
文摘This paper deals with almost sure and moment exponential stability of a class of predictor- corrector methods applied to the stochastic differential equations of Ito-type. Stability criteria for this type of methods are derived. The methods are shown to maintain almost sure and moment exponential stability for all sufficiently small timesteps under appropriate conditions. A numerical experiment further testifies these theoretical results.
基金supported by the National Natural Science Foundation of China(Grant Nos.11272051 and 11302172)
文摘The stability of the first-order and second-order solution moments for a Harrison-type predator-prey model with parametric Gaussian white noise is analyzed in this paper. The moment equations of the system solution are obtained under Ito interpretations. The delay-independent stable condition of the first-order moment is identical to that of the deterministic delayed system, and the delay-independent stable condition of the second-order moment depends on the noise intensities. The corresponding critical time delays are determined once the stabilities of moments lose. Further, when the time delays are greater than the critical time delays, the system solution becomes unstable with the increase of noise intensities. Finally, some numerical simulations are given to verify the theoretical results.
文摘In this paper we will see that, under certain conditions, the techniques of generalized moment problem will apply to numerically solve an Volterra integral equation of first kind or second kind. Volterra integral equation is transformed into a one-dimensional generalized moment problem, and shall apply the moment problem techniques to find a numerical approximation of the solution. Specifically you will see that solving the Volterra integral equation of first kind f（t） = {a^t K（t, s）x（s）ds a ≤ t ≤ b or solve the Volterra integral equation of the second kind x（t） =f（t）＋{a^t K（t,s）x（s）ds a ≤ t ≤ b is equivalent to solving a generalized moment problem of the form un = {a^b gn（s）x（s）ds n = 0,1,2… This shall apply for to find the solution of an integrodifferential equation of the form x＇（t） = f（t） ＋ {a^t K（t,s）x（s）ds for a ≤ t ≤ b and x（a） = a0 Also considering the nonlinear integral equation： f（x）= {fa^x y（x-t）y（t）dt This integral equation is transformed a two-dimensional generalized moment problem. In all cases, we will find an approximated solution and bounds for the error of the estimated solution using the techniques ofgeneralized moment problem.
文摘It will be shown that finding solutions from the Poisson and Klein-Gordon equations under Neumann conditions are equivalent to solving an integral equation, which can be treated as a generalized two-dimensional moment problem over a domain that is considered rectangular. The method consists to solve the integral equation numerically using the two-dimensional inverse moments problem techniques. We illustrate the different cases with examples.
文摘We considerer parabolic partial differential equations: under the conditions , on a region . We will see that an approximate solution can be found using the techniques of generalized inverse moments problem and also bounds for the error of estimated solution. First we transform the parabolic partial differential equation to the integral equation . Using the inverse moments problem techniques we obtain an approximate solution of . Then we find a numerical approximation of when solving the integral equation , because solving the previous integral equation is equivalent to solving the equation .
文摘Several available methods, known in literatures, are available for solving nth order differential equations and their complexities differ based on the accuracy of the solution. A successful method, known to researcher in the area of computational electromagnetic and called the Method of Moment (MoM) is found to have its way in this domain and can be used in solving boundary value problems where differential equations are resulting. A simplified version of this method is adopted in this paper to address this problem, and two differential equations examples are considered to clarify the approach and present the simplicity of the method. As illustrated in this paper, this approach can be introduced along with other methods, and can be considered as an attractive way to solve differential equations and other boundary value problems.
文摘The derivation of moment equations for the theoretical description of electrons is of interest for modelling of gas discharge plasmas and semiconductor devices. Usually, certain artificial closure assumptions are applied in order to derive a closed system of moment equations from the electron Boltzmann equation. Here, a novel four-moment model for the description of electrons in nonthermal plasmas is derived by an expansion of the electron velocity distribution function in Legendre polynomials. The proposed system of partial differential equations is consistently closed by definition of transport coefficients that are determined by solving the electron Boltzmann equation and are then used in the fluid calculations as function of the mean electron energy. It is shown that the four-moment model can be simplified to a new drift-diffusion approximation for electrons without loss of accuracy, if the characteristic frequency of the electric field alteration in the discharge is small in comparison with the momentum dissipation frequency of the electrons. Results obtained by the proposed fluid models are compared to those of a conventional drift-diffusion approximation as well as to kinetic results using the example of low pressure argon plasmas. It is shown that the results provided by the new approaches are in good agreement with kinetic results and strongly improve the accuracy of fluid descriptions of gas discharges.
文摘In this paper, with the Kronecker's product and Kronecker's sum of matrices, the 2nd order moment equations of linear Ito stochastic systems are dervided. Based on the moment equations obtained, a necessary and sufficient condition for the mean-square asymptotic stability of linear Ito stochastic systems is obtained.For the time-invariant stochastic systems,the necessary and sufficient condition is just the same as the Hurwitz property of certain matrices related to the coefficient matrices of the systems. An algorithm STILSS is given for testing the mean-square asymptotic stability of time-invariant linear Ito stochastic systems.
基金Supported by National Natural Science Foundation of China under Grant Nos.10902085 and 10902062
文摘In this paper, the asymptotical p-moment stabifity of stochastic impulsive differential equations is studied, and a comparison theory to ensure the asymptotieal p-moment stability for trivial solution of this system is established, from which we can find out whether a stochastic impulsive differential system is stable just from a deterministic comparison system. As an application of this theory, we control the chaos of stochastic Chen system using impulsive method, and a stable region is deduced too. Finally, numerical simulations verify the feasibility of our method.
基金This work was supported in part by NSF/ITR Grant EAR-0110289through a scholarship granted to the lead author by CONACYT of Mexico.
文摘Morales-Casique et al.(Adv.Water Res.,29(2006),pp.1238-1255)developed exact first and second nonlocal moment equations for advective-dispersive transport in finite,randomly heterogeneous geologic media.The velocity and concentration in these equations are generally nonstationary due to trends in heterogeneity,conditioning on site data and the influence of forcing terms.Morales-Casique et al.(Adv.Water Res.,29(2006),pp.1399-1418)solved the Laplace transformed versions of these equations recursively to second order in the standard deviationσY of(natural)log hydraulic conductivity,and iteratively to higher-order,by finite elements followed by numerical inversion of the Laplace transform.They did the same for a space-localized version of the mean transport equation.Here we recount briefly their theory and algorithms;compare the numerical performance of the Laplace-transform finite element scheme with that of a high-accuracy ULTIMATE-QUICKEST algorithm coupled with an alternating split operator approach;and review some computational results due to Morales-Casique et al.(Adv.Water Res.,29(2006),pp.1399-1418)to shed light on the accuracy and computational efficiency of their recursive and iterative solutions in comparison to conditional Monte Carlo simulations in two spatial dimensions.
基金supported by National Natural Science Foundation of China(11571190)the Fundamental Research Funds for the Central Universities+3 种基金supported by the China Scholarship Council(201807315008)National Natural Science Foundation of China(11501565)the Youth Project of Humanities and Social Sciences of Ministry of Education(19YJCZH251)supported by National Natural Science Foundation of China(11701084 and 11671084)
文摘In this article, we consider the long time behavior of the solutions to stochastic wave equations driven by a non-Gaussian Lévy process. We shall prove that under some appropriate conditions, the exponential stability of the solutions holds. Finally, we give two examples to illustrate our results.
