In this paper,we present the semi-implicit Euler（SIE）numerical solution for stochastic pantograph equations with jumps and prove that the SIE approximation solution converges to the exact solution in the mean-square...In this paper,we present the semi-implicit Euler（SIE）numerical solution for stochastic pantograph equations with jumps and prove that the SIE approximation solution converges to the exact solution in the mean-square sense under the Local Lipschitz condition.展开更多
Several efficient analytical methods have been developed to solve the solid-state diffusion problem, for constant diffusion coefficient problems. However, these methods cannot be applied for concentration-dependent di...Several efficient analytical methods have been developed to solve the solid-state diffusion problem, for constant diffusion coefficient problems. However, these methods cannot be applied for concentration-dependent diffusion coefficient problems and numerical methods are used instead. Herein, grid-based numerical methods derived from the control volume discretization are presented to resolve the characteristic nonlinear system of partial differential equations. A novel hybrid backward Euler control volume (HBECV) method is presented which requires only one iteration to reach an implicit solution. The HBECV results are shown to be stable and accurate for a moderate number of grid points. The computational speed and accuracy of the HBECV, justify its use in battery simulations, in which the solid-state diffusion coefficient is a strong function of the concentration.展开更多
In this paper, we consider strong convergence and almost sure exponential stability of the backward Euler-Maruyama method for nonlinear hybrid stochastic differential equations with time-variable delay. Under the loca...In this paper, we consider strong convergence and almost sure exponential stability of the backward Euler-Maruyama method for nonlinear hybrid stochastic differential equations with time-variable delay. Under the local Lipschitz condition and polynomial growth condition, it is proved that the backward Euler-Maruyama method is strongly convergent. Additionally, the moment estimates and almost sure exponential stability for the analytical solution are proved. Also, under the appropriate condition, we show that the numerical solutions for the backward Euler-Maruyama methods are almost surely exponentially stable. A numerical experiment is given to illustrate the computational effectiveness and the theoretical results of the method.展开更多
In the literature （Tan and Wang, 2010）, Tan and Wang investigated the convergence of the split-step backward Euler （SSBE） method for linear stochastic delay integro-differential equations （SDIDEs） and proved the...In the literature （Tan and Wang, 2010）, Tan and Wang investigated the convergence of the split-step backward Euler （SSBE） method for linear stochastic delay integro-differential equations （SDIDEs） and proved the mean-square stability of SSBE method under some condition. Unfortu- nately, the main result of stability derived by the condition is somewhat restrictive to be applied for practical application. This paper improves the corresponding results. The authors not only prove the mean-square stability of the numerical method but also prove the general mean-square stability of the numerical method. Furthermore, an example is given to illustrate the theory.展开更多
In this paper,we consider the Shliomis ferrofluid model and study its numerical approximation.We investigate a first-order energy-stable fully discrete finite element scheme for solving the simplified ferrohydrodynami...In this paper,we consider the Shliomis ferrofluid model and study its numerical approximation.We investigate a first-order energy-stable fully discrete finite element scheme for solving the simplified ferrohydrodynamics(SFHD)equations.First,we establish the well-posedness and some regularity results for the solution of the SFHD model.Next we study the Euler semi-implicit time-discrete scheme for the SFHD systems and derive the L^(2)-H^(1)error estimates for the time-discrete solution.Moreover,certain regularity results for the time-discrete solution are proved rigorously.With the help of these regularity results,we prove the unconditional L^(2)-H^(1)error estimates for the finite element solution of the SFHD model.Finally,some three-dimensional numerical examples are carried out to demonstrate both the accuracy and efficiency of the fully discrete finite element scheme.展开更多
基金Supported by the NSF of the Higher Education Institutions of Jiangsu Province(10KJD110006)Supported by the grant of Jiangsu Institute of Education(Jsjy2009zd03)Supported by the Qing Lan Project of Jiangsu Province(2010)
文摘In this paper,we present the semi-implicit Euler（SIE）numerical solution for stochastic pantograph equations with jumps and prove that the SIE approximation solution converges to the exact solution in the mean-square sense under the Local Lipschitz condition.
文摘Several efficient analytical methods have been developed to solve the solid-state diffusion problem, for constant diffusion coefficient problems. However, these methods cannot be applied for concentration-dependent diffusion coefficient problems and numerical methods are used instead. Herein, grid-based numerical methods derived from the control volume discretization are presented to resolve the characteristic nonlinear system of partial differential equations. A novel hybrid backward Euler control volume (HBECV) method is presented which requires only one iteration to reach an implicit solution. The HBECV results are shown to be stable and accurate for a moderate number of grid points. The computational speed and accuracy of the HBECV, justify its use in battery simulations, in which the solid-state diffusion coefficient is a strong function of the concentration.
基金supported by National Natural Science Foundation of China (Grant No. 11571128)
文摘In this paper, we consider strong convergence and almost sure exponential stability of the backward Euler-Maruyama method for nonlinear hybrid stochastic differential equations with time-variable delay. Under the local Lipschitz condition and polynomial growth condition, it is proved that the backward Euler-Maruyama method is strongly convergent. Additionally, the moment estimates and almost sure exponential stability for the analytical solution are proved. Also, under the appropriate condition, we show that the numerical solutions for the backward Euler-Maruyama methods are almost surely exponentially stable. A numerical experiment is given to illustrate the computational effectiveness and the theoretical results of the method.
基金supported by the Fundamental Research Funds for the Central Universities under Grant No. 2012089:31541111213China Postdoctoral Science Foundation Funded Project under Grant No.2012M511615the State Key Program of National Natural Science of China under Grant No.61134012
文摘In the literature （Tan and Wang, 2010）, Tan and Wang investigated the convergence of the split-step backward Euler （SSBE） method for linear stochastic delay integro-differential equations （SDIDEs） and proved the mean-square stability of SSBE method under some condition. Unfortu- nately, the main result of stability derived by the condition is somewhat restrictive to be applied for practical application. This paper improves the corresponding results. The authors not only prove the mean-square stability of the numerical method but also prove the general mean-square stability of the numerical method. Furthermore, an example is given to illustrate the theory.
基金supported by the National Natural Science Foundation of China(Nos.12271514,11871467,12161141017)the National Key Research and Development Program of China(2023YFC3705701).
文摘In this paper,we consider the Shliomis ferrofluid model and study its numerical approximation.We investigate a first-order energy-stable fully discrete finite element scheme for solving the simplified ferrohydrodynamics(SFHD)equations.First,we establish the well-posedness and some regularity results for the solution of the SFHD model.Next we study the Euler semi-implicit time-discrete scheme for the SFHD systems and derive the L^(2)-H^(1)error estimates for the time-discrete solution.Moreover,certain regularity results for the time-discrete solution are proved rigorously.With the help of these regularity results,we prove the unconditional L^(2)-H^(1)error estimates for the finite element solution of the SFHD model.Finally,some three-dimensional numerical examples are carried out to demonstrate both the accuracy and efficiency of the fully discrete finite element scheme.
