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Numerical Discretization-Based Kernel Type Estimation Methods for ordinary differential equation Models

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Acta Mathematica Sinica,English Series 2015年第8期31卷 1233-1254页

作者： Tao HU Yan Ping QIU Heng Jian CUI Li Hong CHEN School of Mathematical Sciences & BCMIIS Capital Normal University School of Mathematical Sciences Beijing Normal University Fujian College of Water Conservancy and Eletric Power

We consider the problem of parameter estimation in both linear and nonlinear ordinary differential equation（ODE） models. Nonlinear ODE models are widely used in applications. But their analytic solutions are usually not available. Thus regular methods usually depend on repetitive use of numerical solutions which bring huge computational cost. We proposed a new two-stage approach which includes a smoothing method（kernel smoothing or local polynomial fitting） in the first stage, and a numerical discretization method（Eulers discretization method, the trapezoidal discretization method,or the Runge–Kutta discretization method） in the second stage. Through numerical simulations, we find the proposed method gains a proper balance between estimation accuracy and computational *** properties are also presented, which show the consistency and asymptotic normality of estimators under some mild conditions. The proposed method is compared to existing methods in term of accuracy and computational cost. The simulation results show that the estimators with local linear smoothing in the first stage and trapezoidal discretization in the second stage have the lowest average relative errors. We apply the proposed method to HIV dynamics data to illustrate the practicability of the estimator.

关键词： Nonparametric regression kernel smoothing local polynomial fitting parametric identification ordinary differential equation nume

On Bifurcations of an ordinary differential equation

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Advances in Manufacturing 2000年第S1期4卷 4-6页

作者： LI Chang-pin College of Sciences, Shanghai University, Shanghai 200436, China College of Sciences Shanghai University Shanghai 200436 China

Biforations of an ordinary differential equation with two-point boundary value condition are investigated. Using the singularity theory based on the Liapunov-Schmidt reduction, we have obtained some characterization r... 详细信息

关键词： bifurcations ordinary differential equation singularity theory Liapunov-Schlnidt reduction

New Derivation of ordinary differential equations for Transient Free-Surface Green Functions

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China Ocean Engineering 2001年第4期16卷 499-508页

作者：段文洋戴遗山 College of Shipbuilding Engineering Harbin Engineering University Harbin 150001 China

Based on the Laplace transform, a direct derivation of the ordinary differential equations for the three-dimensional transient free-surface Green function in marine hydrodynamics is presented. The results for the 3D Green function and all its spatial derivatives are a set of fourth-order ordinary differential equations, which are identical with that of Clement (1998). All of these results may be used to accelerate numerical computation for the time-domain boundary element method in marine hydrodynamics.

关键词： marine hydrodynamics time-domain Green function ordinary differential equation Laplace transform

Rotational Hypersurfaces with Constant Gauss-Kronecker Curvature

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Chinese Annals of Mathematics,Series B 2022年第3期43卷 343-358页

作者： Yuhang LIU Yunchu DAI Beijing International Center for Mathematical Research Peking UniversityBeijing 100871China Department of Mathematics Stanford UniversityCalifornia 94305USA

The authors study rotational hypersurfaces with constant Gauss-Kronecker curvature in R^(n).They solve the ODE associated with the generating curve of such hypersurface using integral expressions and obtain several geometric properties of such *** particular,they discover a class of non-compact rotational hypersurfaces with constant and negative Gauss-Kronecker curvature and finite volume,which can be seen as the higher-dimensional generalization of the pseudo-sphere.

关键词： differential geometry Gauss-Kronecker curvature ordinary differential equation

Mathematical modeling of malaria transmission dynamics in humans with mobility and control states

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Infectious Disease Modelling 2023年第4期8卷 1015-1031页

作者： Gbenga Adegbite Sunday Edeki Itunuoluwa Isewon Jerry Emmanuel Titilope Dokunmu Solomon Rotimi Jelili Oyelade Ezekiel Adebiyi Covenant University Bioinformatics Research Covenant UniversityOtaNigeria Department of Computer and Information Sciences Covenant UniversityOtaNigeria Department of Mathematics Covenant UniversityOtaNigeria Department of Biochemistry Covenant UniversityOtaNigeria Covenant Applied Informatics and Communications-African Centre of Excellence Covenant UniversityOtaOgun StateNigeria Division of Applied Bioinformatics German Cancer Research Center(DKFZ)HeidelbergGermany

Malaria importation is one of the hypothetical drivers of malaria transmission dynamics across the *** studies on malaria importation focused on the effect of the use of conventional malaria control strategies as approved by the World Health Organization(WHO)on malaria transmission dynamics but did not capture the effect of the use of traditional malaria control strategies by vigilant *** order to handle the aforementioned situation,a novel system of ordinary differential equations(ODEs)was developed comprising the human and the malaria vector *** of the system was carried out to assess its quantitative *** novel computational algorithm used to solve the developed system of ODEs was implemented and benchmarked with the existing Runge-Kutta numerical solution ***,simulations of different vigilant conditions useful to control malaria were carried *** novel system of malaria models was well-posed and epidemiologically meaningful based on its quantitative *** novel algorithm performed relatively better in terms of model simulation accuracy than *** the best model-fit condition of 98%vigilance to the use of conventional and traditional malaria control strategies,this study revealed that malaria importation has a persistent impact on malaria transmission *** lieu of this,this study opined that total vigilance to the use of the WHO-approved and traditional malaria management tools would be the most effective control strategy against malaria importation.

关键词： Malaria importation Traditional malaria control ordinary differential equation Quantitative properties Novel algorithm Runge-Kutta

EXISTENCE OF POSITIVE SOLUTION TO TWO-POINT BOUNDARY VALUE PROBLEM FOR A SYSTEM OF SECOND ORDER ordinary differential equationS

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Annals of differential equations 2009年第1期25卷 74-79页

作者： Xing Meihong, Zhang Kemei (Dept. of Math., Qufu Normal University, Qufu 273165, Shandong) Dept. of Math. Qufu Normal University Qufu 273165 Shandong

In this paper, we consider a two-point boundary value problem for a system of second order ordinary differential equations. Under some conditions, we show the existence of positive solution to the system of second ord... 详细信息

关键词： mixed monotone operator infinite boundary value problems positive solution ordinary differential equation

An Adaptive Time-Step Backward Differentiation Algorithm to Solve Stiff ordinary differential equations: Application to Solve Activated Sludge Models

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American Journal of Computational Mathematics 2016年第4期6卷 298-312页

作者： Jamal Alikhani Bahareh Shoghli Ujjal Kumar Bhowmik Arash Massoudieh Department of Electrical Engineering and Computer Science The Catholic University of America Washington DC USA Department of Civil Engineering The Catholic University of America Washington DC USA Department of Civil Engineering University of North Dakota Grand Forks ND USA

A backward differentiation formula (BDF) has been shown to be an effective way to solve a system of ordinary differential equations (ODEs) that have some degree of stiffness. However, sometimes, due to high-frequency variations in the external time series of boundary conditions, a small time-step is required to solve the ODE system throughout the entire simulation period, which can lead to a high computational cost, slower response, and need for more memory resources. One possible strategy to overcome this problem is to dynamically adjust the time-step with respect to the system’s stiffness. Therefore, small time-steps can be applied when needed, and larger time-steps can be used when allowable. This paper presents a new algorithm for adjusting the dynamic time-step based on a BDF discretization method. The parameters used to dynamically adjust the size of the time-step can be optimally specified to result in a minimum computation time and reasonable accuracy for a particular case of ODEs. The proposed algorithm was applied to solve the system of ODEs obtained from an activated sludge model (ASM) for biological wastewater treatment processes. The algorithm was tested for various solver parameters, and the optimum set of three adjustable parameters that represented minimum computation time was identified. In addition, the accuracy of the algorithm was evaluated for various sets of solver parameters.

关键词： Adaptive Time-Step Backward Differentiation Formula Activated Sludge Model ordinary differential equation Stiffness Computation Time

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常微分方程初值问题连续有限元的超收敛性

高等学校计算数学学报 2004年第1期26卷 91-96页

作者：杨禄源汤琼株洲工学院信息与计算科学系株洲工学院信息与计算科学系株洲 412008

1 引言及算法考虑一阶非线性常微分方程初值问题u′=f（t,u）,t∈I=[0,T],u（0）=u;,（1）其中f（t,u）是t,u的适当光滑函数。我们知道,常微分方程初值问题的数值解法不仅本身有独立的兴趣,它也是抛物与双曲方程时间离散的基础。目前... 详细信息

1 引言及算法考虑一阶非线性常微分方程初值问题u′=f（t,u）,t∈I=[0,T],u（0）=u;,（1）其中f（t,u）是t,u的适当光滑函数。我们知道,常微分方程初值问题的数值解法不仅本身有独立的兴趣,它也是抛物与双曲方程时间离散的基础。目前已有许多数值方法,如

关键词： ordinary differential equation Initial value problem continuous finite element superconvergence conservation law.

Distributed accelerated optimization algorithms:Insights from an ODE

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Science China(Technological Sciences) 2020年第9期63卷 1647-1655页

作者： CHEN RuiJuan YANG Tao CHAI Tian You School of Artificial Intelligence and Automation Hluazhong University of Science and TechnologyWuhan 430074China State Key Laboratory of Synthetical Automation for Process Industries Northeastern UniversityShenyang 110819China

In this paper, we consider the distributed optimization problem, where the goal is to minimize the global objective function formed by a sum of agents' local smooth and strongly convex objective functions, over undirected connected graphs. Several distributed accelerated algorithms have been proposed for solving such a problem in the existing literature. In this paper, we provide insights for understanding these existing distributed algorithms from an ordinary differential equation(ODE) point of view. More specifically, we first derive an equivalent second-order ODE, which is the exact limit of these existing algorithms by taking the small step-size. Moreover, focusing on the quadratic objective functions, we show that the solution of the resulting ODE exponentially converges to the unique global optimal solution. The theoretical results are validated and illustrated by numerical simulations.

关键词： distributed accelerated optimization algorithms exponential convergence ordinary differential equation