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International Journal of Modeling, Simulation, and Scientific Computing 2020年第1期11卷 42-57页

作者： Lei Zhang Chaofeng Zhang Mengya Liu School of Physics and Electronics Henan University KaifengHenan 475004P.R.China

According to the relationship between truncation error and step size of two implicit second-order-derivative multistep formulas based on Hermite interpolation polynomial,a variable-order and variable-step-size numerical method for solving differential equations is *** stability properties of the formulas are discussed and the stability regions are *** deduced methods are applied to a simulation *** results show that the numerical method can satisfy calculation accuracy,reduce the number of calculation steps and accelerate calculation speed.

关键词： Numerical method variable step size variable order hermite interpolation ordinary differential equations

ON THE BOUNDARY VALUE PROBLEMS FOR ordinary differential equations WITH TURNING POINTS

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Applied Mathematics and Mechanics(English Edition) 1991年第2期12卷 121-129页

作者：江福汝 1. Shanghai Institute of Applied Mathematics and Mechanics Shanghai University of Technology Shanghai

In this paper, we consider the boundary value problems of the form ey″ - f(x, e)y′ + g(x, e)y=0 (-a&lex&leb, 0 详细信息

In this paper, we consider the boundary value problems of the form ey″ - f(x, e)y′ + g(x, e)y=0 (-a&lex&leb, 0

关键词： ordinary differential equations singular perturbation turning point problem

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ASYMPTOTIC SOLUTIONS OF BOUNDARY VALUE PROBLEMS FOR THIRD-ORDER ordinary differential equations WITH TURNING POINTS

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Applied Mathematics and Mechanics(English Edition) 2001年第4期22卷 394-403页

作者： JIANG Fu-ru(江福汝) JIN Qi-nian(金其年) Shanghai Institute of Applied Mathematics and Mechanics Shanghai UniversityShanghai 200072 P R China Department of Mathematics Nanjing University Nanjing 210000 P R China

Boundary value problem; for third-order ordinary differential equations with turning points are studied as follows : epsilon gamma ' ' + f(x ; epsilon) gamma ' + g(x ; epsilon) gamma ' +h(x ; epsilon) gamma = 0 (- a < x < b, 0 epsilon 1), where f(x ; 0) has several multiple zero points in ( - n, b). the necessary conditions for exhibiting resonance is given, and the uniformly valid asymptotic solutions and the estimations of remainder terms are obtained.

关键词： boundary value problems ordinary differential equations turning points asymptotic solutions

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Interval of effective time-step size for the numerical computation of nonlinear ordinary differential equations

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Atmospheric and Oceanic Science Letters 2017年第1期10卷 17-20页

作者： CAO Jing LI Jian-Ping ZHANG Xin-Yuan College of Science Tianjin University of Technolog College of Global Change and Earth System Science（GCESS） Beijing Normal Universit Joint Center for Global Change Studie Tianjin Free Trade Zone Customs Hous

The computational uncertainty principle states that the numerical computation of nonlinear ordinary differential equations（ODEs） should use appropriately sized time steps to obtain reliable ***,the interval of effective step size（IES） has not been thoroughly explored *** this paper,by using a general estimation for the total error of the numerical solutions of ODEs,a method is proposed for determining an approximate IES by translating the functions for truncation and rounding *** also illustrates this process with an ***,the relationship between the IES and its approximation is found,and the relative error of the approximation with respect to the IES is *** addition,variation in the IES with increasing integration time is studied,which can provide an explanation for the observed numerical *** findings contribute to computational step-size choice for reliable numerical solutions.

关键词： ordinary differential equations interval of effective step size computational uncertainty principle integration time relative error

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A Kind of Arc-length Method for ordinary differential equations

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Communications in Nonlinear Science and Numerical Simulation 1997年第3期 15-20页

作者： Jike WU, Grafton W. H. HUI & Hongli DING (Department of Mechanics & Engineering Science,Peking University, Peking, CHINA) (Department of Mathematics, Hong Kong University of Science and Technology, CHINA)

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ＡＫｉｎｄｏｆＡｒｃ－ｌｅｎｇｔｈＭｅｔｈｏｄｆｏｒＯｒｄｉｎａｒｙＤｉｆｅｒｅｎｔｉａｌＥｑｕａｔｉｏｎｓ１ＪｉｋｅＷＵ，ＧｒａｆｔｏｎＷ．Ｈ．ＨＵＩ＆ＨｏｎｇｌｉＤＩＮＧ（ＤｅｐａｒｔｍｅｎｔｏｆＭｅｃｈａｎｉｃｓ＆ＥｎｇｉｎｅｅｒｉｎｇＳｃｉｅ...

关键词： ordinary differential equations Numerical Method Arc length method Stiff Equation Burger′s Equation

An Approach to Parallel Simulation of ordinary differential equations

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Journal of Software Engineering and Applications 2016年第5期9卷 250-290页

作者： Joshua D. Carl Gautam Biswas Department of Electrical Engineering and Computer Science Milwaukee School of Engineering Milwaukee WI USA Department of Electrical Engineering and Computer Science Vanderbilt University Nashville TN USA

Cyber-physical systems (CPS) represent a class of complex engineered systems where functionality and behavior emerge through the interaction between the computational and physical domains. Simulation provides design engineers with quick and accurate feedback on the behaviors generated by their designs. However, as systems become more complex, simulating their behaviors becomes computation all complex. But, most modern simulation environments still execute on a single thread, which does not take advantage of the processing power available on modern multi-core CPUs. This paper investigates methods to partition and simulate differential equation-based models of cyber-physical systems using multiple threads on multi-core CPUs that can share data across threads. We describe model partitioning methods using fixed step and variable step numerical in-tegration methods that consider the multi-layer cache structure of these CPUs to avoid simulation performance degradation due to cache conflicts. We study the effectiveness of each parallel simu-lation algorithm by calculating the relative speedup compared to a serial simulation applied to a series of large electric circuit models. We also develop a series of guidelines for maximizing performance when developing parallel simulation software intended for use on multi-core CPUs.

关键词： Parallel and Multi-Thread Programming ordinary differential equations Simulation

A Novel Method for Solving ordinary differential equations with Artificial Neural Networks

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Applied Mathematics 2021年第10期12卷 900-918页

作者： Roseline N. Okereke Olaniyi S. Maliki Ben I. Oruh Department of Mathematics Michael Okpara University of Agriculture Umudike Nigeria

This research work investigates the use of Artificial Neural Network (ANN) based on models for solving first and second order linear constant coefficient ordinary differential equations with initial conditions. In particular, we employ a feed-forward Multilayer Perceptron Neural Network (MLPNN), but bypass the standard back-propagation algorithm for updating the intrinsic weights. A trial solution of the differential equation is written as a sum of two parts. The first part satisfies the initial or boundary conditions and contains no adjustable parameters. The second part involves a feed-forward neural network to be trained to satisfy the differential equation. Numerous works have appeared in recent times regarding the solution of differential equations using ANN, however majority of these employed a single hidden layer perceptron model, incorporating a back-propagation algorithm for weight updation. For the homogeneous case, we assume a solution in exponential form and compute a polynomial approximation using statistical regression. From here we pick the unknown coefficients as the weights from input layer to hidden layer of the associated neural network trial solution. To get the weights from hidden layer to the output layer, we form algebraic equations incorporating the default sign of the differential equations. We then apply the Gaussian Radial Basis function (GRBF) approximation model to achieve our objective. The weights obtained in this manner need not be adjusted. We proceed to develop a Neural Network algorithm using MathCAD software, which enables us to slightly adjust the intrinsic biases. We compare the convergence and the accuracy of our results with analytic solutions, as well as well-known numerical methods and obtain satisfactory results for our example ODE problems.

关键词： ordinary differential equations Multilayer Perceptron Neural Networks Gaussian Radial Basis Function Network Training MathCAD (Computer Aided Design) 14 IBM-SPSS (Statistical Package for Social Science) 23

New Numerical Integration Formulations for ordinary differential equations

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Advances in Pure Mathematics 2024年第8期14卷 650-666页

作者： Serdar Beji Faculty of Naval Architecture and Ocean Engineering Istanbul Technical University Istanbul Trkiye

An entirely new framework is established for developing various single- and multi-step formulations for the numerical integration of ordinary differential equations. Besides polynomials, unconventional base-functions with trigonometric and exponential terms satisfying different conditions are employed to generate a number of formulations. Performances of the new schemes are tested against well-known numerical integrators for selected test cases with quite satisfactory results. Convergence and stability issues of the new formulations are not addressed as the treatment of these aspects requires a separate work. The general approach introduced herein opens a wide vista for producing virtually unlimited number of formulations.

关键词： Single- and Multi-Step Numerical Integration Unconventional Base-Functions ordinary differential equations

The Hyers-Ulam Stability and Hyers-Ulam Instability for Some Nonhomogeneous ordinary differential equations

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Chinese Quarterly Journal of Mathematics 2024年第4期39卷 420-430页

作者： DENG Jing-tao School Of Science University of Science and Technology LiaoningAnShan 114051China

In this paper,we get a necessary and sufficient condition such that a class of differential inequalities *** this necessary and sufficient condition,we prove that a class of first order nonhomogeneous ordinary differential equations have the Hyers-Ulam *** then,we prove that some first order nonhomogeneous ordinary differential equations and some second order nonhomogeneous ordinary differential equations do not have the Hyers-Ulam instability under some suitable conditions.

关键词： Hyers-Ulam stability Hyers-Ulam instability ordinary differential equations Necessary and sufficient condition

SVM for Solving ordinary and Partial differential equations with Regular Boundary

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SVM for Solving Ordinary and Partial Differential Equations ...