Existence of Periodic Solutions for Odd Order Ordinary Differential Equations via the Homotopy Method

This paper deals with the problems of finding periodic solutions for the third order ordinary differential equations of the form (1) where T is a fixed positive number and f satisfies some additional conditions which will be stated later.The periodicity problem has been one of main topics in the qualitative theory of ordinary

Hojman's theorem of the third-order ordinary differential equation

This paper extends Hojman＇s conservation law to the third-order differential equation. A new conserved quantity is constructed based on the Lie group of transformation generators of the equations of motion. The generators contain variations of the time and generalized coordinates. Two independent non-trivial conserved quantities of the third-order ordinary differential equation are obtained. A simple example is presented to illustrate the applications of the results.

HIGH ACCURACY FINITE VOLUME ELEMENT METHOD FOR TWO-POINT BOUNDARY VALUE PROBLEM OF SECOND ORDER ORDINARY DIFFERENTIAL EQUATIONS

In this paper, a high accuracy finite volume element method is presented for two-point boundary value problem of second order ordinary differential equation, which differs from the high order generalized difference methods. It is proved that the method has optimal order error estimate O(h3) in H1 norm. Finally, two examples show that the method is effective.

HERMITE MATRIX POLYNOMIALS AND SECOND ORDER MATRIX DIFFERENTIAL EQUATIONS

In this paper we introduce the class of Hermite's matrix polynomials which appear as finite series solutions of second order matrix differential equations Y'-xAY'+BY=0.An explicit expression for the Hermite matrix polynomials,the orthogonality property and a Rodrigues' formula are given.

Solving the Nonlinear Variable Order Fractional Differential Equations by Using Euler Wavelets

An Euler wavelets method is proposed to solve a class of nonlinear variable order fractional differential equations in this paper.The properties of Euler wavelets and their operational matrix together with a family of piecewise functions are first presented.Then they are utilized to reduce the problem to the solution of a nonlinear system of algebraic equations.And the convergence of the Euler wavelets basis is given.The method is computationally attractive and some numerical examples are provided to illustrate its high accuracy.

SINGULAR PERTURBATION OF BOUNDARY VALUE PROBLEMS FOR SECOND ORDER NONLINEAR ORDINARY DIFFERENTIAL EQUATIONS ON INFINITE INTERVAL(Ⅰ)

In this paper existence, uniqueness and asymptotic estimations of solutions of the boundary value problems on infinite interval for the second order nonlinear equation depending singularly on a small parameterare examined, where are constants, and i=0,1.

A UNIFORMLY CONVERGENT SECOND ORDER DIFFERENCE SCHEME FOR A SINGULARLY PERTURBED SELF-ADJOINT ORDINARY DIFFERENTIAL EQUATION IN CONSERVATION FORM

In this paper, based on the idea of El-Mistikawy and Werle[1] we construct a difference scheme for a singularly perturbed self-adjoint ordinary differential equation in conservation form. We prove that it is a uniformly convergent second order scheme.

Criteria for System of Three Second-Order Ordinary Differential Equations to Be Reduced to a Linear System via Restricted Class of Point Transformation

This paper is devoted to the study of the linearization problem of system of three second-order ordinary differential equations and . The necessary conditions for linearization by general point transformation and are found. The sufficient conditions for linearization by restricted class of point transformation and are obtained. Moreover, the procedure for obtaining the linearizing transformation is provided in explicit forms. Examples demonstrating the procedure of using the linearization theorems are presented.

Solution of Second-Order Ordinary Differential Equations via Simulated Annealing

In this paper, we approach the problem of obtaining approximate solution of second-order initial value problems by converting it to an optimization problem. It is assumed that the solution can be approximated by a polynomial. The coefficients of the polynomial are then optimized using simulated annealing technique. Numerical examples with good results show the accuracy of the proposed approach compared with some existing methods.

Noether's and Poisson's methods for solving differential equation x_s^((m))=F_s(t,x_k^((m-2)) ,x_k^((m-1)))

This paper studies integration of a higher-order differential equation which can be reduced to a second-order ordinary differential equation. The solution of the second-order equation can be obtained by the Noether method and the Poisson method. Then the solution of the higher-order equation can be obtained by integrating the solution of the second-order equation.

Numerical Treatment of Initial Value Problems of Nonlinear Ordinary Differential Equations by Duan-Rach-Wazwaz Modified Adomian Decomposition Method

We employ the Duan-Rach-Wazwaz modified Adomian decomposition method for solving initial value problems for the systems of nonlinear ordinary differential equations numerically. In order to confirm practicality, robustness and reliability of the method, we compare the results from the modified Adomian decomposition method with those from the MATHEMATICA solutions and also from the fourth-order Runge Kutta method solutions in some cases. Furthermore, we apply Padé approximants technique to improve the solutions of the modified decomposition method whenever the exact solutions exist.

Application of Hyperbola Function Method to the Family of Third Order Korteweg-de Vries Equations

In this work, we apply a hyperbola function method to solve the nonlinear family of third order Korteweg-de Vries equations. Exact travelling wave solutions are obtained and expressed in terms of hyperbolic functions and trigonometric functions. The method used is a promising method to solve other nonlinear evaluation equations.

Linearization of Emden Differential Equation via the Generalized Sundman Transformations

The Emden differential equation is one of the most widely studied and challenging nonlinear dynamics equations in literature. It finds applications in various areas of study such as celestial mechanics, fluid mechanics, Steller structure, isothermal gas spheres, thermionic currents and so on. Because of the importance of the equation, the method of generalized Sundman transformation (GST) as proposed by Nakpim and Meleshko is used for linearizing the Emden differential equation. The Emden differential equation considered here is a modification of the equation given by Berkovic. The results obtained in this paper imply that the Emden equation cannot be linearized by a point transformation. The general solution of the modified Emden equation is also obtained.

Parallel Evolutionary Modeling for Nonlinear Ordinary Differential Equations

We introduce a new parallel evolutionary algorithm in modeling dynamic systems by nonlinear higher-order ordinary differential equations (NHODEs). The NHODEs models are much more universal than the traditional linear models. In order to accelerate the modeling process, we propose and realize a parallel evolutionary algorithm using distributed CORBA object on the heterogeneous networking. Some numerical experiments show that the new algorithm is feasible and efficient.

On the Existence of Positive Solution for a Nonlinear Third-order Three-point Boundary Value Problem

An existence theorem of positive solution is established for a nonlinear third-order three-point boundary value problem. Here, we concentrated on the case that the nonlinear term is neither superlinear nor sublinear, and is not asymptotic at zero and infinity.

An Explicit Single-Step Nonlinear Numerical Method for First Order Initial Value Problems (IVPs)

Interest in the construction of efficient methods for solving initial value problems that have some peculiar properties with it or its solution is recently gaining wide popularity. Based on the assumption that the solution is representable by nonlinear trigonometric expressions, this work presents an explicit single-step nonlinear method for solving first order initial value problems whose solution possesses singularity. The stability and convergence properties of the constructed scheme are also presented. Implementation of the new method on some standard test problems compared with those discussed in the literature proved its accuracy and efficiency.

精细积分方法的发展与扩展应用

钟万勰院士于1991年首先提出计算矩阵指数的精细积分方法,其要点是2N类算法和增量存储。精细积分方法可给出矩阵指数在计算机意义上的精确解,为常微分方程的数值计算提供了高精度、高稳定性的算法,现已成功应用于结构动力响应、随机振动、热传导以及最优控制等众多领域。本文首先介绍矩阵指数精细积分方法的提出、基本思想和发展;然后依次介绍在时不变/时变线性微分方程、非线性微分方程以及大规模问题求解中发展起来的各种精细积分方法,分析了其优缺点和适用范围;最后介绍了精细积分方法的基本思想在两点边值问题、椭圆函数和病态代数方程等问题的扩展应用,进一步展示了该思想的特色。

Variable-step-size second-order-derivative multistep method for solving first-order ordinary differential equations in system simulation

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 designed.The stability properties of the formulas are discussed and the stability regions are analyzed.The deduced methods are applied to a simulation problem.The results show that the numerical method can satisfy calculation accuracy,reduce the number of calculation steps and accelerate calculation speed.

分数阶共振向量边值问题解的存在性

研究了一类带有系数矩阵的分数阶共振边值问题.首先,定义2个合适的Banach空间;然后,在Banach空间中构造恰当的算子并使用仿伪逆矩阵的性质及Mawhin的重合度理论,得到了其解的存在性;最后,给出一个例子来验证主要结果.

一类三阶两点边值问题解的存在性

考察了三阶非线性常微分方程边值问题{u'''(t)=f(t,u(t),u'(t),u''(t)),a.e.0<t<1,u(0)=u'(0)=u'(1)=0,其中f:[0,1]×R3→R满足Carathéodory条件。在非线性项f满足适当增长性条件下,三阶非线性常微分方程边值问题至少存在1个解。基于Leray-Schauder不动点定理证明了主要结果。