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 me...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.展开更多
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 ...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.展开更多
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 pol...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.展开更多
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 uniform...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.展开更多
This paper deals with the construction of Heun’s method of random initial value problems. Sufficient conditions for their mean square convergence are established. Main statistical properties of the approximations pro...This paper deals with the construction of Heun’s method of random initial value problems. Sufficient conditions for their mean square convergence are established. Main statistical properties of the approximations processes are computed in several illustrative examples.展开更多
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 numeric...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.展开更多
An iterative process of positive solution for BVP w'+h(t)f(w)=0, w(0)=w(1)= 0 is established, where h(t) is allowed to changes sign on [0,1]. The process starts from a simple function.
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 mechanic...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.展开更多
微分方程的计算求解在计算机工程上有重要的理论意义和应用价值。针对传统数值解法计算复杂度高、解的形式离散等问题,本文基于微分方程的回归方程观点与解法,应用统计回归方法求解二阶常微分方程,并给出基于中心支持向量机(proximal su...微分方程的计算求解在计算机工程上有重要的理论意义和应用价值。针对传统数值解法计算复杂度高、解的形式离散等问题,本文基于微分方程的回归方程观点与解法,应用统计回归方法求解二阶常微分方程,并给出基于中心支持向量机(proximal support vector machine,P-SVM)在常微分方程的初值和边值问题上的近似解求法。通过在目标优化函数中添加偏置项,构建P-SVM回归模型,从而避免大规模求解线性方程组,得到结构简洁的最优解表达式。模型通过最小化训练样本点的均方误差和,在保证精度的同时,有效提高了近似解的计算速度。此外,形式简洁固定的解析解表达式也便于在实际应用中进行定性分析和性质研究。数值试验结果验证了P-SVM方法是一种高效可行的常微分方程求解方法。展开更多
基金heprojectissupportedbyNNSFofChina (No .1 9972 0 39) .
文摘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.
文摘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.
文摘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.
文摘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.
文摘This paper deals with the construction of Heun’s method of random initial value problems. Sufficient conditions for their mean square convergence are established. Main statistical properties of the approximations processes are computed in several illustrative examples.
基金supported by the National Natural Science Foundation of China Under Grant No.61773008.
文摘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.
文摘An iterative process of positive solution for BVP w'+h(t)f(w)=0, w(0)=w(1)= 0 is established, where h(t) is allowed to changes sign on [0,1]. The process starts from a simple function.
文摘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.
文摘微分方程的计算求解在计算机工程上有重要的理论意义和应用价值。针对传统数值解法计算复杂度高、解的形式离散等问题,本文基于微分方程的回归方程观点与解法,应用统计回归方法求解二阶常微分方程,并给出基于中心支持向量机(proximal support vector machine,P-SVM)在常微分方程的初值和边值问题上的近似解求法。通过在目标优化函数中添加偏置项,构建P-SVM回归模型,从而避免大规模求解线性方程组,得到结构简洁的最优解表达式。模型通过最小化训练样本点的均方误差和,在保证精度的同时,有效提高了近似解的计算速度。此外,形式简洁固定的解析解表达式也便于在实际应用中进行定性分析和性质研究。数值试验结果验证了P-SVM方法是一种高效可行的常微分方程求解方法。
