Asymptotics of solutions of nonautonomous second-order ordinary differential equations close to linear equations

We find asymptotic representations for certain classes of solutions of nonautonomous second-order differential equations that are close, in a certain sense, to linear equations. __________ Translated from Neliniini Kolyvannya, Vol. 11, No. 2, pp. 230–241, April–June, 2008.     

The solution for the generalized riccati algebraic equations of linear equality constraint system

Based on the dynamic equation, the performance functional and the system constraint equation of time-invariant discrete LQ control problem, the generalized Riccati equations of linear equality constraint system are obtained according to the minimum principle, then a deep discussion about the above equations is given, and finally numerical example is shown in this paper. Project supported by the National Natural Science Youth Foundation of China     

The explicit solution of homogeneous linear oridinary differential equations with constant coefficients

In this paper by using tensor analysis we give the explicit expressions of the solution of the initial-value problem of homogeneous linear differential equations with constant coefficients and the n th-order homogeneous linear differential equation with constant coefficients. In fact, we give the general formula for calculating the elements of the matrix exp [At]. We also give the results when the characteristic equation has the repeated roots. The present method is simpler and better than, the other methods.     

Conditions for the unique solvability of the initial-value problem for linear second-order differential equations with argument deviations

We establish new conditions under which the initial-value problem for a system of linear second-order differential equations with argument deviations has a unique solution, which depends monotonically on additive perturbations of the problem. __________ Translated from Neliniini Kolyvannya, Vol. 9, No. 4, pp. 535–547, October–December, 2006.     

子域精细积分及偏微分方程数值解

对于偏微分方程半解析法的方程，精细时程积分虽然能求出高度准确的解，但往往面临矩阵尺度太大的困难；另一方面差分法虽然有带宽小的优点，但有稳定性及精度方面的问题．本文提出子域精细积分法，既可利用精细积分的数值优点，又有带宽小的好处．数值例题表明了子域精细积分法的效能．     

Averaging method for the solution of non-linear differential equations with periodic non-harmonic solutions

While Krylov and Bogolyubov used harmonic functions in their averaging method for the approximate solution of weakly non-linear differential equations with oscillatory solution, we apply a similar averaging technique using Jacobi elliptic functions. These functions are also periodic and are exact solutions of strongly non-linear differential equations. The method is used to solve non-linear differential equations with linear and non-linear small dissipative terms and/or with time dependent parameters. It is also shown that quite general dissipative terms can be transformed into time-dependent parameters. As a special example, the Langevin (collisional) equation of motion of electrons in a neutralizing ion background under the influence of a time and space-dependent electric field is presented. The method may also be used for non-linear control theory, dynamic and parametric stabilization of non-linear oscillations in plasma physics, etc.     

A second-order accurate fluid-in-cell(FLIC) method for the 2D shallow water equations with topography

The fluid-in-cell(FLIC)approach of Gentry et al.(1966)is extended to second-order accuracy in space and applied to solve the 2D shallow water equations with topography.The FLIC method can be interpreted in a finite volume sense,it therefore conserves both water mass and momentum.Like the original FLIC method the second-order FLIC method presented here is able to handle wetting-drying fronts without any special treatment.Moreover,the resulting method is shock capturing and well-balanced,satisfying both the C-and extended C-properties exactly.     

Functional differential equations of mixed type: The linear autonomous case

Functional differential equations of mixed type (MFDE) are introduced; in these equations of functional type, the time derivative may depend both on past and future values of the variables. Here the linear autonomous case is considered. We study the spectrum of the (unbounded) operator, and construct continuous semigroups on the stable, center, and unstable subspaces.     

2-D solution for free vibrations of parabolic shells using generalized differential quadrature method

The Generalized Differential Quadrature (GDQ) procedure is developed for the free vibration analysis of complete parabolic shells of revolution and parabolic shell panels. The First-order Shear Deformation Theory (FSDT) is used to analyze the above moderately thick structural elements. The treatment is conducted within the theory of linear elasticity, when the material behaviour is assumed to be homogeneous and isotropic. The governing equations of motion, written in terms of internal resultants, are expressed as functions of five kinematic parameters, by using the constitutive and kinematic relationships. The solution is given in terms of generalized displacement components of the points lying on the middle surface of the shell. The discretization of the system by means of the Differential Quadrature (DQ) technique leads to a standard linear eigenvalue problem, where two independent variables are involved. The results are obtained taking the meridional and circumferential co-ordinates into account, without using the Fourier modal expansion methodology. Several examples of parabolic shell elements are presented to illustrate the validity and the accuracy of GDQ method. Numerical solutions are compared with the ones obtained using commercial programs such as Abaqus, Ansys, Femap/Nastran, Straus, Pro/Mechanica. Very good agreement is observed. Furthermore, the convergence rate of natural frequencies is shown to be very fast and the stability of the numerical methodology is very good. The accuracy of the method is sensitive to the number of sampling points used, to their distribution and to the boundary conditions. Different typologies of non-uniform grid point distributions are considered. The effect of the distribution choice of sampling points on the accuracy of GDQ solution is investigated. New numerical results are presented.     

On the numerical solution of partial differential equations of the elliptic type. I

Summary The potential equation and some connected problems in which the unknown function is given on the boundary is solved by using the properties of a special class of matrices which have the same structure as the coefficient matrix of the system of linear difference equations resulting from the differential equation.     

Bounded solutions of linear differential equations in a banach space

For a linear inhomogeneous differential equation in a Banach space, we find a criterion for the existence of solutions that are bounded on the entire real axis under the assumption that the homogeneous equation admits an exponential dichotomy on the semiaxes. This result is a generalization of the Palmer lemma to the case of infinite-dimensional spaces. We consider examples of countable systems of ordinary differential equations that have bounded solutions. __________ Translated from Neliniini Kolyvannya, Vol. 9, No. 1, pp. 3–14, January–March, 2006.     

基于局部DQ-RBF不可压缩N-S方程的数值求解

以RBF作为DQ方法的基函数,将迎风机制引入DQ-RBF中,建立了二维不可压缩黏性N-S方程数值求解模型,采用Levenberg-Marquardt算法求解非线性方程组.求解时分析了形状参数对求解精度的影响,改进了边界速度的处理方法.对平板Couette流及有限宽台阶绕流流动问题进行了数值求解.比较了本文方法和FLUE...     

On a representation of the solution of the linear elasticity equations

A new representation of the stress tensor in the linear theory of elasticity is proposed. The representation satisfies the equilibrium equations and the compatibility conditions for strains. In this representation, the stress tensor is expressed in terms of a harmonic vector. The second boundary-value problem for an elastic half-space and elastic layer is considered as an example.Translated from Prikladnaya Mekhanika, Vol. 40, No. 11, pp. 85–91, November 2004.This revised version was published online in April 2005 with a corrected cover date.     

Numerical solution of the generalized Serre equations with the MacCormack finite-difference scheme

This paper describes a two-dimensional numerical model to solve the generalized Serre equations. In order to solve the system equations, written in the conservative form, we use an explicit finite-difference method based on the MacCormack time-splitting scheme. The numerical method and the computational model are validated by comparing one- and two-dimensional numerical solutions with theoretical and experimental results. Finally, the two-dimensional model (in a horizontal plane) is tested in a domain with complicated boundary conditions.     

求解线性方程组的一种新方法

将线性方程组的一般系数矩阵转化为对称正定矩阵，从而把原线性方程组的求解问题转化为一个等价变分问题的极少值点寻优问题，借助对分寻优法进行求解。算例结果表明，本文方法不仅对于良态线性方程组的求解问题是有效的，而且对于病态线性方程组的求解问题同样是有效的。