一类高振荡常微分方程数值解法的误差分析

以特殊的线性振荡方程y″ g(t)y=0(其中limt→∞g(t)= ∞)为例讨论了高振荡微分方程数值解的问题.分析了梯形格式的整体截断误差,并对梯形格式做了修改,讨论了修改后格式的局部截断误差对整体截断误差的影响,最后给出了数值结果.     

A new Legendre wavelet operational matrix of derivative and its applications in solving the singular ordinary differential equations

In the present paper, a new Legendre wavelet operational matrix of derivative is presented. Shifted Legendre polynomials and their properties are employed for deriving a general procedure for forming this matrix. The application of the proposed operational matrix for solving initial and boundary value problems is explained. Then the scheme is tested for linear and nonlinear singular examples. The obtained results demonstrate efficiency and capability of the proposed method.     

常微分方程中几个例子的说明

在本文中,我们指出常微分方程教材[3]中一个关于变量分离方程的表述欠妥的问题,并针对一些课程内容的教学思路进行了改进。     

Robust periodic stability implies uniform exponential stability of Markovian jump linear systems and random linear ordinary differential equations

In this paper, we mainly show the following two statements.

(1)

A discrete-time topological Markovian jump linear system is uniformly exponentially stable if and only if it is robustly periodically stable, by using a Gel?fand–Berger–Wang formula proved here.     

Integraph solution of differential equations

In a previous article¹ a continuously recording integraph was described, by means of which differential equations, involving only one integration, could be solved. The present article describes a revision of this machine such that an equation involving two successive integrations, corresponding to practically any second-order total differential equation, with all terminal conditions included, can be solved. The need for a workable means of solving the differential equations involving empirical and discontinuous coefficients which occur repeatedly in electrical engineering and physics is recalled. In the machine described such solutions are effected by means of suitable interlinked integrating devices, the result being plotted continuously as a function of the independent variable. Tests and simple solutions show the over-all error to be approximately 1 or 2 per cent. The various sources of this error are discussed.     

Mean escape time of switched Riccati differential equations

Riccati differential equations are a class of first-order quadratic ordinary differential equations and have various applications in systems and control theory. In this study, we analyzed a switched Riccati differential equation driven by a Poisson-like stochastic signal. We specifically focused on computing the mean escape time of the switched Riccati differential equation. The contribution of this study is twofold. We first show that, under the assumption that the subsystems described as deterministic Riccati differential equations escape in finite time regardless of their initial state, the mean escape time of the switched Riccati differential equation admits a power series expression. To further expand the applicability of this result, we then present an approximate formula to compute the escape time of deterministic Riccati differential equations. Numerical simulations were performed to illustrate the obtained results.     

Method of solving differential equations with constant coefficients

The equation developed here is fundamental for an understanding of the behavior of inhomogeneous insulation, especially when the insulating material is of considerable extent in the form of continuous flat sheets, or cables, or discontinuous form as in pole-line construction. The “unit” may then be measured in terms of area in the case of sheets, and in terms of length in the case of cables. It is of basic importance in insulation research, testing, and operation. It furnishes the logical basis for the establishing of safety factors.     

Stability and boundedness properties of certain second-order differential equations

In this paper, we investigated the differential equation

     

Steady-state invariant control systems under disturbances satisfying differential equations

This paper deals with the problem of regulating the output of a linear multivariable system affected by disturbances satisfying a given linear differential equation. Conditions are given for the existence of a linear controller such that the overall control system is asymptotically stable and the regulated output goes asymptotically to zero for any disturbance of the given class. Only the regulated outputs are assumed to be measured. Moreover, procedures are presented for the synthesis of the controller.     

Solutions of some partial differential equations (with tables)

A simple method has been demonstrated for obtaining some solutions of the principal equations of mathematical physics. Tables of solutions have been included in the Appendix so that problems in field theory can be solved with a minimum of labor.     

Representation and realization of operational differential equations with time-varying coefficients

An algebraic treatment of operational differential equations with time-varying coefficients is presented in terms of skew rings of differential polynomials defined over a Noetherian ring. Included in this framework are delay differential equations with time- varying coefficients. The operator equations are characterized by transfer matrices which are utilized to construct realizations given by first-order vector differential equations with operator coefficients. It is shown that the realization of matrix equations can be reduced to the realization of scalar equations. Finally, a simple procedure is derived for realizing scalar equations.     

On the asymptotic behavior of solutions of certain second-order differential equations

In this paper, the second order non-linear differential equation

     

Optimal control of stochastic functional neutral differential equations with time lag in control

In this work, we consider an optimal control problem of a class of stochastic differential equations driven by additive noise with aftereffect appearing in control. We develop a semigroup theory of the driving deterministic neutral system and identify explicitly the adjoint operator of the corresponding infinitesimal generator. We formulate the time delay equation under consideration into an infinite dimensional stochastic control system without time lag by means of the adjoint theory established. Consequently, we can deal with the associated optimal control problem through the study of a Hamilton–Jacob–Bellman (HJB) equation. Last, we present an example whose optimal control can be explicitly determined to illustrate our theory.     

Noise suppresses exponential growth for neutral stochastic differential delay equations

Our aim in this paper is to investigate the polynomial growth for a class of neutral stochastic differential delay systems. We reveal that environmental noise will suppress the exponential growth if it is sufficiently strong. That is, the given system (neutral ordinary differential delay equation) whose solution grows exponentially and becomes a new system (neutral stochastic differential delay equation) whose solution will grow polynomially with probability one. We also provide two examples to illustrate the main result.