Solution of the two-dimensional heat equation for a square in terms of elliptic functions

In this study, an alternative method is presented for the solution of two-dimensional heat equation in a square region. In this method, the solution function of the problem is based on the Green function, and therefore on elliptic functions. The numerical results have been computed for an application example and compared with the results of the well-known method of separation of variables and the finite elements method. The comparison study shows that the present method gives better results than those of the other methods.     

Estimate for the root-location of linear systems via the lyapunov matrix equation

It is well known that the Lyapunov matrix equation can be utilized to obtain the estimate for the transient behavior of linear constant systems. This paper shows that the equation gives information not only on the maximum real part of the characteristic roots of system matrix but also on other extremal values to these roots.     

Numerical solving initial value problem for Fredholm type linear integro-differential equation system

In this paper, we introduce a numerical method for solving initial value problems for a system of linear integro-differential equations. The main idea is based on the interpolations of unknown functions at distinct interpolation points. We next use Clenshaw-Curtis quadrature formulae required in the approximation of the integral equations. The technique is very effective and simple. In the end, to show the efficiency of this method, we present some numerical examples.     

Lyapunov functionals vs Lyapunov functions for various types of stability of hybrid stochastic differential equation

In this paper, the effects of using Lyapunov like-functionals vs using Lyapunov like-functions coupled with partial differential inequalities on various types of stability criteria of hybrid stochastic parabolic differential equation of Itô type are investigated. A concluding remark will highlight the advantages of using the two approaches and the major differences between the utilization of two approaches for this specific type of differential equations with diffusion term.     

Numerical solutions of optimal control for linear Volterra integrodifferential systems via hybrid functions

By applying hybrid functions of general block-pulse functions and Legendre polynomials, linear Volterra integrodifferential systems are converted into a system of algebraic equations. The approximate solutions of linear Volterra integrodifferential systems are derived. Using the results we obtain the optimal control and state as well as the optimal value of the objective functional. The numerical examples illustrate that the algorithms are valid.     

Numerical solutions of optimal control for linear time-varying systems with delays via hybrid functions

By applying hybrid functions of general block-pulse functions and Legendre polynomials, the linear-quadratic problem of linear time-varying systems with delays are transformed into the optimization problem of multivariate functions. The approximate solutions of the optimal control and state as well as the optimal value of the objective functional are derived. The numerical examples illustrate that the algorithms are valid.     

Controller design via the synthesis equation

A comprehensive and efficient frequency domain methodology for the design of controllers for high order single-input-single-output (SISO) plants is presented. The unique advantage of the methodology is that it enables the designer to readily visualize the optimum structure for the controller in the early stage of design. This eliminates time consuming and often frustrating trial-and-error iterations in controller design which is characteristic of many of the current methods. The methodology has been motivated by the need for a streamlined procedure to directly synthesize a low order, robust controller with which the controlled step response can be made to closely match that of a prespecified reference model. The method does not require order reduction of the original model. Also, only output feedback is utilized. The method is applicable to non-minimum phase, unstable, and irrational plant transfer functions. It is ideally suited to design local controllers for the subsystems so that they replicate low order models with compatible dynamic characteristics. Examples are given to demonstrate the method.     

Solution of a Scaled System via Chebyshev Polynomials

A new time-domain approach to the derivation of a Chebyshev scale matrix is presented. The derived Chebyshev scale matrix, together with the Chebyshev integration matrix, is used to analyze differential equations containing terms with a scaled argument. The results are expressed in terms of Chebyshev series. As illustrated in the included examples, the Chebyshev series solution converges faster than that represented in Laguerre series.     

DSP-based synchronous control of dual linear motors via Sugeno type fuzzy neural network compensator

A digital signal processor (DSP)-based complementary sliding mode control (CSMC) with Sugeno type fuzzy neural network (SFNN) compensator is proposed in this study for the synchronous control of a dual linear motors servo system installed in a gantry position stage. The dual linear motors servo system comprises two parallel permanent magnet linear synchronous motors (PMLSMs). The dynamics of the single-axis motion system with a lumped uncertainty which contains parameter variations, external disturbances and nonlinear friction force is briefly introduced first. Then, a CSMC is designed to guarantee the precision position tracking requirement in single-axis control for the dual linear motors. Moreover, to enhance the robustness to uncertainties and to eliminate the synchronous error of dual linear motors, the CSMC with a SFNN compensator is proposed where the SFNN compensator is designed mainly to compensate the synchronous error. Furthermore, to increase the control performance of the proposed intelligent control approach, a 32-bit floating-point DSP, TMS320VC33, is adopted for the implementation of the proposed CSMC and SFNN. In addition, some experimental results are illustrated to show the validity of the proposed control approach.     

一类非共振四阶椭圆方程解的存在性

本文利用山路引理讨论了空间E=H2（Ω）∩H0（Ω）中一类非共振四阶椭圆方程在（C）条件下非平凡解的存在性问题。     

Solution of integral equations using a set of block pulse functions

A set of the block pulse functions is applied to solve the Fredholm's and the Volterra's integral equations of the second kind. An algebraic equation in matrix form which is equivalent to the solution of the integral equation is developed. The approximate results are easily obtained by a few computations. An accurate solution canbe evaluated in a digital computer by solving the algebraic equation. Two examples are given.     

An approximate analytic solution of the nonlinear Riccati differential equation

In this paper, a hybrid method which combines the Adomian decomposition method (ADM), the Laplace transform algorithm and the Padé approximant is introduced to solve the approximate analytic solutions of the nonlinear Riccati differential equations. This hybrid method demonstrates accurate and reliable results, and has a great improvement in the ADM truncated series solution which diverges rapidly as the applicable domain increases. Three examples herein are given to demonstrate good accuracy and fast convergence in comparison with the exact solution.     

Topological formulation of general linear network functions

This paper first discusses the corrections of the errors which Jong and Zobrist made in Ref. (1) and next describes the sign determination suitable forthese topological formulas. The technique for sign determination is based on the node partitioning algorithm which is also presented.     

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。     

Periodic solutions for a fourth-order p-Laplacian neutral functional differential equation

By means of Mawhin's continuation theorem, we study a kind of fourth-order p-Laplacian neutral functional differential equation with a deviating argument in the form:

(φ_p(x(t)−cx(t−δ))^″)^″=f(x(t))x^′(t)+g(t,x(t−τ(t,|x|_∞)))+e(t).     

Existence of three periodic solutions for a nonlinear first order functional differential equation

In this paper, we use Leggett-Williams multiple fixed point theorem to obtain different sufficient conditions for the existence of at least three nonnegative periodic solutions of the first order functional differential equation of the form

y^′(t)=-a(t)y(t)+λf(t,y(h(t))).     

Initial conditioned solutions of a second-order nonlinear conservative differential equation with a periodically varying coefficient

The exact solution of the equation

$\text{d}{}^2\text{x}\text{dt}{}^2\text{+dx+d′f(wt)x}{}^3\text{=0,}$

where d, d' and w are positive constants, and ?(wt) is a rectangular periodic function of time is discussed. The equation describes approximately the transversal movement of a particle in an alternating gradient accelerator. The exact solution is obtained in the form of a composite recurrent relation containing five particular solutions. Each of these solutions corresponds to a specific well-defined area of the phase plane of the initial conditions. The dynamical behaviour and the stability of the movement are examined analytically.     

An equation of state for a system with a single type of transformation

Based on theory of a previous paper, the writer has developed an equation of state for a system with a single type of transformation. This equation is of the form

$\text{h=A+Bv+Cp+Dpv?T(E+Fv+Gp+Hpv)}$

where h = ε + pv is the total heat, p the pressure, v the specific volume, T the temperature, and p, v, T are considered independent variables. A, B, C, etc., are constants for the system. The latent eat at constant (p, T) is given by

$\text{λp,T=(v}{}_2\text{?v}{}_1\text{)(}\text{?h}\text{?v}\text{)P,T= (v}{}_2\text{?v}{}_1\text{)[(B?TF)+p(D?TH)]}$

. These equations are checked with data on saturated and superheated ammonia, and the agreement is good to within a few tenths of a per cent. Also, checks with data on saturated and superheated steam show agreement within several per cent.     

一种基于完全线性变换法则的差分方程解析解算法

利用传统方法很难在计算机上实现差分方程的解析解求解,本文提出了一种获得差分方程解析解的线性算法,该算法的基础是完全线形变化法。其核心操作为降维处理,对高阶差分方程进行逐次降阶运算,直至获得其解析解表达式。本质上,该算法属于Z变换法的一种矩阵法变形。算法的线性特征使得其容易移植到计算机上实现差分方程的解析解运算,而非传统的数值迭代解。