应用数学和力学

1990 Vol. 11, No. 11

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Yeh Kai-yuan, Ji Zhen-yi

1990, 11(11): 937-946.

Abstract(2186) PDF(501)

Abstract:
In this paper, a new method, exact element method for constructing finite element, is presented.It can be applied to solve nonpositive definite or positive definite partial differential equation with arbitrary variable coefficient under arbitrary boundary condition.Its convergence is proved and its united formula for solving partial differential equation is given.By the present method, a noncompatible element can be obtained and the compatibility conditions between elements can be treated very easily.Comparing the exact element method with the general finite element method with the same degrees of freedom, the high convergence rate of the high order derivatives of solution can be obtained.Three numerical examples are given at the end of this paper, which indicate all results can converge to exact solution and have higher numerical precision.

Fixed Point Index of Uniform Limit of Set-Valued Strict Set-Contractive Mappings and Its Applications

Ding Xie-ping, Lan Kun-quan

1990, 11(11): 947-959.

Abstract(2107) PDF(525)

Abstract:
In this paper, the class of uniform limit mappings of set-valued, strick set-contractive mappings is discussed.Furthermore, the fixed point index theory for the uniform limit mappings is established.Using the fixed point index theory, some positive fixed point theorems are proved.Our theorems generalize some results in [1,4,5,7].

Some Further Generalizations of Ky Fan’s Minimax Inequality and Its Applications to Variational Inequalities

Zhang Shi-sheng, Yang Gan-shan

1990, 11(11): 961-968.

Abstract(2658) PDF(570)

Abstract:
The purpose of this paper is to introduce the concept of generalized KKM mapping and to obtain some general version of the famous KKM theorem and Ky Fan's minimax inequality.As applications, we utilize the results presented in this paper to study the saddle point problem and the existence problem of solutions for a class of quasi-variational inequalities.The results obtained in this paper extend and improve some recent results of [1-6].

Singular Perturbation of Boundary Value Problem of Systems for Quasilinear Ordinary Differential Equations

Lin Zong-chi, Lin Su-rong

1990, 11(11): 969-976.

Abstract(2150) PDF(675)

Abstract:
In this paper, we study the singular perturbation of boundary value problem of systems for quasilinear ordinary differential equations: x′=f(t,x,y,ε),εy″=g(t,x,y,ε)y′+h(t,x,y,ε), x(0,ε)=A(ε),y(0,ε)=B(ε),y(1,ε)=C(ε) where x,f,y,h,A,B and C belong to Rⁿ and a is a diagonal matrix.Under the appropriate assumptions, using the technique of diagonalization and the theory of differential inequalities we obtain the existence of solution and its componentwise uniformly valid asymptotic estimation.

The Method of the Reciprocal Theorem of Force Vibration for the Elastic Thin Rectangular Plates (Ⅱ)——Rectangular Plates with Two Adjacent Clamped Edges

Fu Bao-lian, Li Nong

1990, 11(11): 977-988.

Abstract(2450) PDF(694)

Abstract:
In this paper, applying the method of reciprocal theorem, we give the distributions of the amplitude of bending moments along clamped edges and the amplitude of deflections along free edges of rectangular plates with two adjacent clamped edges under harmonic distributed and concentrated loads.

Numerical Solution of Nonlinear Ordinary Differential Equation for a Turning Point Problem

Lin Peng-cheng, Bai Qing-yuan

1990, 11(11): 989-998.

Abstract(2379) PDF(568)

Abstract:
By using the method in [3], several useful estimations of the derivatives of the solution of the boundary value problem for a nonlinear ordinary differential equation with a turning point are obtained.With the help of the technique in [4], the uniform convergence on the small parameter ε for a difference scheme is proved.At the end of this paper, a numerical example is given.The numerical result coincides with theoretical analysis.

Boundary and Interior Layer Behavior for Singularly Perturbed Vector Problem

Zhang Xiang

1990, 11(11): 999-1005.

Abstract(2175) PDF(655)

Abstract:
In this paper, we consider the vector nonlinear boundary value problem:εy″=f(x,y,z,y',ε), y(0)=A₁ y(1)=B₁ εz″=f(x,y,z,z',ε), z(0)=A₂ z(1)=B₂ where ε>0 is a small parameter,0≤x≤1 f and g are continuous functions in R⁴. Under appropriate assumptions, by means of the differential inequalities, we demonstrate the existence and estimation, involving boundary and interior layers, of the solutions to the above problem.

Fractal Geometry Derived from Geometric Inversion

Zhang Yong-ping, Xie He-ping

1990, 11(11): 1007-1012.

Abstract(2179) PDF(907)

Abstract:
In this paper, the conception of fractal geometry derived from geometric inversion is introduced.A ramified self-inverse fractal with symmetry and a self-inverse fractal dust set are constructed.The authors extend the conception of the fractal osculation and propose a new notion——fractal envelope.Finally, two examples of self-inverse fractal (soap and egg) are given.

High-Order Boundary Conditions for the Problems of Laplace Equation in Infinite Region and Their Application

Huang He-ning, Wang Fa-jun

1990, 11(11): 1013-1018.

Abstract(2482) PDF(636)

Abstract:
The high-order boundary conditions for the problems cf Laplace equation in infinite region have been developed.The improvement in accuracy for numerical solution is achieved by imposing the high-order boundary conditions on the exterior boundarv of a reduced finite region in which the numerical method is used.So both the computing efforts and the required storage in computer are reduced.The numerical examples show that the 1st-order boundary condition approaches to the exact boundary condition and it is clearly superior to the traditional boundary condition and the 2nd-order boundary condition.

The Inefficiency of the Least Squares Estimator and Its Bound

Yang Hu

1990, 11(11): 1019-1025.

Abstract(3078) PDF(687)

Abstract:
It was suggested by Pantanen^[1] that the mean squared error may be used to measure the inefficiency of the least squares estimator.Styan^[2] and Rao^[3] et al.discussed this inefficiency and it's bound later.In this paper we propose a new inefficiency of the least squares estimator with the measure of generalized variance and obtain its bound.