应用数学和力学

1984 Vol. 5, No. 5

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Singular Perturbation of Nonlinear Boundary Value Problem

K. W. Chang, Lin Zong-chi

1984, 5(5): 603-612.

Abstract(1843) PDF(548)

Abstract:
In this paper we consider the boundary value problemwhere εy"=f(x,y,y',ε,μ)(μ0(ε,μ)y(x,ε,μ)|(x=1-μ)=φ₁(ε,μ) where ε,µ,are two positive parameters. Under ƒy'≤-k<0 and other suitable restrictions,there exists a solution and it satisfied where yo,o(x) is solution of reduced problem ƒ(x,y,y')=0(01(0,0) while y_1-j,j(x)(j=0,1…,i;i=1,2…,m)'can be obtained successively from certain linear equations.

Energy Principle of Three-Dimensional J Integral

Chen Zhi-da

1984, 5(5): 613-617.

Abstract(1782) PDF(451)

Abstract:
Various expressions of three-dimensional J integral are proposed. We derive here three-dimensional J integral by means of general potential energy principle and Green's theo-rem. Its physical meaning and application are shown in the paper,the results are true both for infinitesimal and finite deformation.

Steady Heat Conduction of a Disc with Non-Homogeneity and Variable Thickness

Yeh Kai-yuan, Liu Ping

1984, 5(5): 619-624.

Abstract(1775) PDF(506)

Abstract:
In this paper,we use the stepped reduction method suggested by Yeh Kai-yuan in 1965 to obtain the general oslution of steady heat conduction of a disc with nonhomogeneity and variable thickness. Through an illustrative example,the error of Yeh's method is analysed. The result shows that it is effective for solving ordinary differential equation with variable coefficients.

Phase Transition in Cavitation Problem and the Speed Approaching to Equilibrium

Cai Shu-tang, Wu Den-kai

1984, 5(5): 625-631.

Abstract(2000) PDF(455)

Abstract:
In the previous article^[9],we mentioned that the cavita-tion phenomenon cannot be described both by the fast expansion and compression of a single gas bubble,because the actual fact is departed far from these kinds of description but close to the equilibrium condition. In this article,we shall go on to tackle the problem,which was discussed in the previous paper^[9]. We get the speed and the relaxiation time of approaching to the equilibrium state of a gas bubble,and give out the criterion that the gas bubble may be considered as in equilibrium condition in cavitation phenomenon.

The Method of Weighted Difference for Singular Perturbation Problem

Wu Chi-guang

1984, 5(5): 633-638.

Abstract(1423) PDF(581)

Abstract:
In this paper,we introduce a new difference approximation to the first order derivative u and give a class of uniformly convergent difference schemes.

The Application of Jacobi Factor in the Stability Theory of Ordinary Differential Equations

Li Li

1984, 5(5): 639-646.

Abstract(1641) PDF(464)

Abstract:
In this paper,we obtain some unstable theorems concerning the autonomous system,nonautonomous system and limit cycle by using the properties of Jacobi factor.

New Theory for Equations of Non-Fuchsian Type Representation Theorem of Tree Series Solution(I)

Dong Ming-de

1984, 5(5): 647-663.

Abstract(1652) PDF(539)

Abstract:
In the analytic theory of differential equations the exact explicit analytic solution has not been obtained for equations of the non-Fuchsian type(Poincare's problem). The new theory proposed in this paper for the first time affords a general method of finding exact analytic expres-sion for irregular integrals.By discarding the assumption of formal solution of classical theory,our method consists in deriving a cor-respondence relation from the equation itself and providing the analytic structure of irregular integrals naturally by the residue theorem. Irregular integrals are made up of three parts: noncontracted part,represented by ordinary recursion series,all-and semi-contracted part by the so-called tree series. Tree series solutions belong to analytic function of the new kind with recursion series as the special case only.

Extension of the Mac-Millan’s Equations to Nonlinear Nonholonomic Mechanical Systems

Mei Feng-xiang

1984, 5(5): 665-672.

Abstract(1596) PDF(607)

Abstract:
In this article,the Mac-Millan's equations are extended to the most general nonholonomic mechanical systems and the generalized Mac-Millan's equations for nonlinear nonholonomic systems are obtained. And then the equivalence between the generalized Mac-Millan's equations and the generalized Chaplygin's equations is demonstrated. Finally an example is given.

Mechanical Failure in Haversian Systems on the Basis of Microstructure

Sun Jia-ju, Geng Jie, Zhang Xiao-ru

1984, 5(5): 673-682.

Abstract(1944) PDF(489)

Abstract:
The aim of this paper is to observe variations in the mechanical properties of Haversian systems including Haversian lamellae and canals,by means of the compressive test of five bone specimens subjected to different loads corresponding to 2KN,3KN,dKN,5KN and 6KN respectively. From a series of related microscopic pictures it is clearly seen that:(1) in the lower level of loading(2KN),the Haversian canals much the same as the one unsubjected to loads,and conditions of Harversian lamellae are also the same,but,the individual Harversian canal was slightly bent;(2) in the case of the higher levels of loading,it is first deformed in the weaked portion of the Haversian lamellae;(3) with increasing loads,the fracture shapes of the Haversian system are very complex,but the fractures always take place in the cement line between osteons.At the same time,deformed dimensions of the Harversian lamellae and canals were measured. The observed phenomena were qualitatively interpreted by the theory of the linear viscoelasticity.

The Inversion of Circulant Matrix

Wang Ying-jian

1984, 5(5): 683-690.

Abstract(1985) PDF(602)

Abstract:
The method of Inverting the circulant matrix and the expression of inversion of the circulant matrix are obtained in this paper.

Perturbation Solution of the Weak-Nonlinear Differential Equation with δ-Function

Liu Zheng-rong, Wei Si-rong

1984, 5(5): 691-697.

Abstract(1786) PDF(482)

Abstract:
In this paper,starting from some fundamental properties of Heaviside function and δ-function,making use of singular perturbation methods we provide a method of finding the asymptotic analytic solution of equationwhere M(u)=eƒ+λ#948;(t-a) where M is an n-order linear differential operator,ƒ(u) is a polynomial. By means of this method,we discuss some examples concretely. The results can be explained satisfactorily in physics. If we deal with linear problem by this method,the result will agree with that drawn from theorem of impulse.

General Random Fixed Point Theorem and Its Applications

Ding Xie-ping

1984, 5(5): 699-708.

Abstract(1851) PDF(661)

Abstract:
In this paper,we obtain a general random fixed point theorem which generalizes the main results of Engle^[4,7] and Bocsan^[8]. The usefulness of this theorem seems to lie in the fact that unlike many random fixed point theorems obtained by using special methods^[1,4,5-13] can be proved by using out general theorem(Theorem 1 and Corollaries 1 and 2). Finally,we indicate some possible applications of our results to nonlinear random integral and differential equations.

Perturbation Finite Element Method for Solving Geometrically Non-linear Problems of Axisymmetrical Shell

Xie Zhi-cheng, Yang Xue-zhong, Chien Zhen-dong, Liu Yan, Zhang Li-ping

1984, 5(5): 709-722.

Abstract(1888) PDF(545)

Abstract:
In analysing the geometrically nonlinear problem of an axisymmetrical thin-walled shell,the paper combines the perturbation method with the finite element method by introducing the former into the variational equation to obtain a series of linear equations of different orders and then solving the equations with the latter. It is well-known that the finite element method can be used to deal with difficult problems as in the case of structures with complicated shapes or boundary conditions,and the perturbation method can change the nonlinear problems into linear ones. Evidently the combination of the two methods will give an efficient solution to many difficult nonlinear problems and clear away some obstacles resulted from using any of the two methods solely.The paper derives all the formulas concerning an axisym-metric. shell of large deformation by means of the perturbation finite element method and gives two numerical examples,the results of which show good convergence characteristics.

Solutions of the Plane-Strain Problems in a Power Hardening and Incompressible Material

Lin Bai-song

1984, 5(5): 723-730.

Abstract(1776) PDF(470)

Abstract:
According to Iliushin's small elastic-plastic deformation theory,in this paper,we derive the basic equations of plane-strain problems in a power hardening and incompressible material.In addition,this paper presents two methods to solvethese basic equations,i.e. the displacement function_stress method and the stress function_strain method. Twoexamples have been calculated to illustrate the application of these two methods.

The Mathematical Modelling of Near Coast Shallow Water Circulation

Lü Yu-lin, Lai Guo-zhang

1984, 5(5): 731-742.

Abstract(1780) PDF(422)

Abstract:
This paper presents a finite element method to solve the shallow water circulation problem numerically.Considering the Coriolis effect,bottom friction and eddy viscosity,the continuity equation and momentum equation are integrated vertically. Using Galerkin weighted residual method,the weak variational formulation is derived for the finite element analysis. The split-time method is applied for the numerical integration instead of iteration for nonlinear terms. Moreover,an artificial smooth approach is proposed to suppress the short wavelength noise.In order to save computer storage units,a densed storage scheme is set up,where all the zero elements in large scaled and sparse matrices are excluded.

Pansymmetry and Fixed Pansystems Theorems

Gao Long-ying, Wang Shu-ji

1984, 5(5): 743-747.

Abstract(1711) PDF(418)

Abstract:
In this paper,we continue the research of fixed pansys-tems theorems presented in ref [1],and discuss the existence,the structure and the number of fixed subsets.

The General Variational Principle on the Cantilever Beam of Circular Cross Section Including Transverse Shear Deformation

LI Jia-ren, Zhang Shen-xue

1984, 5(5): 749-754.

Abstract(1413) PDF(487)

Abstract:
In this paper we derive the approximate theory on the straight cantilever beam of a same circular cross section including transverse shear deformation,using the general variational principle with two class variates,and we present the expression with two class variates containing two general displacement,of general complementary energy,corresponding to the theory.