Method of high-order lagrange multiplier and generalized variational principles of elasticity with more general forms of functionals

It is known[1]that the minimum principles of potential energy andcomplementary energy are the conditional variation principles underrespective conditions of constraints.By means of the method of La-grange multipliers,we are able to reduce the functionals of condi-tional variation principles to new functionals of non-conditionalvariation principles.This method can be described as follows:Mul-tiply undetermined Lagrange multipliers by various constraints,andadd these products to the original functionals.Considering these un-determined Lagrange multipliers and the original variables in thesenew functionals as independent variables of variation,we can see thatthe stationary conditions of these functionals give these undeter-mined Lagrange multipliers in terms of original variables.The sub-stitutions of these results for Lagrange multipliers into the abovefunctionals lead to the functionals of these non-conditional varia-tion principles.However,in certain cases,some of the undetermined Lagrangemultipliers ma     

REPRESENTATION THEOREM AND ONE-ITERATION THEOREM FOR FREDHOLM INTEGRAL EQUATION OF THE FIRST KIND Ax=y

In this paper,two theorems are presented.The representation theorem states:if the Fredholm integral equation of the first kind Ax=y,with bounded L_2 kernel,has a uniquesolution(?),then(?),whereThe one-iteration theorem states:(?)can be achieved in one iteration by(?)=x_0 g_0 A*(y-Ax_0)iff one of the following conditions is satisfied.     

THE HLDER CONTINUITY FOR THE GRADIENT OF SOLUTIONS TO ONE-SIDED OBSTACLE PROBLEMS

The Hölder continuity is proved for the gradient of the solution Jo the one-sided obstacle problem of the following variational inequality in the case 1相似文献   

Singularly perturbed nonlinear second order elliptic equation

In this paper, we study singular perturbation problems of some semi-linear second order elliptic equations with nonlinear boundary value conditions: where ε is a small positive parameter and u/ l is a directional derivative, which lies on an oblique vector (x,ε). We have given a construction of the asymptotic solutions and proof of their asymptotic correctness, which is based on the principle of contraction mapping.     

Multiplicity results for a fourth-order boundary value problem

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Singular perturbation of nonlinear vector boundary value problem

In this paper we study the perturbed boundary value problem of the form dx/dt=f(x,y,t;ε), εdy/dt=g(x,y,t;ε), a_1(ε)x(0,ε)+a_2(ε)y(0,ε)=a(ε) b_1(ε)x(1,ε)+εb_2(ε)y(1,ε)=β(ε)in whichx,f,β∈E~m, y,g,a∈E~n, 0<ε(?)1and a_1(ε), a_2(ε), b_2(ε)and b_2(ε) are matrices of the appropriate size. Under the condition that g_y(t) is nonsingular and other suitable restrictions, the existence of the solution is proved, the asymptotic expansion of solution of order n is constructed, and the remainder term is estimated.     

Necessary and sufficient conditions for the absolute stability of discrete type lurie control system

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Doubly Curved Shallow Shells with the Rectangular Base Elastically Supported by Edge Arched Beams and Tie-Rods (Ⅱ)

In order to study the boundary conditions of integrated form(91a,d),firstlywe make use of trigonometrical series to express N_ξη(0,η),Q_ξ^*(0,η)of(97C,D).Suppose .     

The floquet theory for quasi-periodic linear systems

In this paper we establish the Floquet theory for the quasi-perio-dic systemwhere A(u_1,u_2,…,u_m)is an n×n periodic matrix function of u_1,u_2.…,u_mwith period 2π,and it is of C~τ,τ=(N 1)τ_0,τ_0=2(m 1).N=(1/2)n(n 1).Meanwhile,we define the characteristic exponential roots β_1,β_2,…,β_nof(0.1),and assume thatwhere K(ω),K(ω,β)>0.k_μ,j_v.are integers,all the integers k_1,k_2,…,k_m.are not zero,i~2=-1,Then there exists aquasi-periodic linear transformation,which carries(0.1)into a li-near system with constant coefficients.     

Extended graphical representation of rational fractions with applications to cybernetics

In this paper,we discuss the extended graphical representation ofthe fraction of a complex variable sWhere K is confined to be real.Three figures of the above fraction canbe used in feedback systems as well as to study the properties of figuresfor any one coefficient of a characteristic equation as a real parameter.It is easy to prove the following theorem:K_1=f(n)(s)/F(d)(s)and K_2=F(d)(s)/f(n)(s)have the same root locus.By this graphical theory,we find out that if the zeros and polesof a fraction are alternatively placed on the axis x,then there is nocomplex root locus of this fraction,therefore the state of such a sys-tem is always non-oscillatory.Using these figures of this,fraction,wecan discuss its stable interval systematically.     

The traction problem in elastodynamics

In continuum mechanics, Cauchy's six equations are incomplete, and the famous Cauchy's laws of motion where ρ(?), ρb, T and divT are continuous are also incomplete. The first six equations are incomplete because the geometrical representation of deformation at a given point is as yet incomplete, and the last two laws are incomplete because b,T and divT are frame-indifferent, but ρ(?) is not, and T is a symmetric, as Cauchy interpreted himself. Therefore, we say, the last two laws can't accommodate to the asymmetric tensor. The purpose of this paper is to complete Cauchy's laws of motion by postulating an asymmetric tensor for the underlying traction field of 3-dimensional space on a general framing.     

On the global asymptotic stability of periodic solutions of a class of ecological system

In this paper, we have made reaearches on the mathematical models which have three populations of mutual action: We have obtained the sufficient conditions respectively for the systems(*)and(**)for existence and uniqueness of single positive periodic solutions which are globally asymptotically stable.     

弹性理论中的临界变分及消除方法

临界变分现象是拉氏乘子法的固有特性，钱伟长应用高阶拉氏乘子消除了临界变分现象。本文将提出一种新的方法－凑合反推法，这种方法摒充了拉氏乘子法，把拉氏乘子所在的项目一个待定函数Ｆ代替。这样构成的泛函，作者称之为试泛函。而待定函数Ｆ的识别类似于拉氏乘子的识别。通过该法可以方便地构造出各种多变量广义变分原理，并且可以消除临界变分现象。     

Singular perturbation of boundary value problems for second order nonlinear ordinary differential equations on infinite interval (I)

In this paper existence, uniqueness and asymptotic estimations of solutions of the boundary value problems on infinite interval for the second order nonlinear equation depending singularly on a small parameter ε>0 are examined, where α_i, β are constants, and i=0,1.     

Semi-inverse method and generalized variational principles with multi-variables in elasticity

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The application of the asymptotic method to a clas of strongly nonlinear systems

In this paper, according to the form of the asymptotic solution of papers [1, 2], the asymptotic method is extended to the following a class of more general strong nonlinear vibration systems where g and f are the nonlinear analytical-functions of x and x, and ε>0 is a small parameter. We assume that the derivative system corresponding to ε=0 has periodic solution. The recurrence equations of the asymptotic solution for the system(0.1)are deduced in this paper, and they are applied to practical examples.     

Variational principles and generalized variational principles for nonlinear elasticity with finite displacement

In a previous paper (1979)^[1], the minimum potential energy principle and stationary complementary energy principle for nonlinear elasticity with finite displacement, together with various complete and incomplete generalized principles were studied. However, the statements and proofs of these principles were not so clearly stated about their constraint conditions and their Euler equations. In somecases, the Euler equations have been mistaken as constraint conditions. For example, the stress displacement relation should be considered as Euler equation in complementary energy principle but have been mistaken as constraint conditions in variation. That is to say, in the above mentioned paper, the number of constraint conditions exceeds the necessary requirement. Furthermore, in all these variational principles, the stress-strain relation never participate in the variation process as constraints, i.e., they may act as a constraint in the sense that, after the set of Euler equations is solved, the stress-strain relation may be used to derive the stresses from known strains, or to derive the strains from known stresses. This point was not clearly mentioned in the previous paper (1979)^[1]. In this paper, the high order Lagrange multiplier method (1983)^[2] is used to construct the corresponding generalized variational principle in more general form. Throughout this paper, V/.V. Novozhilov's results (1958)^[3] for nonlinear elasticity are used.     

On singular perturbation boundary-value problem of coupling type system of convection-diffusion equations

In this paper we consider the singular perturbation boundary-value problem of thefollowing coupling type system of convection-diffusion equationsWe advance two methods:the first one is the initial value solving method,by which theoriginal boundary-value problem is changed into a series of unperturbed initial-valueproblems of the first order ordinary differential equation or system so that an asymptoticexpansion is obtained;the second one is the boundary-value solving method,by which theoriginal problem is changed into a few boundary-value problems having no phenomenon ofboundary-layer so that the exact solution can be obtained and any classical numericalmethods can be used to obtain the numerical solution of consismethods can be used to obtainthe numerical solution of consistant high accuracy with respect to the perturbationparameterε     

On the method of Lagrange multiplier and others

The fundamentals for the correct use of the method of Lagrange multiplier are presented and illustrated by examples. Some misunderstandings of the method are clarified. Equivalent variational principles are defined. It is pointed out that for a given problem of mechanics, there may be many equivalent and unequivalent variational principles. The functional of the so called generalized variational principles of elasticity with more general forms^[16] are linear combinations of the well known functionals given by Reissner and Hu-Washizu.     

The criterion algorithm of relation of implication between periodic orbits (I)

In recent years, there is a wide interest in Sarkovskii’s theorem ami the related study. According to Sarkovskii’s theoren if the continuous self-mapf of the closed interval has a 3-pcriodic orbit, then fmust has an n-pcriodic orbit for any positive integer n. But f can not has all n-periodic orbits for some n.For example, let Evidently, f has only one kind of 3-periodic orbit in the two kinds of 3-periodic orbits. This explains that it isn’t far enough to uncover the relation between periodic orbits by information which Sarkovskii’s theorem has offered. In this paper, we raise the concept of type of periodic orbits, and give a feasible algorithm which decides the relation of implication between two periodic orbits.