ON SOLVING HIGH-ORDER SOLUTIONS OF CHIEN’S PERTURBATION METHOD TO STUDY CONVERGENCE BY COMPUTER

In this paper.we have obtained the analytical formula of arbitrary n-th orderperturbation solution which perturbation parameter is central deflection for circular thinplate under a concentrated load by means of its integral equation.All unkown constants ofevery perturbation solution can be determined by computer.Hence higher orderperturbation solution is obtained.Based on the solution of higher order perturbation,asymptotical property and suitable interval of Chien’s perturbation method are alsodiscussed.     

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     

Dirichlet Problem of Higher Order Elliptic Equation with Perturbation both in Differential Operator and in Boundary

This paper is the continuation of article[7].It gives further results about the asymptoticexpression for the solution of higher order elliptic equation in the case of boundary perturbationcombined with operator perturbation.When unperturbed problem Ao is not on the spectrum,theasymptotic expression for the solution of perturbation problem Aεmay be expanded with respectto the small parameterε.While Ao is on the spectrum,the asymptotic expression of the solutioncontains negative powers of the small parameterε.The approximation of arbitrary order to thesolution is considered and the recursive formula for the general term and the estimation of remain-der term are given.     

High order symplectic conservative perturbation method for time-varying Hamiltonian system

This paper presents a high order symplectic conservative perturbation method for linear time-varying Hamiltonian system.Firstly,the dynamic equation of Hamiltonian system is gradually changed into a high order perturbation equation,which is solved approximately by resolving the Hamiltonian coefficient matrix into a "major component" and a "high order small quantity" and using perturbation transformation technique,then the solution to the original equation of Hamiltonian system is determined through a series of inverse transform.Because the transfer matrix determined by the method in this paper is the product of a series of exponential matrixes,the transfer matrix is a symplectic matrix;furthermore,the exponential matrices can be calculated accurately by the precise time integration method,so the method presented in this paper has fine accuracy,efficiency and stability.The examples show that the proposed method can also give good results even though a large time step is selected,and with the increase of the perturbation order,the perturbation solutions tend to exact solutions rapidly.     

Singular perturbation for the weakly nonlinear reaction diffusion equation with boundary perturbation

In this paper, a class of nonlinear singularly perturbed initial boundary value problems for reaction diffusion equations with boundary perturbation are considered under suitable conditions. Firstly, by dint of the regular perturbation method, the outer solution of the original problem is obtained. Secondly, by using the stretched variable and the expansion theory of power series the initial layer of the solution is constructed. And then, by using the theory of differential inequalities, the asymptotic behavior of the solution for the initial boundary value problems is studied. Finally, using some relational inequalities the existence and uniqueness of solution for the original problem and the uniformly valid asymptotic estimation are discussed.     

THE BOUNDED SOLUTION OF A CLASS OF DIFFERENTIAL-DIFFERENCE EQUATION OF ADVANCED TYPE AND ITS ASYMPTOTIC BEHAVIOR

In this paper, we consider the differential-difference equation of advanced type with perturbation term. It is shown that if the bounded solution of the reduced equation has negative exponential order and the perturbation term f satisfies certain condition, then the bounded solution of the perturbation equation has negative exponential order.     

PARALLEL COMPUTING FOR STATIC RESPONSE ANALYSIS OF STRUCTURES WITH UNCERTAIN-BUT-BOUNDED PARAMETERS

The vertex solution for estimation on the static displacement bounds of structures with uncertain-but-bounded parameters is studied in this paper. For the linear static problem, when there are uncertain interval parameters in the stiffness matrix and the vector of applied forces, the static response may be an interval. Based on the interval operations, the interval solution obtained by the vertex solution is more accurate and more credible than other methods （such as the perturbation method）. However, the vertex solution method by traditional serial computing usually needs large computational efforts, especially for large structures. In order to avoid its disadvantages of large calculation and much runtime, its parallel computing which can be used in large-scale computing is presented in this paper. Two kinds of parallel computing algorithms are proposed based on the vertex solution. The parallel computing will solve many interval problems which cannot be resolved by traditional interval analysis methods.     

An approximate analytical solution of the cylindrical detonation wave generated by the linear explosion

The solution of the cylindrical detonation wave generated by the linear explosion was obtained by numerical method in ref.[1].In this paper,when the ratio of specific heatγ》1 by using the enlargement coordinate method,the first-order analytical solutions are obtained.The perturbation parameter isε=1/γ~2 .The correction of these solutions is checked at the end of this paper.     

Head-on collision between two mKdV solitary waves in a two-layer fluid system

In this paper, based on the equations presented in [2], the head-on collision between two solitary waves described by the modified KdV equation (the mKdV equation, for short) is investigated by using the reductive perturbation method combined with the PLK method. These waves propagate at the interface of a two-fluid system, in which the density ratio of the two fluids equals the square of the depth ratio of the fluids. The second order perturbation solution is obtained. It is found that in the case of disregarding the nonuniform phase shift, the solitary waves preserve their original profiles after collision, which agrees with Fornberg and Whitham's numerical result of overtaking collision161 whereas after considering the nonuniform phase shift, the wave profiles may deform after collision.     

Singular perturbation of initial value problem for a nonlinear second order systems

In this paper,the singular perturbation of initial value problem for nonlinearsecond order vector differential equationsε~rx″=f(t,x,x′,ε)x(0,ε)=a,x′(0,ε)=βis discussed,where r>0 is an arbitrary constant,ε>0 is a small parameter,x,f,aandβ∈R~n.Under suitable assumptions,by using the method of many-parameterexpansion and the technique of diagonalization,the existence of the solution of pertur-bation problem is proved and its uniformly valid asymptotic expansion of higher order isderived.     

A Universal Matrix Perturbation Technique for Complex Modes

ＩｎｔｒｏｄｕｃｔｉｏｎＭａｔｒｉｘｐｅｒｔｕｒｂａｔｉｏｎｍｅｔｈｏｄｓｆｏｒｔｈｅｄｙｎａｍｉｃｒｅａｎａｌｙｓｉｓｏｆｓｅｌｆ＿ａｄｊｏｉｎｔｓｙｓｔｅｍｓｈａｖｅｂｅｅｎｗｅｌｌｄｅｖｅｌｏｐｅｄ[1] .Ｈｏｗｅｖｅｒ,ｍａｎｙｓｙｓｔｅｍｓｇｉｖｅｒｉｓｅｔｏｇｅｎｅｒａｌｎｏｎ＿ｓｅｌｆ＿ａｄｊｏｉｎｔｆｏｒｍｕｌａｔｉｏｎｓ.Ｉｍｐｏｒｔａｎｔｅｘａｍｐｌｅｓａｒｅａｅｒｏｅｌａｓｔｉｃｓｔａｂｉｌｉｔｙｏｆｓｙｓｔｅｍｓ,ａｒｂｉｔｒａｒｉｌｙｄａｍｐｅｄｏｒｇｙｒｏｓｃｏｐｉｃｓｙｓ…     

Singular perturbations in free convection

A second order solution to the problem of free convection from a vertical surface has been obtained using the method of perturbations. It has been shown that the perturbation solution gives divergent solutions at the leading edge, but is otherwise satisfactory. Further, using the first order outer solution, a coordinate system, optimal in the sense ofKaplun, has been obtained. It is shown that this coordinate system presents a free convection flow field which is more satisfactory than that given by the classical Pohlhausen solution.     

Singular perturbation of general boundary value problem for nonlinear differential equation system

I.IntroductionAllphysicalsystemsarenonlineartosomeextent.Actually,Lineal.systemisimaginarymodelwherenonlinearfactorisomittedinnonlinearsystem.Insolvingtheautocontl'ol,nonlinearoscillationtheory,theboundarystagnationproblenloffluidIncchanicsandsollleproblemsofsemi-conducttheoryandquantummechanicsetc'.weOnlyncedtosolvethefollowingproblem,whichisnonlineardifferentialequationsystemwithtilesll,allparanletel'inhighestorderderivativeandnonlinearboundaryconditions.whereE>0isasmallparameter,teR,x,fi…     

A dynamical study of the evolution of pressure waves propagating through a semi-infinite region of homogeneous gas combustion subject to a time-harmonic signal at the boundary

In this paper, we study the evolution of pressure waves propagating in a region of gas combustion subject to a time-harmonic signal at the boundary. The problem is modeled by a non-linear, hyperbolic partial differential equation. Steady-state behavior is investigated using the perturbation method to ensure that enough time has passed for transient effects to have dissipated. The zeroth-order and first two approximations are obtained. Furthermore, the behavior of the following quantities is investigated, with particular attention paid to the low and high-frequency limits: the location of the peak of the first-order approximation, dispersion relations and phase speeds. Additionally, a maximum value of the perturbation parameter is determined ensuring boundedness of the solution. Next, approximate solutions are obtained in the low and high frequency limits and a comparison is made with the corresponding perturbation solution. Finally, the solution obtained from the perturbation method is compared with the long-time solution obtained by a non-standard finite-difference scheme.     

The bounded solution of a class of differential-difference equation of advanced type and its asymptotic behavior

In this paper, we consider the differential-difference equation of advanced type with perturbation term. It is shown that if the bounded solution of the reduced equation has negative exponential order and the perturbation term f satisfies certain condition, then the bounded solution of the perturbation equation has negative exponential order.     

Perturbation solution to heat conduction in melting or solidification with heat generation

The Stefan problem involving a source term is considered in this technical note. As an example, planar solidification with time-dependent heat generation in a semi-infinite plane is solved by use of a perturbation technique. The perturbation solution is validated by reducing the problem to the case without heat generation whose exact solution is available. An application to the case with constant heat generation is presented, for which a closed-form solution is obtained. The effects of heat generation and Stefan number on the evolution of solidification are examined using the perturbation solution.     

脱层屈曲的高阶摄动解

本文用摄动法求解了三维复合材料中有纤维搭桥的圆形脱层的屈曲问题。搭桥纤维被简化为连续分布于脱层面上的线性弹簧。用Taylor级数展开法确定了每一级摄动方程用Bessel函数表达的解析解．给出了脱层屈曲临界载荷、挠度曲线与脱层半径、搭桥因子等的封闭关系式。根据这一结果并考虑脱层只能向一个方向挠曲的约束条件，可以进一步探讨脱层屈曲的特征。     

Thermoelastic stresses in a cylinder or disk with cubic anisotropy

The thermoelastic stresses in a crystal in the shape of a circular cylinder or disk are considered. The crystal is a cubically-orthotropic linear elastic solid, with three independent elastic properties. The cubic anisotropy renders the problem asymmetric, despite the axisymmetry of the geometry and thermal loading. This problem is motivated by a thermoelastic model used for certain crystal growth processes. Two simplifying assumptions are made here: (a) the problem is two-dimensional with plane strain or plain stress conditions, and (b) the elastic properties do not depend on the temperature. A new Fourier-type perturbation method is devised and an analytic asymptotic solution of a closed form is obtained, based on the weak cubic anisotropy of the crystal as a perturbation parameter. A general solution technique is described which yields the asymptotic solution up to a desired order. Numerical results are presented for typical parameter values.