Singular perturbation solution of boundary-value problem for a secord-order differential-difference equation

In this paper, the method of two-variables expansion is used to construct boundary layer terms of asymptotic solution of the boundary-value problem for a second-order DDE. The n-order formal asymptotic solution is obtained and the error is estimated. Thus the existence of uniformly valid asymptotic solution is proved.     

轴对称正交异性圆环壳的齐次完全渐近解

承受轴对称载荷的正交异性圆环壳的静力分析,归结为求解一非齐次二阶复变量方程.当所含参数μ较大时,常采用渐近解法.因方程含一阶转点,所以求全域一致有效且达到薄壳理论精度的完全渐近解较为困难.过去,齐次解只求到一级近似.本文采用广义Airy函数方法,求出了高级近似.这样,轴对称正交异性圆环壳的齐次解第一次有了达到薄壳理论精度的完全的渐近展开.     

Effect of Heat Generation on Forced Convection Through a Porous Saturated Duct

The effect of heat generation on the flow characteristics of the fully developed forced convection through a porous duct is investigated analytically on the basis of Brinkman?CForchheimer model. The duct is bounded by two isoflux plates. For solving momentum equation a regular asymptotic expansion method is used for hyper-porous materials and a matched asymptotic expansion method is used for low-porous materials. This solution permits a uniform solution for the energy equation to find the temperature distribution as well as Nusselt number. A numerical solution is found here to check the accuracy of the asymptotic one.     

Asymptotics of the Critical Regimes of Internal Gravity Wave Generation

The problem of constructing an asymptotic representation of the solution of the internal gravity wave field excited by a source moving at a velocity close to the maximum group velocity of the individual wave mode is considered. For the critical regimes of individual mode generation the asymptotic representation of the solution obtained is expressed in terms of a zero-order Macdonald function. The results of numerical calculations based on the exact and asymptotic formulas are given.     

Free convection boundary-layer flow above a nearly horizontal surface in a porous medium with newtonian heating

In this paper the steady free convection boundary-layer along a semi-infinite, slightly inclined (both positive and negative) to the horizontal plate embedded in a porous medium with the flow generated by Newtonian heating has been investigated. The asymptotic solution near the leading edge and the full numerical solution along the whole plate domain have been obtained numerically, whilst the asymptotic solution far downstream along the plate has been obtained analytically. For a positive inclination the full numerical solution is in agreement with the asymptotic solutions. However, for a negatively inclined plate, only the small asymptotic solution near the leading edge of the plate can be predicted giving an insight that the model for a negatively inclined plate, whilst mathematically interesting, is not physically realistic.     

The asymptotic expression of the solution of the Cauchy's problem for a higher order linear ordinary differential equation when the limit equation has singularity

In this paper we consider the asymptotic expression of the solution of the Cauchy's problem for a higher order equation when the limit equation has singularity. In order to construct the asymptotic expression of the solution, the region is divided into three sub-areas. In every small region, the solution of the differential equation is different. Project supported by the National Natural Science Foundation of China     

Perturbation method in the problem of large deflections of circular plates with nonuniform thickness

In this paper the perturbation method about two parameters is applied to the problem of large deflection of a cricular plate with exponentially varying thickness under uniform pressure. An asymptotic solution up to the third-order is derived. In comparison with the exact solutions in special cases, the asymptotic solution shows a precise accuracy.     

On the long-time limit of the Garvin–Alterman–Loewenthal solution for a buried blast load

The Garvin–Alterman–Loewenthal solution refers to the problem of a line blast load suddenly applied in the interior of an elastic half-space. It is expected that the long-time asymptotic limit of this solution should be equal to the solution of a related static problem. This expectation is justified here. First, the solution of the static problem is constructed. Then, the asymptotic limit of the transient problem is found, correcting previously published results.     

The asymptotic solution of a kind of Stefan problem

In this paper, a kind of Stefan problem subject to general initial condition is studied. The slab considered is divided into three regions. There is a different time scale in each one. By means of PLK method or like multi-scales method, the asymptotic solution of each one is obtained. Finally we discuss the asymptotic solution and draw some conclusions.     

Stress-strain state in the vicinity of a crack tip under mixed loading

A method is proposed to calculate the eigenvalues of the class of nonlinear eigenvalue problems resulting from the problem of determining the stress-strain state in the vicinity of a crack tip in power-law materials over the entire range of mixed modes of deformation, from the opening mode to pure shear. The proposed approach was used to found eigenvalues of the problem that differ from the well-known eigenvalue corresponding to the Hutchinson-Rice-Rosengren solution. The resulting asymptotic form of the stress field is a self-similar intermediate asymptotic solution of the problem of a crack in a damaged medium under mixed loading. Using the new asymptotic form of the stress field and introducing a self-similar variable, we obtained an asymptotic solution of the problem of a crack in a damaged medium and constructed the regions of dispersed material near the crack.     

Analysis of the transpiration cooling of a thin porous plate in a hot laminar convective flow

This paper deals with the asymptotic and numerical analysis for the steady-state transpiration cooling of a thin porous flat plate in a laminar hot convective flow, taking into account the streamwise heat conduction through the plate. For high conductivity plates, a regular perturbation analysis has been carried out, yielding a three-term asymptotic solution for the distribution of plate temperature. In the limit of a very poorly conducting plate, a singular perturbation technique, based on matched asymptotic expansions, is employed to solve the governing equations. We also solved the equations numerically using a quasilinearization technique. The numerical results are in good agreement with the asymptotic solution close to the asymptotic limits studied.     

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.     

Exponential dichotomies of nonlinear discrete systems and its application to numerical analysis and computation

In this pepar we consider the upwind difference scheme of a kind of boundary value problems for nonlinear, second order, ordinary differential equations. Singular perturbation method is applied to construct the asymptotic approximation of the solution to the upwind difference equation. Using the theory of exponential dichotomies we show that the solution of an order-reduced equation is a good approximation of the solution to the upwind difference equation except near boundaries. We construct correctors which yield asymptotic approximations by adding them to the solution of the order-reduced equation. Finally, some numerical examples are illustrated.     

具有内域的双层加筋圆柱壳动响应特性简

采用外域双渐近法模拟外部流体域,内域双渐近法模拟内部流体域,采用非线性有限元软件ABAQUS 模拟结构,建立流固耦合数值模型并验证了其有效性. 利用建立的数值模型研究内部流体对双层加筋圆柱壳动响应的影响. 研究结果表明：一阶双渐近的解在中后期振荡周期变短、幅值变小,而二阶双渐近的解与解析解吻合良好;内部流体的存在减弱了双层加筋圆柱壳外壳的塑性变形,对其内壳塑性变形的影响较小;内部流体的存在减弱了双层加筋圆柱壳的速度响应.     

Stationary points iteration method for periodic solution to nonlinear system

The value method which is used to obtain the periodic solution to nonlinear system is mentioned in this article. Different point reflection is defined in the nonlinear autonomous and nonautonomous system firstly and then that linear reflection obtained from the inserting value of nonlinear reflection is asymptotic to original nonlinear reflection. The stationary points obtained by linear reflection are regarded as the asymptotic solution of the stationary points of original system. If this asymptotic solution of the stationary points is not satisfactorily accurate it can be used as the initial point of the next reflection. In addition, a corresponding method of researching the stability of periodic solution is put forward in this article.     

Minimizing sequences of variational problems with small parameters

A class of variational problems with small parameters is studied. Their zeroth-order asymptotic solutions are constructed. It is shown that the zeroth-order asymptotic solution is just the minimizing sequence of variational problems as the small parameter approaches to zero.     

Ⅲ型扩展裂纹尖端场渐近解的研究

本文采用弹性-粘塑性模型对反平面剪切扩展裂纹问题进行了研究。文中通过对动态扩展问题和准静态扩展问题的分析得到了各自的渐近解,并且证明了准静态扩展解是动态扩展解的特殊情况,从而使二者统一起来。     

具有内域的双层加筋圆柱壳动响应特性

采用外域双渐近法模拟外部流体域,内域双渐近法模拟内部流体域,采用非线性有限元软件ABAQUS 模拟结构,建立流固耦合数值模型并验证了其有效性. 利用建立的数值模型研究内部流体对双层加筋圆柱壳动响应的影响. 研究结果表明:一阶双渐近的解在中后期振荡周期变短、幅值变小,而二阶双渐近的解与解析解吻合良好;内部流体的存在减弱了双层加筋圆柱壳外壳的塑性变形,对其内壳塑性变形的影响较小;内部流体的存在减弱了双层加筋圆柱壳的速度响应.      

A model of two-velocity particles transport in a porous medium

A one-dimensional flow of suspension with two types of solid particles moving with different velocities in a porous medium is considered. A mathematical model of deep bed filtration which generalizes the known equations of mass balance and particle capture kinetics for a flow of fluid with identical particles is developed. The exact solution is evaluated at the filter inlet and on the concentration front of fast suspended and retained particles, asymptotic solutions are provided in certain vicinities of these lines. A global asymptotic solution to the problem with a small limit deposit is constructed. The asymptotics rapidly converges to the numerical solution.     

High-order asymptotics and perturbation problems for 3D interfacial cracks

We present an asymptotic algorithm for analysis of a singularly perturbed problem in a domain containing an interfacial crack. The crack is assumed to be flat and its front, initially straight, is perturbed in the plane containing the crack. The aim of the work is to determine the asymptotic representation of the stress-intensity factors near the edge of the crack. Mathematically, the limit problem is reduced to the analysis of a matrix, 3×3, Wiener-Hopf problem, and its solution generates the “weight matrix-function” characterised by a special singular solution near the crack edge. The two-term asymptotic representation for the weight function components is required by the asymptotic algorithm, together with two-term asymptotics for stress components associated with the physical fields near the edge of the crack. The particular feature of the solution is the coupling between the normal opening mode (Mode-I), and the shear modes (Mode-II and Mode-III), and the oscillatory behaviour of certain stress components near the crack edge. Explicit asymptotic formulae for the stress-intensity factors are obtained at the edge of a “wavy crack” at an interface.