A second-order difference scheme for a parameterized singular perturbation problem

In this paper we consider a singularly perturbed quasilinear boundary value problem depending on a parameter. The problem is discretized using a hybrid difference scheme on Shishkin-type meshes. We show that the scheme is second-order convergent, in the discrete maximum norm, independent of singular perturbation parameter. Numerical experiments support these theoretical results.     

带可变参数四阶奇异边值问题正解的存在性

本文研究了带有一个可变参数的四阶奇异边值问题,利用Banach空间的不动点理论,在非常弱的条件下,获得了带可变参数四阶奇异边值问题存在正解的结果.     

一类含小参数的Hill方程的自由振动

本文研究一类含小参数的Hill方程的初值问题,利用边值问题可解性条件及摄动理论中的伸缩参数法,给出寻求该初值问题近似周期解的方法,并以Mathieu方程为例作了具体计算.     

Boundary-value problem for a class of second-order parameter-dependent dynamic equations on a time scale

In this study, we consider a boundary value problem generated by a second-order dynamic equation on a time scale and boundary conditions depending on the spectral parameter. We give some properties of the solutions and obtain a formulation of the number of eigenvalues of the problem.     

Existence and nonexistence results for a class of fractional boundary value problems

In this paper, we consider a fractional boundary value problem involving Riemann-Liouville fractional derivative and depending on a parameter. we obtain the existence and nonexistence results of positive solutions when the nonlinear term satisfies different requirements of superlinearity, sublinearity and the parameter lies in some intervals.     

Stability of supersonic contact discontinuity for two-dimensional steady compressible Euler flows in a finite nozzle

In this paper, we study the stability of supersonic contact discontinuity for the two-dimensional steady compressible Euler flows in a finitely long nozzle of varying cross-sections. We formulate the problem as an initial–boundary value problem with the contact discontinuity as a free boundary. To deal with the free boundary value problem, we employ the Lagrangian transformation to straighten the contact discontinuity and then the free boundary value problem becomes a fixed boundary value problem. We develop an iteration scheme and establish some novel estimates of solutions for the first order of hyperbolic equations on a cornered domain. Finally, by using the inverse Lagrangian transformation and under the assumption that the incoming flows and the nozzle walls are smooth perturbations of the background state, we prove that the original free boundary problem admits a unique weak solution which is a small perturbation of the background state and the solution consists of two smooth supersonic flows separated by a smooth contact discontinuity.     

A method of solving a nonlinear boundary value problem with a parameter for a loaded differential equation

A nonlinear loaded differential equation with a parameter on a finite interval is studied. The interval is partitioned by the load points, at which the values of the solution to the equation are set as additional parameters. A nonlinear boundary value problem for the considered equation is reduced to a nonlinear multipoint boundary value problem for the system of nonlinear ordinary differential equations with parameters. For fixed parameters, we obtain the Cauchy problems for ordinary differential equations on the subintervals. Substituting the values of the solutions to these problems into the boundary condition and continuity conditions at the partition points, we compose a system of nonlinear algebraic equations in parameters. A method of solving the boundary value problem with a parameter is proposed. The method is based on finding the solution to the system of nonlinear algebraic equations composed.     

Some homogenization problems for the system of elasticity with nonlinear boundary conditions in perforated domains

The system of linear elasticity is considered in a perforated domain with an ε-periodic structure. External forces nonlinearly depending on the displacements are applied to the surface of the cavities (or channels), while the body is fixed along the outer portion of its boundary. We investigate the asymptotic behavior of solutions to such boundary value problems asε→0 and construct the limit problem, according to the external surface forces and their dependence on the parameter ε. In some cases, this dependence results in the homogenized problem having the form of a variational inequality over a certain closed convex cone in a Sobolev space. This cone is described in terms of the functions involved in the nonlinear boundary conditions on the perforated boundary. A homogenization theorem is also proved for some unilateral problems with boundary conditions of Signorini type for the system of elasticity in a perforated domain. We discuss some cases when the homogenized tensor may depend on the functions specifying the boundary conditions.     

On output functionals of boundary value problems on stochastic domains

We consider the computation of output functionals of random solutions to elliptic boundary value problems in domains with random boundary perturbations. We use a second‐order shape calculus to linearize the problem around a fixed nominal domain. For known mean and two‐point correlation function of the boundary perturbation, we derive, with leading order, deterministic expressions for the mean and the variance of the random output functional. These expressions include the solution of the boundary value problem on the nominal domain and a further, deterministic solution of the so‐called adjoint equation. The theoretical findings are supported and quantified by numerical experiments. Copyright © 2009 John Wiley & Sons, Ltd.     

A note on a parameterized singular perturbation problem

We consider a uniform finite difference method on Shishkin mesh for a quasilinear first-order singularly perturbed boundary value problem (BVP) depending on a parameter. We prove that the method is first-order convergent except for a logarithmic factor, in the discrete maximum norm, independently of the perturbation parameter. Numerical experiments support these theoretical results.     

On a class of singular boundary value problems with singular perturbation

In this paper we investigate a class of singular second order differential equations with singular perturbation subject to three-point boundary value conditions, whose solution exhibits a couple of boundary layers at two endpoints. We first establish a lower–upper solutions theorem by using the Schauder fixed point theorem. By the asymptotic expansions and the lower–upper solutions theorem we obtain the existence, asymptotic estimates and uniqueness for the proposed problem. Several examples are given for illustrating our results.     

Singular numbers of the monodromy operator and sufficient conditions of the asymptotic stability of periodic system of differential equations with fixed delay

To obtain sufficient conditions for the asymptotic stability of linear periodic systems with fixed delay commensurable with the period of coefficients, singular numbers of the monodromy operator are used. To find these numbers, a self-adjoint boundary value problem for ordinary differential equations is applied. We study the motion of eigenvalues of this boundary value problem under a variation of a parameter. Obtaining sufficient conditions for the asymptotic stability is reduced to finding the bifurcation value of the parameter for the boundary value problem.     

一类四阶差分方程多点边值问题的正解

为解决多点支撑弹性梁的正解的存在性问题,运用锥上不动点指数理论,研究一类含参四阶差分方程多点边值问题.获得了当参数在一定范围内取值时正解的存在性结果,得到了正解存在的充分条件.     

半线性系统的内层现象

本文研究半线性系统
εy″=f(t,y,ε) (ay(a,ε)=A(ε), y(b,ε)=B(ε)
的奇摄动边值问题的内层现象,其中ε>0是一个小参数,y、f、A和B是n-维向量函数。这种向量边值问题似乎还没有研究过,尽管纯量的边值问题已广泛地研究过。在适当地假设下如同纯量问题一样我们得到解的存在、同时使用适当的不等式,这个解的估计也得到。     

A Hybrid Method for Low Reynolds Number Flow Past an Asymmetric Cylindrical Body

Low Reynolds number fluid flow past a cylindrical body of arbitrary shape in an unbounded, two-dimensional domain is a singular perturbation problem involving an infinite logarithmic expansion in the small parameter ε, representing the Reynolds number. We apply a hybrid asymptotic–numerical method to compute the drag coefficient, C _D and lift coefficient C _L to within all logarithmic terms. The hybrid method solution involves a matrix M , depending only on the shape of the body, which we compute using a boundary integral method. We illustrate the hybrid method results on an elliptic object and on a more complicated profile.     

Uniform analyticity and exponential decay of the semigroup associated with a thermoelastic plate equation with perturbed boundary conditions

In a bounded domain, we consider an Euler–Bernoulli-type thermoelastic plate equation with perturbed boundary conditions. The boundary conditions are such that when the perturbation parameter goes to infinity, we recover the hinged boundary conditions, while one recovers the clamped boundary conditions when the perturbation parameter goes to zero. Relying on resolvent estimates, we show that the underlying semigroup is uniformly, with respect to the perturbation parameter, analytic and exponentially stable. The main features of our proof are appropriate decompositions of the components of the system and the use of Lions? interpolation inequalities.     

Stability of Displacement to the Second Fundamental Problem in Plane Elasticity

In this article, by using the stability of Cauchy type integral when the smooth perturbation for integral curve and the Sobolev type perturbation for kernel density happen, we discuss the stability of the second fundamental problem in plane elasticity when the smooth perturbation for the boundary of the elastic domain (unit disk) and the Sobolev type perturbation for the displacement happen. And the error estimate of the displacement between the second fundamental problem and its perturbed problem is obtained.     

Invariant Integrals for the Equilibrium Problem for a Plate with a Crack

We consider the equilibrium problem for a plate with a crack. The equilibrium of a plate is described by the biharmonic equation. Stress free boundary conditions are given on the crack faces. We introduce a perturbation of the domain in order to obtain an invariant Cherepanov–Rice-type integral which gives the energy release rate upon the quasistatic growth of a crack. We obtain a formula for the derivative of the energy functional with respect to the perturbation parameter which is useful in forecasting the development of a crack (for example, in study of local stability of a crack). The derivative of the energy functional is representable as an invariant integral along a sufficiently smooth closed contour. We construct some invariant integrals for the particular perturbations of a domain: translation of the whole cut and local translation along the cut.     

含多个小参数的高阶拟线性椭圆型方程一般边值问题的奇摄动

本文应用М.Н.Вишик和Л.А.Люстерник[1]的渐近方法以及泛函分析的不动点原理研究了方程与边界摄动相结合的高阶拟线性椭圆型方程一般边值问题的奇摄动,证明了摄动问题解的存在且唯一,给出解的渐近展开式和有关的余项估计.     

Properties of positive solutions to a class of four-point boundary value problem of Caputo fractional differential equations with a parameter

In this paper, by using fixed point theorems of concave operators in partial ordering Banach spaces, we establish the existence and uniqueness of positive solutions to a class of four-point boundary value problem of Caputo fractional differential equations for any given parameter. Moreover, we present some pleasant properties of positive solutions to the boundary value problem dependent on the parameter. In the end, two examples are given to illustrate our main results.