ASYMPTOTIC BEHAVIOUR OF EIGENVALUES FOR THE DISCONTINUOUS BOUNDARY-VALUE PROBLEM WITH FUNCTIONAL-TRANSMISSION CONDITIONS

In this study, the boundary-value problem with eigenvalue parameter generated by the differential equation with discontinuous coefficients and boundary conditions which contains not only endpoints of the considered interval, but also point of discontinuity and linear functionals is investigated. So, the problem is not pure boundary-value. The authors single out a class of linear functionals and find simple algebraic conditions on coefficients, which garantee the existence of infinit number eigenvalues. Also the asymptotic formulas for eigenvalues are found.

热传导方程Robin系数反问题解的唯一性及正则化解的存在性

Robin系数在热传导模型中刻画了热传导区域边界上的热交换,是一类非常重要的参数,本文基于某小时段温度测量值反演热传导模型中的Robin系数.首先,在边界值以及测量值满足一定的光滑性条件时,给出了反问题解的唯一性;其次,基于Tikhonov正则化思想,通过构造目标泛函将反问题转化为求目标泛函的极小值,并证明了泛函极小元的存在性.

Solvability on boundary-value problems of elasticity of three-dimensional quasicrystals

Weak solution （or generalized solution） for the boundary-value problems of partial differential equations of elasticity of 3D （three-dimensional） quasicrystals is given, in which the matrix expression is used. In terms of Korn inequality and theory of function space, we prove the uniqueness of the weak solution. This gives an extension of existence theorem of solution for classical elasticity to that of quasicrystals, and develops the weak solution theory of elasticity of 2D quasicrystals given by the second author of the paper and his students.

Study on the Existence of Sign-Changing Solutions of Case Theory Based a Class of Differential Equations Boundary-Value Problems

By using the fixed point theorem under the case structure, we study the existence of sign-changing solutions of A class of second-order differential equations three-point boundary-value problems, and a positive solution and a negative solution are obtained respectively, so as to popularize and improve some results that have been known.

Logarithmic Sine and Cosine Transforms and Their Applications to Boundary-Value Problems Connected with Sectionally-Harmonic Functions

Let stand for the polar coordinates in R2, ?be a given constant while satisfies the Laplace equation in the wedge-shaped domain or . Here αj(j = 1,2,...,n + 1) denote certain angles such that αj αj(j = 1,2,...,n + 1). It is known that if r = a satisfies homogeneous boundary conditions on all boundary lines ?in addition to non-homogeneous ones on the circular boundary , then an explicit expression of in terms of eigen-functions can be found through the classical method of separation of variables. But when the boundary?condition given on the circular boundary r = a is homogeneous, it is not possible to define a discrete set of eigen-functions. In this paper one shows that if the homogeneous condition in question is of the Dirichlet (or Neumann) type, then the logarithmic sine transform (or logarithmic cosine transform) defined by (or ) may be effective in solving the problem. The inverses of these transformations are expressed through the same kernels on or . Some properties of these transforms are also given in four theorems. An illustrative example, connected with the heat transfer in a two-part wedge domain, shows their effectiveness in getting exact solution. In the example in question the lateral boundaries are assumed to be non-conducting, which are expressed through Neumann type boundary conditions. The application of the method gives also the necessary condition for the solvability of the problem (the already known existence condition!). This kind of problems arise in various domain of applications such as electrostatics, magneto-statics, hydrostatics, heat transfer, mass transfer, acoustics, elasticity, etc.

A Numerical Method for Singular Boundary-Value Problems

This note is concerned with an iterative method for the solution of singular boundary value problems. It can be considered as a predictor-corrector method. Sufficient conditions for the convergence of the method are introduced. A number of numerical examples are used to study the applicability of the method.

AN EFFICIENT FINITE DIFFERENCE METHOD FOR STOCHASTIC LINEAR SECOND-ORDER BOUNDARY-VALUE PROBLEMS DRIVEN BY ADDITIVE WHITE NOISES

In this paper,we develop and analyze a finite difference method for linear second-order stochastic boundary-value problems(SBVPs)driven by additive white noises.First we regularize the noise by the Wong-Zakai approximation and introduce a sequence of linear second-order SBVPs.We prove that the solution of the SBVP with regularized noise converges to the solution of the original SBVP with convergence order O(h)in the meansquare sense.To obtain a numerical solution,we apply the finite difference method to the stochastic BVP whose noise is piecewise constant approximation of the original noise.The approximate SBVP with regularized noise is shown to have better regularity than the original problem,which facilitates the convergence proof for the proposed scheme.Convergence analysis is presented based on the standard finite difference method for deterministic problems.More specifically,we prove that the finite difference solution converges at O(h)in the mean-square sense,when the second-order accurate three-point formulas to approximate the first and second derivatives are used.Finally,we present several numerical examples to validate the efficiency and accuracy of the proposed scheme.

A Robin Problem for Quasi-linear System

In this paper, a Robin problem for quasi-linear system is considered. Under the appropriate assumptions, the existence of solution for the problem is proved and the asymptotic behavior of the solution is studied using the theory of differential inequalities.

Spherical Harmonic Solution of the Robin Problem for the Helmholtz Equation in a Supershaped Shell

The Robin problem for the Helmholtz equation in normal-polar shells is addressed by using a suitable spherical harmonic expansion technique. Attention is in particular focused on the wide class of domains whose boundaries are defined by a generalized version of the so-called “superformula” introduced by Gielis. A dedicated numerical procedure based on the computer algebra system Mathematica? is developed in order to validate the proposed methodology. In this way, highly accurate approximations of the solution, featuring properties similar to the classical ones, are obtained.

The spike layer solution to singular perturbation Robin boundary value problem for higher order elliptic equation

The singularly perturbed Robin boundary value problem for the higher order elliptic equation is considered.Under suitable conditions,the existence and asymptotic behavior of solution to the boundary value problems are studied.The uniform validity of its asymptotic expansion is proved by using the fixed point theorem.

Numerical Solutions to the Robin Inverse Problem with Nonnegativity Constraints

We present iterative numerical methods for solving the inverse problem of recovering the nonnegative Robin coefficient from partial boundary measurement of the solution to the Laplace equation. Based on the boundary integral equation formulation of the problem, nonnegativity constraints in the form of a penalty term are incorporated conveniently into least-squares iteration schemes for solving the inverse problem. Numerical implementation and examples are presented to illustrate the effectiveness of this strategy in improving recovery results.

Quasilinear Robin Problem with Boundary Perturbation

有边界不安的 quasilinear 罗宾问题的一个类是 considered.Under 合适的条件，用微分不平等的理论，边界值问题的答案的 asymptotic 行为被学习。

The Nonlinear Predator-Prey Singularly Perturbed Robin Initial Boundary Value Problems for Reaction Diffusion System

The nonlinear predator-prey singularly perturbed Robin initial boundary value problems for reaction diffusion systems were considered. Under suitable conditions, using theory of differential inequalities the existence and asymptotic behavior of solution for initial boundary value problems were studied.

DIAGONALIZATION METHOD FOR A SINGULARLY PERTURBED VECTOR ROBIN PROBLEM

In this paper the method and technique of the diagonalization are employed to transform a vector second-order nonlinear system into two first-order approximate diagonalized systems. The existence and the asymptotic behavior of the solutions are obtained for a vector second-order nonlinear Robin problem of singular perturbation type.

位势方程的Robin问题解的研究

位势方程是偏微分方程理论研究中的一类典型的二阶椭圆型方程。对位势方程的Robin问题作了深入研究,运用散度定理证明了该问题在C^(1)(-Ω)∩C^(2)(Ω)中解的唯一性,并运用Green函数法导出该问题的Green函数G(x,y)满足的条件和该问题的解的表达式。

半线性方程ROBIN问题的角层解(英文)

本文讨论了一类半线性方程Robin边值问题.利用微分不等式理论,研究了边值问题角层解的存在性和渐近性态.

椭圆方程Robin问题的第二基本估计

讨论椭圆方程的Robin问题-Δu+au=f,inΩ, u n+αu=0,on Ω,这里a≥0,α≥0,且至少有一个不恒为0.若Ω是光滑凸域,弱解u∈H2(Ω),证明了第二基本估计‖u‖2,Ω≤c‖f‖0,Ω.

非线性种群反应扩散系统奇摄动ROBIN问题(英文)

研究了一类具有非线性种群反应扩散系统奇摄动Robin初始边值问题.在适当的条件下,利用微分不等式理论,讨论了问题解的存在性和渐近性态.

具有多重解的非线性Robin问题的奇摄动(英文)

本文利用边界层法 ,研究了具有多重解的非线性Robin问题εx″+ f(t,x)x′+ g(t,x) =0 ,0 ≤t≤ 1 ,x′( 0 ,ε) -ax( 0 ,ε) =A ,x′( 1 ,ε) +bx( 1 ,ε) =B其中ε为正的小参数 .在适当的假设下 ,我们通过给出外部解展开式系数的一般表达式 ,得到了退化问题的边值为某方程的多重根时的渐近解 ,推广了有关结果 .