三阶奇摄动非线性边值问题

利用微分不等式理论,研究了某一类三阶奇摄动非线性边值问题。以二阶边值问题的已知结果为基础,引入Volterra型积分算子,建立了三阶非线性边值问题的上下解方法。在适当条件下,构造出具体的上下解,得出解的存在性和渐进估计。结果表明这种技巧也为三阶奇摄动边值问题的研究提出了崭新的思路。最后举例验证文中定理的正确性。     

具有共振的2n阶m点边值问题的可解性

对具有共振的高阶多点边值问题进行研究.首先在具有2n-1阶连续导数的函数全体所成的空间X的子集上定义了指数为0的Fredholm算子L,并在X上定义了投影算子P,使得算子L在其定义域和P的核的交集上是可逆的.然后,在Lebesgue可积函数全体所成的空间Y上定义了投影算子Q,使得L的逆与I-Q及非线性项f的复合是紧算子,其中,I是Y上的恒同算子.最后通过赋予f一定的增长条件,利用Mawhin的重合度理论,证明了具有共振的2n阶m点边值问题至少存在一个解,并给出一个例子验证这一结果.在这里不要求f具有连续性.     

Existence of Positive Solutions for a Nonlinear Second Order Periodic Boundary Value Problem

By using the first eigenvalue corresponding to the relevant linear operator and the topological degree theorem, sufficient conditions for the existence of positive solutions for a nonlinear second order periodic boundary value problem are given. Our results improve and generalize some preliminary works.     

Full discretization of a nonlinear parabolic problem containing volterra operators and an unknown dirichlet boundary condition

The reconstruction of an unknown solely time‐dependent Dirichlet boundary condition in a nonlinear parabolic problem containing a linear and a nonlinear Volterra operator is considered. The inverse problem is converted into a variational problem in which the unknown Dirichlet condition is eliminated using a given integral overdetermination. A time‐discrete recurrent approximation scheme is designed, using Backward Euler's method. The convergence of the approximations towards a solution of the variational problem is proved under appropriate assumptions on the data and on the Volterra operators. The uniqueness of this solution is shown in the case that the nonlinear Volterra operator satisfies a particular inequality. Moreover, the Finite Element Method is used to discretize the time‐discrete approximation scheme in space. Finally, full‐discrete error estimates are derived for a particular choice of the finite elements. The corresponding convergence rates are supported by a numerical experiment. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 1444–1460, 2015     

Boundary element methods for potential problems with nonlinear boundary conditions

Galerkin boundary element methods for the solution of novel first kind Steklov-Poincaré and hypersingular operator boundary integral equations with nonlinear perturbations are investigated to solve potential type problems in two- and three-dimensional Lipschitz domains with nonlinear boundary conditions. For the numerical solution of the resulting Newton iterate linear boundary integral equations, we propose practical variants of the Galerkin scheme and give corresponding error estimates. We also discuss the actual implementation process with suitable preconditioners and propose an optimal hybrid solution strategy.

     

Linear and nonlinear abstract equations with parameters

This paper focuses on nonlocal boundary value problems for linear and nonlinear abstract elliptic equations in Banach spaces. Here equations and boundary conditions contain certain parameters. The uniform separability of the linear problem and the existence and uniqueness of maximal regular solution of nonlinear problem are obtained in L_p spaces. For linear case the discreteness of spectrum of corresponding parameter dependent differential operator is obtained. The behavior of solution when the parameter approaches zero and its smoothness with respect to the parameter is established. Moreover, we show the estimate for analytic semigroups in terms of interpolation spaces. This fact can be used to obtain maximal regularity properties for abstract boundary value problems.     

On the coupled BEM and FEM for a nonlinear exterior Dirichlet problem in R2

Summary In this paper we apply the coupling of boundary integral and finite element methods to solve a nonlinear exterior Dirichlet problem in the plane. Specifically, the boundary value problem consists of a nonlinear second order elliptic equation in divergence form in a bounded inner region, and the Laplace equation in the corresponding unbounded exterior region, in addition to appropriate boundary and transmission conditions. The main feature of the coupling method utilized here consists in the reduction of the nonlinear exterior boundary value problem to an equivalent monotone operator equation. We provide sufficient conditions for the coefficients of the nonlinear elliptic equation from which existence, uniqueness and approximation results are established. Then, we consider the case where the corresponding operator is strongly monotone and Lipschitz-continuous, and derive asymptotic error estimates for a boundary-finite element solution. We prove the unique solvability of the discrete operator equations, and based on a Strang type abstract error estimate, we show the strong convergence of the approximated solutions. Moreover, under additional regularity assumptions on the solution of the continous operator equation, the asymptotic rate of convergenceO (h) is obtained.The first author's research was partly supported by the U.S. Army Research Office through the Mathematical Science Institute of Cornell University, by the Universidad de Concepción through the Facultad de Ciencias, Dirección de Investigación and Vicerretoria, and by FONDECYT-Chile through Project 91-386.     

二阶非线性椭圆型方程带位移的边值问题

In this paper, we consider the boundary value problem with the shift for nonlinear uniformly elliptic equations of second order in a multiply connected domain. For this sake, we propose a modified boundary value problem for nonlinear elliptic systems of first order equations, and give a priori estimates of solutions for the modified boundary value problem. Afterwards we prove by using the Schauder fixedpoint theorem that this boundary value problem with some conditions has a solution. The result obtained is the generlization of the corresponding theorem on the Poincare boundary value problem.     

Inverse nodal problem for differential pencils

An inverse nodal problem is studied for the diffusion operator with real-valued coefficients on a finite interval with Dirichlet boundary conditions. The oscillation of the eigenfunctions corresponding to large modulus eigenvalues is established and an asymptotic of the nodal points is obtained. The uniqueness theorem is proved and a constructive procedure for solving the inverse problem is given.     

Boundary augmented Lagrangian method for the Signorini problem

An augmented Lagrangian method, based on boundary variational formulations and fixed point method, is designed and analyzed for the Signorini problem of the Laplacian. Using the equivalence between Signorini boundary conditions and a fixed-point problem, we develop a new iterative algorithm that formulates the Signorini problem as a sequence of corresponding variational equations with the Steklov-Poincaré operator. Both theoretical results and numerical experiments show that the method presented is efficient.     

An initial boundary value problem for a pseudoparabolic equation with a nonlinear boundary condition

An initial boundary value problem for a quasilinear equation of pseudoparabolic type with a nonlinear boundary condition of the Neumann–Dirichlet type is investigated in this work. From a physical point of view, the initial boundary value problem considered here is a mathematical model of quasistationary processes in semiconductors and magnets, which takes into account a wide variety of physical factors. Many approximate methods are suitable for finding eigenvalues and eigenfunctions in problems where the boundary conditions are linear with respect to the desired function and its derivatives. Among these methods, the Galerkin method leads to the simplest calculations. On the basis of a priori estimates, we prove a local existence theorem and uniqueness for a weak generalized solution of the initial boundary value problem for the quasilinear pseudoparabolic equation. A special place in the theory of nonlinear equations is occupied by the study of unbounded solutions, or, as they are called in another way, blow-up regimes. Nonlinear evolutionary problems admitting unbounded solutions are globally unsolvable. In the article, sufficient conditions for the blow-up of a solution in a finite time in a limited area with a nonlinear Neumann–Dirichlet boundary condition are obtained.     

Boundary value problems for a quasilinear parabolic equation with an unknown coefficient

The work is connected with the mathematical modeling of physical–chemical processes in which inner characteristics of materials are subjected to changes. The considered nonlinear parabolic models consist of a boundary value problem for a quasilinear parabolic equation with an unknown coefficient multiplying the derivative with respect to time and, moreover, involve an additional relationship for a time dependence of this coefficient. For such a system, conditions of unique solvability in a class of smooth functions are studied on the basis of the Rothe method. The proposed approach involves the proof of a priori estimates in the difference-continuous Hölder spaces for the corresponding differential-difference nonlinear system that approximates the original system by the Rothe method. These estimates allow one to establish the existence of the smooth solutions and to obtain the error estimates of the approximate solutions.As examples of applications of the considered nonlinear boundary value problems, the models of destruction of heat-protective composite under the influence of high temperature heating are discussed.     

Application of the Factorization Method to Recover Cuts with Oblique Derivative Boundary Condition

Direct and inverse problems for the scattering of cracks with mixed oblique derivative boundary conditions from the incident plane wave are considered, which describe the scattering phenomenons such as the scattering of tidal waves by spits or reefs. The solvability of the direct scattering problem is proven by using the boundary integral equation method. In order to show the equivalent boundary integral system is Fredholm of index zero, some relationships concerning the tangential potential operator is used. Due to the mixed oblique derivative boundary conditions, we cannot employ the factorization method in a usual manner to reconstruct the cracks. An alternative technique is used in the theoretical analysis such that the far field operator can be factorized in an appropriate form and fulfills the range identity theorem. Finally, we present some numerical examples to demonstrate the feasibility and effectiveness of the factorization method.     

具有复杂边界条件的杆的振动分析

本文研究一端带有集中质量并支以弹簧另一端作支承运动的杆的纵向振动.由于这个问题的边界条件比较复杂,且要考虑阻尼,因此本文只求稳态周期解.首先分析线性系统;然后考虑材料非线性,用摄动法求具有非线性边界条件的非线性方程的近似解析解.     

Nonreflecting Boundary Conditions for Nonlinear Problems of Fluid Dynamics

External problems of fluid dynamics are modeled on artificially bounded regions. A method is proposed for applying nonreflecting boundary conditions to the linearized Euler equations obtained from the original nonlinear problem. The nonreflecting conditions are modified to allow for viscosity and nonhomogeneity of unperturbed background flow in linear Euler, Navier–Stokes, and quasi-fluid-dynamic models. One-dimensional test problems are computed in the linear and nonlinear cases.     

奇异二阶周期边值问题正解的存在性

利用不动点指数理论研究奇异二阶周期边值问题.在有关其线性算子方程对应的第一特征值的条件下得到边值问题正解的存在性,推广和改进了最近文献的结果.     

Numerical Solution of the Inverse Nonequilibrium Sorption Problem with a Time-Dependent Boundary Condition

We consider the quasi-linear problem of nonequilibrium sorption dynamics with external-diffusion kinetics and a boundary condition that contains the time derivative of a solution component. A numerical method is proposed for describing the inverse problem to recover the nonlinear parameter of the system of differential equations—the inverse of the sorption isotherm. Convergence of the difference scheme for the direct problem is proved. Numerical solutions of both the direct and the inverse problem are obtained for various parameter values.     

Positive Solutions to a Fourth Order Boundary Value Problem

The authors consider a fourth order two-point boundary value problem. Some a priori estimates to positive solutions for the problem are obtained. Explicit sufficient conditions for existence and nonexistence of positive solutions to the problem are established. An example to illustrate the results is included.     

具有滑动边界条件Stokes问题的自适应Uzawa块松弛算法

对一类具有非线性滑动边界条件的Stokes问题,得到了求其数值解的自适应Uzawa块松弛算法（SUBRM）.通过该问题导出的变分问题,引入辅助变量将原问题转化为一个基于增广Lagrange函数表示的鞍点问题,并采用Uzawa块松弛算法（UBRM）求解.为了提高算法性能,提出利用迭代函数自动选取合适罚参数的自适应法则.该算法的优点是每次迭代只需计算一个线性问题,同时显式计算辅助变量.对算法的收敛性进行了理论分析,最后用数值结果验证了该算法的可行性和有效性.     

Inverse Scattering Problems for Sturm–Liouville Operators with Spectral Parameter Dependent on Boundary Conditions

In this paper, we consider the inverse scattering problem for the Sturm–Liouville operator on the half-line [0,∞) with Herglotz function of spectral parameter in the boundary condition. The scattering data of the problem is defined, and its properties are investigated. The main equation is obtained for the solution of the inverse problem and it is shown that the potential is uniquely recovered in terms of the scattering data.