Showing papers on "Neumann boundary condition published in 1988"

On convergence of block-centered finite differences for elliptic-problems

Abstract: We consider linear, selfadjoint, elliptic problems with Neumann boundary conditions in rectangular domains. We demonstrate that with sufficiently smooth data, the discrete $L^2 $-norm errors for tensor product block-centered finite differences in both the approximate solution and its first derivatives are second-order for all nonuniform grids. Extensions to nonselfadjoint and parabolic problems are discussed.

Algebraic Riccati equations with non-smoothing observation arising in hyperbolic and Euler-Bernoulli boundary control problems

Abstract: This paper considers the optimal quadratic cost problem (regulator problem) for a class of abstract differential equations with unbounded operators which, under the same unified framework, model in particular «concrete» boundary control problems for partial differential equations defined on a bounded open domain of any dimension, including: second order hyperbolic scalar equations with control in the Dirichlet or in the Neumann boundary conditions; first order hyperbolic systems with boundary control; and Euler-Bernoulli (plate) equations with (for instance) control(s) in the Dirichlet and/or Neumann boundary conditions. The observation operator in the quadratic cost functional is assumed to be non-smoothing (in particular, it may be the identity operator), a case which introduces technical difficulties due to the low regularity of the solutions. The paper studies existence and uniqueness of the resulting algebraic (operator) Riccati equation, as well as the relationship between exact controllability and the property that the Riccati operator be an isomorphism, a distinctive feature of the dynamics in question (emphatically not true for, say, parabolic boundary control problems). This isomorphism allows one to introduce a «dual» Riccati equation, corresponding to a «dual» optimal control problem. Properties between the original and the «dual» problem are also investigated.

Existence and stability of transition layers

Abstract: For a second order nonautonomous singularly perturbed ordinary differential equation with Neumann boundary conditions, the existence of single transition layer solutions is proved by using the method of Liapunov-Schmidt The method also gives the stability of these solutions as an equilibrium point of a parabolic equation

Existence and uniqueness of solutions of nonlinear Neumann problems

Abstract: Soit D⊂R N un domaine connexe borne lisse et on suppose qu'en chaque point x∈∂D la normale exterieure n est bien definie. Soit a(•) une fonction arbitraire continue de Holder dans D. On etudie les solutions positives du probleme de Neumann non lineaire: Δu+a(x)g(u)=0 dans D, ∂u/∂n=0 sur ∂D, ou g:R + →R est une fonction de classe Cα(R)∩C 1 (R + ), λ∈(0,1)

Asymptotic behavior of solutions of semilinear heat equations on S1

Abstract: We study the dynamical behavior of the initial value problem for the equation u t = u xx + f(u, u x ), x ∈ S 1 =R/Z, t > 0. One of our main results states that any C 1-bounded solution approaches either a single periodic solution or a set of equilibria as t → ∞. We also consider the case where the solution blows up in a finite time and prove that under certain conditions on f the blow-up set of any solution with nonconstant initial data is a finite set.

Non-homogeneous Neumann problems in domains with small holes

Abstract: On etudie le comportement limite des solutions de problemes de Neumann non homogenes dans des ouverts finement perfores. Des estimations a priori detaillees exprimees en fonction des parametres e et r(e) des trous donnent l'ordre de grandeur exact de la norme H 1 des solutions

Exact controllability for wave equation with Neumann boundary control

Abstract: the author presented a summary of very recent results on exact controllability for the wave equation under boundary control exercised either in the Dirichlet or else in the Neumann boundary conditions. For lack of space, the present paper deals exclusively with the Neumann case, while for the Dirichlet case reference is made to [T.1].

On a Class of Functionals Invariant under a Zn Action.

Abstract: : This document considers a system of ordinary differential equations of the form (*) q + V sub q (t,q) = f(t) where f and V are periodic in t, V is periodic in the components of q = (q sub 1,..., q sub m), and the mean value of f vanishes. By showing that a corresponding functional is invariant under a natural Z sub n action, a simple variational argument yields at least n + 1 distinct periodic solutions of (*). More general versions of (*) are also treated as is a class of Neumann problems for semilinear elliptic partial differential equations.

Non‐linear heat conduction in composite bodies: A boundary element formulation

Abstract: The present paper discusses the numerical solution of steady-state non-linear heat conduction problems in composite bodies by using the boundary element method. Two kinds of non-linearities are considered: the temperature dependence of the thermal conductivity and boundary conditions of the radiative type. By introducing the integral of conductivity as a new variable the governing equation of the problem becomes linear in the transform space. Transformed boundary conditions of the Dirichlet and Neumann types are also linear but convective boundary conditions become non-linear. Also, discontinuities arise in the value of the integral of conductivity across the interface between materials with different properties since continuity of temperature is imposed. The problem is numerically solved by discretizing the external and interface boundaries of the region under consideration with constant boundary elements and applying an iterative scheme of the Newton–Raphson type.

On the boundary element method for some nonlinear boundary value problems

Abstract: Here we analyse the boundary element Galerkin method for two-dimensional nonlinear boundary value problems governed by the Laplacian in an interior (or exterior) domain and by highly nonlinear boundary conditions. The underlying boundary integral operator here can be decomposed into the sum of a monotoneous Hammerstein operator and a compact mapping. We show stability and convergence by using Leray-Schauder fixed-point arguments due to Petryshyn and Necas.

Hamiltonian Systems with Eigenvalue Depending Boundary Conditions

Abstract: In earlier papers [DLS1–6] we have described the selfadjoint extensions, in indefinite inner product spaces and with nonempty resolvent sets of a symmetric closed relation S in a Hilbert space ℌ by means of generalized resolvents, characteristic functions and Straus extensions. In this paper we show how these results can be applied when S comes from a 2n×2n Hamiltonian system of ordinary differential equations on an interval [a, b), (1.1) Jy′(t) = (lΔ(t) + H(t)) y(t) + Δ(t)f(t), t ∈[a, b), l∈ℂ, which is regular in a and in the limit point case in b; for further specifications see Section 5. We pay special attention to selfadjoint extensions beyond the given space ℌ, as they give rise to eigenvalue and boundary value problems with boundary conditions of the form (1.2) A(l)y 1 (a) + B(l)y 2 (a) = 0, in which the matrix coefficients A(l) and В(l) depend holomorphically on the eigenvalue parameter l, see Theorem 7.1 below. The eigenvalue problem for such a selfadjoint extension in a larger space can be considered as a linearization of the corresponding boundary value problem (1.1) and (1.2).

Solution of the advection‐dispersion equation with free exit boundary

Abstract: I investigated the exit boundary condition for the advection-dispersion equation and found that in numerical solutions of this equation, using Galekin finite elements, a free exit boundary condition requiring no a priori information is possible, provided the advective component in the numerical equations is of sufficient magnitude relative to the dispersive component. Since the relationship between these two components is controlled by the spatial discretization through the grid Peclet number, the free exit boundary condition can in fact be applied whenever there is a non-zero advective component. The numerical solution in a finite domain with free exit boundary, using a correctly proportioned spatial discrezation, behaves like an infinite-domain solution.

Boundary control of distributed parabolic system with boundary condition involving a time-varying lag

Abstract: An optimal boundary control problem for a distributed-parameter system with a boundary condition involving a time-varying lag is solved. The time horizon is fixed. Making use of the Lions scheme, necessary and sufficient conditions of optimality for the Neumann problem with the quadratic performance functional and constrained control are derived.

Shape sensitivity analysis of boundary optimal control problems for parabolic systems

Abstract: This paper is concerned with the shape sensitivity analysis of a class of boundary control, constrained, optimal control problems for parabolic systems. The notion of Euler and Lagrange derivatives of a boundary optimal control in the direction of a vector field is introduced. The derivatives are obtained in the form of optimal solutions of auxiliary optimal control problems. The method of sensitivity analysis used in this paper is based on related results on the differential stability of metric projections in Hilbert space onto a convex, closed subset, combined with the material derivative method of shape sensitivity analysis. Parabolic initial-boundary value problems with Dirichlet and Neumann boundary conditions are considered in this paper.

A posteriori error estimate for adaptive finite element mesh generation

Abstract: A method for deriving error estimates for adaptive mesh refinement is presented. It is based on the interelement boundary conditions, namely, the continuity of the normal components of the flux density and the tangential components of the magnetic field intensity. The elements that violate the conditions significantly are considered to have large local errors and are refined. Applications to magnetostatic problems involving Laplace's equation with Dirichlet and Neumann boundary conditions in two-dimensional linear domains are presented. It is shown that an extremely high rate of convergence is obtained. >

Global analytic hypoellipticity of the ?-Neumann problem on circular domains.

Abstract: In this paper we show that if $$D \subseteq \mathbb{C}^n ,n \geqq 2$$ , is a smooth bounded pseudoconvex circular domain with real analytic defining functionr(z) such that $$\sum\limits_{k = 1}^n {z_k \frac{{\partial r}}{{\partial z_k }}} e 0$$ for allz near the boundary, then the solutionu to the $$\bar \partial$$ -Neumann problem, $$square u = (\bar \partial \bar \partial * + \bar \partial *\bar \partial )u = f,$$ is real analytic up to the boundary, if the given formf is real analytic up to the boundary. In particular, if $$D \subseteq \mathbb{C}^n ,n \geqq 2$$ , is a smooth bounded complete Reinhardt pseudoconvex domain with real analytic boundary. Then ▭ is analytic hypoelliptic.

Constriction resistance of circular contacts on coated surfaces - Effect of boundary conditions

Abstract: The thermal constriction resistance of a circular contact spot on a coated half-space is developed for both heat flux and temperature-specified boundary conditions on the contact. Solutions are obtained with the Hankel transform method for flux-specified contacts and with a novel technique of linear superposition for the mixed boundary value problem created by an isothermal contact. A comparison of the results obtained shows that the thermal constriction resistance, which is based on average contact temperature, is insensitive to the contact boundary condition for most practical purposes.

Estimates for the ∂̄-Neumann problem for pseudoconvex domains in C2 of finite type

Abstract: We outline results obtained for the ∂-Neumann problem for an arbitrary pseudoconvex domain in C2 of finite type. We obtain an approximation to the Neumann operator. A number of sharp estimates for the solution of ∂u = f are a consequence; one of these is an extension of the L1 estimate of Henkin and Skoda used to characterize the zero sets of functions of the Nevanlinna class.

Combined integral equation formulation and null‐field method for the exterior acoustic problem

Abstract: The numerical implementation of a combined integral equation and null‐field method used to solve the exterior Neumann problem [D. S. Jones, Q. J. Mech. Appl. Math. XXVII, 129 (1974)] is presented here. The exterior Helmholtz integral equation is solved on the radiating or scattering surface, and the irregular frequencies are eliminated up to a given irregular frequency fM through the use of M additional null‐field equations. An impedance matrix, defined on the object surface, is then obtained that can be used as an exact radiation condition in a finite‐element code. The program and the numerical examples presented here are specialized to axisymmetrical problems. A purely null‐field method is implemented and simple rules are defined that display its failure when applied to high aspect‐ratio objects and (or) in the high‐frequency range. Similar, but less restricting rules are used to specify the numerical limitations of Jones’ technique. Besides, a few theoretical considerations clarify the role played by t...

Lower semicontinuity in domain optimization problems

Abstract: In the present paper, the lower semicontinuity of certain classes of functionals is studied when the domain of integration, which defines the functionals, is not fixed. For this purpose, a certain class of domains introduced by Chenais is employed. For this class of domains, a basic lemma is proved that plays an essential role in the derivations of the lower-semicontinuity theorems. These theorems are applied to the study of the existence of the optimal domain in domain optimization problems; a boundary-value problem of Neumann type or Dirichlet type is the main constraint in these optimization problems.

Constrained integral method for solving moving boundary problems

Abstract: In solving a moving boundary problem by the conventional integral method, the various parameters in the choice of a temperature/concentration profile are expressed in terms of the position of the moving boundary only, while in the present method they are expressed as functions of the position of the moving boundary plus an additional parameter at the fixed surface. This new parameter is taken to be the space derivative, when a Dirichlet condition is prescribed at the fixed end, or to be the function value when a Neumann-type condition is prescribed there. By doing so a control is provided at both ends, i.e. the moving boundary as well as the fixed end. Finally two simultaneous first-order differential equations are obtained which give the position of the moving boundary and the value of the unknown additional parameter in an implicit manner. Two sample problems with different types of boundary conditions at the fixed end are considered for testing the suggested method. The results seem to be in very good agreement with those due to earlier authors who have solved the problem using other techniques.

Scattering of a wave at a periodic boundary: analytical expression for the surface impedance

Abstract: A surface-impedance boundary condition is obtained from first principles for problems involving the scattering of two-dimensional waves by cylindrical periodical surfaces of arbitrary shape. The analysis is performed in the context of the electromagnetic theory of gratings, but it is also applicable to other physical situations, leading to the solution of a two-dimensional Helmholtz equation with high values for index of refraction. The boundary condition deduced here is shown to be analogous to the one suggested by Leontovich for quasi-planar boundaries if Z0 the quantity relating the field and its normal derivative at the boundary and depending only on the constitutive properties of the medium, is replaced by another quantity Z, which also depends on the local curvature of the surface. and on the polarization of the external fields; Z0 is the zero-order term in the expansion of Z in terms of the curvature.

Localization and boundedness of the solutions of the Neumann problem for a filtration equation

Abstract: On etudie l'existence et l'unicite des solutions faibles du probleme de Neumann u t =(α(u)) xx dans D T , u(x, 0)=u 0 (x) sur [0, +∞], (α(u)) x (0, t)=−h(t) sur [0, T], ici α∈C 1 (R)∩C 2 (R−{0}) avec α'(s)>0 si s¬=0, pour T>0, D T =[0, +∞)×[0, T], on suppose que h: [0, T]→R est bornee sur des sous ensembles compacts de [0, T] et u 0 ∈L 1 ([0, ∞]) est telle que (u 0 ) x est bornee sur [0, +∞)