Showing papers on "Free boundary problem published in 1998"

Positive Solutions of Differential, Difference and Integral Equations

Abstract: Preface. Ordinary Differential Equations. 1. First Order Initial Value Problems. 2. Second Order Initial Value Problems. 3. Positone Boundary Value Problems. 4. Semi-positone Boundary Value Problems. 5. Semi-Infinite Interval Problems. 6. Mixed Boundary Value Problems. 7. Singular Boundary Value Problems. 8. General Singular and Nonsingular Boundary Value Problems. 9. Quasilinear Boundary Value Problems. 10. Delay Boundary Value Problems. 11. Coupled System of Boundary Value Problems. 12. Higher Order Sturm-Liouville Boundary Value Problems. 13. (n,p) Boundary Value Problems. 14. Focal Boundary Value Problems. 15. General Focal Boundary Value Problems. 16. Conjugate Boundary Value Problems. Difference Equations. 17. Discrete Second Order Boundary Value Problems. 18. Discrete Higher Order Sturm-Liouville Boundary Value Problems. 19. Discrete (n,p) Boundary Value Problems. 20. Discrete Focal Boundary Value Problems. 21. Discrete Conjugate Boundary Value Problems. Integral and Integrodifferential Equations. 22. Volterra Integral Equations. 23. Hammerstein Integral Equations. 24. First Order Integrodifferential Equations. References. Authors Index. Subject Index.

A meshless local boundary integral equation (LBIE) method for solving nonlinear problems

Abstract: A new meshless method for solving nonlinear boundary value problems, based on the local boundary integral equation (LBIE) method and the moving least squares approximation, is proposed in the present paper. The total formulation and a rate formulation are developed for the implementation of the present method. The present method does not need domain and boundary elements to deal with the volume and boundary integrals, which will cause some difficulties for the conventional boundary element method (BEM) or the field/boundary element method (FBEM), as the volume integrals are inevitable in dealing with nonlinear boundary value problems. This is the same for the element free Galerkin (EFG) method which also needs element-like cells in the entire domain to evaluate volume integrals. The “companion fundamental solution” introduced in Zhu, Zhang and Atluri (1998) is used so that no derivatives of the shape functions are needed to construct the stiffness matrix for the interior nodes, as well as for those nodes with no parts of their local boundaries coinciding with the global boundary of the domain of the problem, where essential boundary conditions are specified. It is shown that the satisfaction of the essential as well as natural boundary conditions is quite simple, and algorithmically very efficient, in the present nonlinear LBIE approach. Numerical examples are presented for several problems, for which exact solutions are available. The present method converges fast to the final solution with reasonably accurate results for both the unknown variable and its derivatives. No post processing procedure is required to compute the derivatives of the unknown variable (as in the conventional FBEM), since the solution from the present method, using the moving least squares approximation, is already smooth enough. The numerical results in these examples show that high rates of convergence for the Sobolev norms ∥·∥0 and ∥·∥1 are achievable, and that the values of the unknown variable and its derivatives are quite accurate.

Convergence of the phase field model to its sharp interface limits

Abstract: We consider the distinguished limits of the phase field equations and prove that the corresponding free boundary problem is attained in each case. These include the classical Stefan model, the surface tension model (with or without kinetics), the surface tension model with zero specific heat, the two phase Hele–Shaw, or quasi-static, model. The Hele–Shaw model is also a limit of the Cahn–Hilliard equation, which is itself a limit of the phase field equations. Also included in the distinguished limits is the motion by mean curvature model that is a limit of the Allen–Cahn equation, which can in turn be attained from the phase field equations.

Numerically Absorbing Boundary Conditions for Quantum Evolution Equations

Abstract: Transparent boundary conditions for the transient Schrodinger equation on a domain Ω can be derived explicitly under the assumption that the given potential V is constant outside of this domain. In 1D these boundary conditions are non-local in time (of memory type). For the Crank-Nicolson finite difference scheme, discrete transparent boundary conditions are derived, and the resulting scheme is proved to be unconditionally stable. A numerical example illustrates the superiority of discrete transparent boundary conditions over existing ad-hoc discretizations of the differential transparent boundary conditions. As an application of these boundary conditions to the modeling of quantum devices, a transient 1D scattering model for mixed quantum states is presented.

Boundary Value Problems and Optimal Boundary Control for the Navier--Stokes System: the Two-Dimensional Case

Abstract: We study optimal boundary control problems for the two-dimensional Navier--Stokes equations in an unbounded domain. Control is effected through the Dirichlet boundary condition and is sought in a subset of the trace space of velocity fields with minimal regularity satisfying the energy estimates. An objective of interest is the drag functional. We first establish three important results for inhomogeneous boundary value problems for the Navier--Stokes equations; namely, we identify the trace space for the velocity fields possessing finite energy, we prove the existence of a solution for the Navier--Stokes equations with boundary data belonging to the trace space, and we identify the space in which the stress vector (along the boundary) of admissible solutions is well defined. Then, we prove the existence of an optimal solution over the control set. Finally, we justify the use of Lagrange multiplier principles, derive an optimality system of equations in the weak sense from which optimal states and controls may be determined, and prove that the optimality system of equations satisfies in appropriate senses a system of partial differential equations with boundary values.

The determination of a discontinuity in a conductivity from a single boundary measurement

Abstract: We consider the determination of the interior domain where D is characterized by a different conductivity from the surrounding medium. This amounts to solving the inverse problem of recovering the piecewise constant conductivity in from boundary data consisting of Cauchy data on the boundary of the exterior domain . We will compute the derivative of the map from the domain D to this data and use this to obtain both qualitative and quantitative measures of the solution of the inverse problem.

A uniqueness result for scattering by infinite rough surfaces

Abstract: Consider the Dirichlet boundary value problem for the Helmholtz equation in a non-locally perturbed half-plane with an unbounded, piecewise Lyapunov boundary. This problem models time-harmonic electromagnetic scattering in transverse magnetic polarization by one-dimensional rough, perfectly conducting surfaces. A radiation condition is introduced for the problem, which is a generalization of the usual one used in the study of diffraction by gratings when the solution is quasi-periodic, and allows a variety of incident fields including an incident plane wave to be included in the results obtained. We show in this paper that the boundary value problem for the scattered field has at most one solution. For the case when the whole boundary is Lyapunov and is a small perturbation of a flat boundary we also prove existence of solution and show a limiting absorption principle.

The Nonlinear Schrodinger Equation on the Half Line

Abstract: The nonlinear Schrodinger equation on the half line with mixed boundary condition is investigated. After a brief introduction to the corresponding classical boundary value problem, the exact second quantized solution of the system is constructed. The construction is based on a new algebraic structure, which is called in what follows boundary algebra and which substitutes, in the presence of boundaries, the familiar Zamolodchikov-Faddeev algebra. The fundamental quantum field theory properties of the solution are established and discussed in detail. The relative scattering operator is derived in the Haag-Ruelle framework, suitably generalized to the case of broken translation invariance in space.

The Cauchy problem and the initial boundary value problem in numerical relativity

Abstract: Some simple ideas on numerical tractability are introduced, and are used to determine under what conditions unconstrained evolution of Einstein's field equations for the Cauchy problem is possible. The initial boundary value problem for Einstein's equations is discussed and a precise description of what boundary data are needed is given. Unconstrained evolution is numerically tractable if and only if the boundary conditions respect the momentum constraints.

Partial Differential Equations and Boundary Value Problems

Abstract: Notations. Preface. 1. Preliminaries. 2. Elliptic Boundary Value Problems. 3. Elliptic Problems in Sobolev Spaces. 4. The Heat Equation. 5. The Wave Equation. Index.

On the numerical solution of an inverse boundary value problem for the heat equation

Abstract: We consider the inverse problem of reconstructing the interior boundary curve of an arbitrary-shaped annulus from overdetermined Cauchy data on the exterior boundary curve. For the approximate solution of this ill-posed and nonlinear problem we propose a regularized Newton method based on a boundary integral equation approach for the initial boundary value problem for the heat equation. A theoretical foundation for this Newton method is given by establishing the differentiability of the initial boundary value problem with respect to the interior boundary curve in the sense of a domain derivative. Numerical examples indicate the feasibility of our method.

A nonlinear elliptic boundary value problem related to corrosion modeling

Abstract: We study a nonlinear boundary value problem arising from electrochemistry. The essential difficulties are due to the strong nonlinear nature of part of the boundary condition. This part of the boundary condition is of an exponential type and is normally in the corrosion literature associated with the names of Butler and Volmer. We examine the questions of existence and uniqueness of solutions to this boundary value problem. In a numerical example we compare the behaviour of the solutions to the nonlinear problem with the behaviour of the solutions to a corresponding linearized problem. In contrast to earlier studies we put a major emphasis on studying parameter values that may be relevant for the case in which part of the boundary is in a transition to passivity—in practice most likely because it is nearly covered by an oxide layer. 0. Introduction. In this paper we shall study a nonlinear boundary value problem arising from galvanic corrosion. Before proceeding to a mathematical analysis of this problem we shall devote this first section to a brief explanation of some background material. Consider the battery cell illustrated in Fig. 1 (see p. 480). It consists of two strips of metal (one of silver and one of zinc) partially emerged in a saltwater solution. It is well known that chemical reactions at each metal strip will result in a current passing between the two. At the silver strip, hydrogen gas is produced, 2H20 + 2e~ -> H2 T +20H\". Here the reactant gains electrons, i.e., one says that a reduction takes place, and the strip is defined to be cathodic. Simultaneously, at the zinc strip, zinc is dissolved into the solution, Zn —> Zn2+ +2e~. The reactant loses electrons, an oxidation takes place, and the strip is defined to be anodic. In electrochemical cells, oxidation and reduction always occur simultaneously. As a result of the chemical reactions, the metal strips change shape. In the voltaic cell Received December 1, 1995. 1991 Mathematics Subject Classification. Primary 35J20, 35J65, 73M99. 479 ©1998 Brown University 480 MICHAEL VOGELIUS and .IIAN-MING XU Fig. 2. A cell formed by rust on iron illustrated above, the driving force for the oxidation-reduction reactions is the chemical potential difference between silver and zinc, and the chemical reactions produce a current. One may conversely use current generated by an outside power source to cause chemical reactions (involving electron transfer) at the metal surfaces. This alternate process is called electrolysis. The quantitative relationship between the electric current and the mass of substances consumed and produced by the chemical reactions has been studied intensively going back to the early work of Faraday. Today, the use of electrochemical reactions plays a significant role in industry. Aside from electrochemical energy conversion, examples can be found in the production of chlorine, aluminum, and other chemicals, and in electroplating and electromachining. On the other hand, electrochemical phenomena also cause great damage, primarily in the form of corrosion. Corrosion is defined as the deterioration of a metal by chemical or electrochemical reaction with its environment. Most corrosion processes are electrochemical. In aqueous media, the action is similar to that taking place in the battery cell shown in Fig. 1, where the zinc electrode is corroded, i.e., metallic zinc is converted into hydrated zinc ions Zn\"+. Figure 2 illustrates a local cell on a piece of iron when it becomes rusted [16]. It is predominantly from the point of view of corrosion detection and corrosion control that we are interested in the problems associated with electrochemistry. To justify the NONLINEAR ELLIPTIC BOUNDARY VALUE PROBLEM 481 commercial interest in these subjects it may be relevant to mention that a study conducted by the National Institute of Standards and Technology, under the request of the U.S. Congress, showed that the effect of corrosion cost the U.S. over 200 billion dollars in 1989, which was approximately 4.2% of the U.S. gross national product of that year [2]The idealized electrochemical system we consider consists of a domain 0, containing an ionic solution (electrolyte). A part of the boundary of fl is electrochemically active, the rest is electrochemically inactive (but may transmit an imposed current). For simplicity we restrict our attention to two-dimensional domains. We emphasize that the entire boundary is in contact with the same bulk solution, i.e., there are no membranes involved. One may imagine that outer circuits carry flows of electrons to and from the electrochemically active boundary parts, but exactly how this is done technically is of no importance for the analysis that we present here. Our main focus is to study the mathematical characteristics of a boundary value problem which incorporates a quite realistic model for the currents on the electrochemically active boundary parts. Since Faraday's law states that the rate of change of the boundary at any point is proportional to the normal current flux across the boundary at the point [5], a realistic modeling of the boundary fluxes thus indirectly provides a good understanding of the shape changes (corrosion) of the boundary. We shall work entirely within the so-called potential model; we refer to [5], [12], [17] and the references therein, for a detailed derivation and justification of this model. Briefly stated the transport processes in the electrolytic solution are modeled by a dilute solution theory, in which the concentrations of the various species of the electrolytic solution are considered spatially constant away from the boundary. In this fashion one obtains a single elliptic equation for the electric potential, , V • (—«V(^>) = 0 in fl, (1) where k, the conductivity of the solution, is given by k = F2^zfuiCi. (2) i In (2), Zi is the charge number for the species i, Ui is its mobility, and a its concentration. Since the concentrations c, are assumed to be constant, so is the conductivity k. Equation (1) therefore simply expresses that cf) is harmonic in fl. Let us now take a closer look at the interface between the electrochemically active boundary part and the electrolyte, where the oxidation-reduction reactions occur. As an aide to understanding the physical situation we consider for a moment a single strip of metal (anodic, say) placed in an electrolytic solution. After some period of time a state of equilibrium has been reached, at which point there is a discontinuity in the electrical potential across the metal-electrolyte interface (free electrons have gathered on the metal strip, metal ions have been released into the solution). The exact value of this potential jump depends on the specific metal and the specific electrolyte. Note that no net current is flowing between the metal and the electrolyte as a result of this potential jump. Since the difference (metai '/'solution) is not directly amenable to measurements, one introduces a reference device, such as a normal hydrogen electrode (NHE), to make a 482 MICHAEL VOGELIUS and JIAN-MING XU Active / \\ Transition

Black hole boundary conditions and coordinate conditions

Abstract: This paper treats boundary conditions on black hole horizons for the full (3+1)D Einstein equations. Following a number of authors, the apparent horizon is employed as the inner boundary on a space slice. It is emphasized that a further condition is necessary for the system to be well posed; the ``prescribed curvature conditions'' are therefore proposed to complete the coordinate conditions at the black hole. These conditions lead to a system of two 2D elliptic differential equations on the inner boundary surface, which coexist nicely to the 3D equation for maximal slicing (or related slicing conditions). The overall 2D-3D system is argued to be well posed and globally well behaved. The importance of ``boundary conditions without boundary values'' is emphasized. This paper is the first of a series.

Chaotic Vibrations of the One-Dimensional Wave Equation Due to a Self-Excitation Boundary Condition.

Abstract: Consider the initial-boundary value problem of the linear wave equation wtt-wxx=0 on an interval. The boundary condition at the left endpoint is linear homogeneous, injecting energy into the system, while the boundary condition at the right endpoint has cubic nonlinearity of a van der Pol type. We show that the interactions of these linear and nonlinear boundary conditions can cause chaos to the Riemann invariants (u,v) of the wave equation when the parameters enter a certain regime. Period-doubling routes to chaos and homoclinic orbits are established. We further show that when the initial data are smooth satisfying certain compatibility conditions at the boundary points, the space-time trajectory or the state of the wave equation, which satisfies another type of the van der Pol boundary condition, can be chaotic. Numerical simulations are also illustrated.

Approximation of boundary conditions for mimetic finite-difference methods

Abstract: The numerical solution of partial differential equations solved with finite-difference approximations that mimic the symmetry properties of the continuum differential operators and satisfy discrete versions of the appropriate integral identities are more likely to produce physically faithful results. Furthermore, those properties are often needed when using the energy method to prove convergence and stability of a particular difference approximation. Unless special care is taken, mimetic difference approximations derived for the interior grid points will fail to preserve the symmetries and identities between the gradient, curl, and divergence operators at the computational boundary. In this paper, we describe how to incorporate boundary conditions into finite-difference methods so the resulting approximations mimic the identities for the differential operators of vector and tensor calculus. The approach is valid for a wide class of partial differential equations of mathematical physics and will be described for Poisson's equation with Dirichlet, Neumann, and Robin boundary conditions. We prove that the resulting difference approximation is symmetric and positive definite for each of these boundary conditions.

Existence and multiplicity of solutions to a p-Laplacian equation with nonlinear boundary condition

Abstract: We study the nonlinear elliptic boundary value problem Au= f(x;u )i n ; Bu=g(x;u )o n @ ; where Ais an operator of p Laplacian type, is an unbounded domain in R N withnon-compactboundary,and fand garesubcritical nonlinearities. We show existence of a nontrivial nonnegative weak solution when both f and g are superlinear. Also we show existence of at least two nonnegative solutions when one of the two functions f, gis sublinear and the other one superlinear. The proofs are based on variational methods applied to weighted function spaces.

Boundary value methods: The third way between linear multistep and Runge-Kutta methods

Abstract: It is well known that the approximation of the solutions of ODEs by means of k-step methods transforms a first-order continuous problem in a kth-order discrete one. Such transformation has the undesired effect of introducing spurious, or parasitic, solutions to be kept under control. It is such control which is responsible of the main drawbacks (e.g., the two Dahlquist barriers) of the classical LMF with respect to Runge-Kutta methods. It is, however, less known that the control of the parasitic solutions is much easier if the problem is transformed into an almost equivalent boundary value problem. Starting from such an idea, a new class of multistep methods, called Boundary Value Methods (BVMs), has been proposed and analyzed in the last few years. Of course, they are free of barriers. Moreover, a block version of such methods presents some similarity with Runge-Kutta schemes, although still maintaining the advantages of being linear methods. In this paper, the recent results on the subject are reviewed.

Homogenization of Boundary-Value Problem in a Locally Periodic Perforated Domain

Abstract: We consider a model homogenization problem for the Poisson equation in a locally periodic perforated domain with the smooth exterior boundary, the Fourier boundary condition being posed on the boundary of the holes. In the paper we construct the leading terms of formal asymptotic expansion. Then, we justify the asymptotics obtained and estimate the residual.