Showing papers on "Free boundary problem published in 1994"

Boundary S matrix and boundary state in two-dimensional integrable quantum field theory

Abstract: We study integrals of motion and factorizable S matrices in two-dimensional integrable field theory with boundary. We propose the "boundary cross-unitarity equation," which is the boundary analog of the crossing-symmetry condition of the "bulk" S matrix. We derive the boundary S matrices for the Ising field theory with boundary magnetic field and for the boundary sine–Gordon model.

Global classical solutions for quasilinear hyperbolic systems

Abstract: Cauchy Problem for Single First Order Equations Cauchy Problem for Reducible Quasilinear Hyperbolic Systems Cauchy Problem for General Quasilinear Hyperbolic Systems Cauchy Problem for Quasilinear Hyperbolic Systems with Dissipation Mixed Initial-Boundary Value Problem with Boundary Dissipation for Quasilinear Hyperbolic Systems Typical Boundary Value Problem and Typical Free Boundary Problem for Reducible Quasilinear Hyperbolic Systems Generalized Riemann Problem for the System of One-Dimensional Isentropic Flow Typical Free Boundary Problem and Generalized Riemann Problem for General Quasilinear Hyperbolic Systems Bibliography Index.

Frechet differentiability of boundary integral operators in inverse acoustic scattering

Abstract: Using integral equation methods to solve the time-harmonic acoustic scattering problem with Dirichlet boundary conditions, it is possible to reduce the solution of the scattering problem to the solution of a boundary integral equation of the second kind. We show the Frechet differentiability of the boundary integral operators which occur. We then use this to prove the Frechet differentiability of the scattered field with respect to the boundary. Finally we characterize the Frechet derivative of the scattered field by a boundary value problem with Dirichlet conditions, in an analogous way to that used by Firsch.

Pattern formation in generalized Turing systems I. Steady-state patterns in systems with mixed boundary conditions

Abstract: Turing's model of pattern formation has been extensively studied analytically and numerically, and there is recent experimental evidence that it may apply in certain chemical systems. The model is based on the assumption that all reacting species obey the same type of boundary condition pointwise on the boundary. We call these scalar boundary conditions. Here we study mixed or nonscalar boundary conditions, under which different species satisfy different boundary conditions at any point on the boundary, and show that qualitatively new phenomena arise in this case. For example, we show that there may be multiple solutions at arbitrarily small lengths under mixed boundary conditions, whereas the solution is unique under homogeneous scalar boundary conditions. Moreover, even when the same solution exists under scalar and mixed boundary conditions, its stability may be different in the two cases. We also show that mixed boundary conditions can reduce the sensitivity of patterns to domain changes.

Nonlinear oscillations of pendant drops

Abstract: Whereas oscillations of free drops have been scrutinized for over a century, oscillations of supported (pendant or sessile) drops have only received limited attention to date. Here, the focus is on the axisymmetric, free oscillations of arbitrary amplitude of a viscous liquid drop of fixed volume V that is pendant from a solid rod of radius R and is surrounded by a dynamically inactive ambient gas. This nonlinear free boundary problem is solved by a method of lines using Galerkin/finite element analysis for discretization in space and an implicit, adaptive finite difference technique for discretization in time. The dynamics of such nonlinear oscillations are governed by four dimensionless groups: (1) a Reynolds number Re, (2) a gravitational Bond number G, (3) dimensionless drop volume V/R3 or some other measure of drop size, and (4) a measure of initial drop deformation a/b. In contrast to free drops whose frequencies of oscillation ω decrease as the amplitudes of their initial deformations increase, the...

On an m -point boundary-value problem for second-order ordinary differential equations

Abstract: where q E (0, 1) is given. We obtain conditions for the existence and uniqueness of a solution for the boundary-value problem (2), using the Leray-Schauder continuation theorem [2]. We give an example of a three-point boundary-value problem where the existence condition is not satisfied and no solution exists. Gupta [3] recently studied the boundary-value problem (2) when Q! = 1. Our results on the three-point boundary-value problem (2) extend the results of Gupta [3], to the case of general CY. (See also [4, 51.) We use the classical spaces C[O, 11, C’[O, 11, Lk[O, 11, and L”[O, l] of continuous, k-times continuously differentiable, measurable real-valued functions whose kth power of the absolute value is Lebesgue integrable on [0, 11, or measurable functions that are essentially bounded

Surface boundary conditions for the numerical solution of the Euler equations

Abstract: We consider the implementation of boundary conditions at solid walls in inviscid Euler solutions by upwind, finite-volume methods. We review some current methods for the implementation of surface boundary conditions and examine their behavior for the problem of an oblique shock reflecting off a planar surface. We show the importance of characteristic boundary conditions for this problem and introduce a method of applying the classical flux-difference splitting of Roe as a characteristic boundary condition. Consideration of the equivalent problem of the intersection of two (equal and opposite) oblique shocks was very illuminating on the role of surface boundary conditions for an inviscid flow and led to the introduction of two new boundary-condition procedures, denoted as the symmetry technique and the curvature-corrected symmetry technique. Examples of the effects of the various surface boundary conditions considered are presented for the supersonic blunt body problem and the subcritical compressible flow over a circular cylinder. Dramatic advantages of the curvature-corrected symmetry technique over the other methods are shown, with regard to numerical entropy generation, total pressure loss, drag and grid convergence.

An inverse problem in thermal imaging

Abstract: This paper examines uniqueness and stability results for an inverse problem in thermal imaging. The goal is to identify an unknown boundary of an object by applying a heat flux and measuring the induced temperature on the boundary of the sample. The problem is studied both in the case in which one has data at every point on the boundary of the region and the case in which only finitely many measurements are available. An inversion procedure is developed and used to study the stability of the inverse problem for various experimental configurations.

Random Walks on Boundary for Solving PDEs

Abstract: Part 1 Random walk algorithms for solving integral solutions: conventional Monte Carlo scheme biased estimators linear-fractional transformations and relations to iterative processes asymptotically unbiased estimators based on singular approximation of the kernel integral equation of the first kind. Part 2 Random walk on boundary algorithms for solving the Laplace equation: Newton potentials and boundary integral equations of the electrostatics the interior Dirichlet problem and isotropic random walk on boundary process solution of the Neumann problem random estimators for the exterior Dirichlet problem third boundary value problem and alternative methods of solving the Dirichlet problem in-homogeneous problems calculation of the derivatives near the boundary normal derivative of a double-layer potential. Part 3 Walk on boundary algorithms for the heat equation: heat potentials and Voltrerra boundary integral equations non-stationary walk on boundary process the Dirichlet problem the Neumann problem third boundary value problem unbiasedness and variance of the walk on boundary algorithms the cost of the walk on boundary algorithms in-homogeneous heat equation calculation of derivatives on the boundary. Part 4 Spatial problems of elasticity: elastopotentials and systems of boundary integral equations of the elasticity theory first boundary value problem and estimators for singular integrals other boundary value problems for the Lame equations and regular integral equations. Part 5 Variants of the random walk on boundary for solving the stationary potential problem: the Robin problem and the ergodic theorem stationary diffusion equation with absorption stabilization methods multiply connected domains. Part 6 Random walk on boundary in nonlinear problems: nonlinear Poisson equation boundary value problem for the Navier-Stokes equation.

Weak solutions of the initial-boundary value problem for the Vlasov–Poisson system

Abstract: In this paper we show the existence of a weak solution of the boundary value problem for the time dependent Vlasov–Poisson system. First, we regularize the system in order to apply a fixed-point theorem. Then we pass to the limit using an energy estimate.

Averaging of Boundary Value Problems with a Singular Perturbation of the Boundary Conditions

Abstract: Boundary value problems with different conditions on alternating small parts of the boundary are considered. An investigation is made of the behavior of solutions of such problems as the small parameter characterizing the period of change of the type of the boundary conditions goes to zero, and estimates are given for the deviation of these solutions from the solutions of the limit problem in various cases. The spectral properties of these problems are studied from a unified point of view on the basis of general methods (see [4], [9]).Bibliography: 20 titles.

Retarded time absorbing boundary conditions

Abstract: Over the past few years simulations of electromagnetic problems in three dimensions using the finite difference time domain (FDTD) method have become increasingly popular. A major problem in such simulations is the truncation of the computational domain. A formulation of this boundary problem using retarded time values of the field inside the computational domain is suggested, and hence the name retarded time absorbing boundary condition (RT-ABC). This formulation allows the boundary to be situated in the near field of the problem and thereby reduces the necessary computational domain, and the present formulation allows error estimates for the numerically calculated fields. >

On diffuse reflection at the boundary for the Boltzmann equation and related equations

Abstract: The paper considers diffuse reflection at the boundary with nonconstant boundary temperature and unbounded velocities. The solutions obtained are proved to conserve mass at the boundary. After a preliminary study of the collisionless case, the main results obtained are existence for the Boltzmann equation in a “DiPerna-Lions framework” with the above boundary conditions in a bounded measure sense, and existence together with uniqueness for the BGK equation with Maxwellian diffusion on the boundary in anL∞ framework.

The falkneer-skan equation: Numerical solutions within group invariance theory

Abstract: The iterative transformation method, defined within the framework of the group invariance theory, is applied to the numerical solution of the Falkner-Skan equation with relevant boundary conditions. In this problem a boundary condition at infinity is imposed which is not suitable for a numerical use. In order to overcome this difficulty we introduce a free boundary formulation of the problem, and we define the iterative transformation method that reducess the free boundary formulation to a sequence of initial value problems. Moreover, as far as the value of the wall shear stress is concerned we propose a numerical test of convergence. The usefulness of our approach is illustrated by considering the wall shear stress for the classical Homann and Hiemenz flows. In the Homann's case we apply the proposed numerical test of convergence, and meaningful numerical results are listed. Moreover, for both cases we compared our results with those reported in literature.

Direct Calculation of the Boundary $S$ Matrix for the Open Heisenberg Chain

Abstract: We calculate the boundary $S$ matrix for the open antiferromagnetic spin $1/2$ isotropic Heisenberg chain with boundary magnetic fields. Our approach, which starts from the model's Bethe Ansatz solution, is an extension of the Korepin-Andrei-Destri method. Our result agrees with the boundary $S$ matrix for the boundary sine-Gordon model with $\beta^2 \rightarrow 8\pi$ and with ``fixed'' boundary conditions.

The boundary integro-differential equations of three-dimensional Neumann problem in linear elasticity

Abstract: In this paper, we mainly consider the three dimensional Neumann problem in linear elasticity, which is reduced to a system of integro-differential equations on the boundary based on a new representation of the derivatives of the double-layer potential. Furthermore a new boundary finite element method for this Neumann problem is presented.

Boundary layer flow of Reiner-Philippoff fluids

Abstract: An analysis is made of the boundary layer flow of Reiner-Philippoff fluids. This work is an extension of a previous analysis by Hansen and Na [A.G. Hansen and T.Y. Na, Similarity solutions of laminar, incompressible boundary layer equations of non-Newtonian fluids. ASME 67-WA/FE-2, presented at the ASME Winter Annual Meeting, November (1967)], where the existence of similar solutions of the boundary layer equations of a class of general non-Newtonian fluids were investigated. It was found that similarity solutions exist only for the case of flow over a 90° wedge and, being similar, the solution of the non-linear boundary layer equations can be reduced to the solution of non-linear ordinary differential equations. In this paper, the more general case of the boundary layer flow of Reiner-Philippoff fluids over other body shapes will be considered. A general formulation is given which makes it possible to solve the boundary layer equations for any body shape by a finite-difference technique. As an example, the classical solution of the boundary layer flow over a flat plate, known as the Blasius solution, will be considered. Numerical results are generated for a series of values of the parameters in the Reiner-Philippoff model.

Exact, nonreflecting boundary conditions for parabolic-type approximations in underwater acoustics

Abstract: In the different parabolic approximations to the reduced wave equation which model acoustic propagation in the ocean, the bottom is usually modeled as an interface and the domain of propagation includes an absorbing layer below the bottom interface. Thus the boundary value problem to be solved has zero boundary conditions at the surface as well as at the bottom boundary. In this paper exact boundary conditions are developed and numerically implemented along the physical bottom boundary if it is horizontal, otherwise, along an artificially placed horizontal computational boundary inside the bottom. These boundary conditions are nonlocal but integrable and can be incorporated in finite difference schemes for the parabolic equations.

On fractional powers of a closed pair of operators and a damped wave equation with dynamic boundary conditions

Abstract: In this paper we are concerned with a wave equation with damping in which one of the boundary conditions is of a dynamic nature. The problem serves to describe the longitudinal vibrations of a homogeneous flexible horizontal rod in which one end is rigidly fixed while the other end is free to move with an attached load. Viscous effects are present both within the rod and at that end of the rod to which the load is attached, with the effect that damping of Kelvin-Voigt type occurs in both the partial differential equation and the dynamic boundary condition. The problem is studied within the framework of the abstract theories of B-evolutions and fractional powers of a closed pair of operators by formulating an abstract evolution problem in the product space X ½ x X with X a Hilbert space and X ½ the domain of a fractional power of a closed pair of operators with domain D in X.

A Sequence of Period Doublings and Chaotic Pulsations in a Free Boundary Problem Modeling Thermal Instabilities

Abstract: A simplified one-sided model associated with combustion and some phase transitions has been solved numerically. The results show a transition from the basic uniform motion of the free boundary to chaotic pulsations via periodic oscillations and a clearly manifested sequence of period doublings. For both numerical and rigorous treatment, the free boundary problem presents clear advantages over the two commonly used classes of models: the free interface (two-sided) models and the models with distributed kinetics. It is argued that, in view of its generic nature and relative simplicity, the problem may serve as a canonical example of the thermal instability leading to a variety of self-oscillatory regimes.