Showing papers on "Free boundary problem published in 1995"
TL;DR: In this article, the exact nonreflecting boundary conditions at artificial boundaries for the numerical solution of the Helmholtz equation and the time dependent wave equation in unbounded domains were developed.
295 citations
TL;DR: An exact nonreflecting boundary condition is derived for solutions of the time dependent wave equation in three space dimensions that holds on a spherical artificial boundary and is local in time, but nonlocal in space.
Abstract: An exact nonreflecting boundary condition is derived for solutions of the time dependent wave equation in three space dimensions. It holds on a spherical artificial boundary and is local in time, but nonlocal in space. It can be reduced to a boundary condition local in space and time for solutions consisting of a finite number of spherical harmonics. The boundary condition is related to the Dirichlet-to-Neumann boundary condition for the Helmholtz equation. It can be used in scattering problems as well as in problems involving nonlinearity in a bounded region of space.
260 citations
TL;DR: In this article, the ground-state energy of integrable 1 + 1 quantum field theories with boundaries is studied, where the boundary is represented by a boundary state, and the thermodynamics involves evaluating scalar products of boundary states with all the states of the theory.
200 citations
180 citations
Book•
01 Jan 1995TL;DR: In this article, a spectral theory of matrix polynomials and pseudoregressive operators on manifold and vector bundles is presented, together with the Lopatinskii condition on the L-condition.
Abstract: Part I. A Spectral Theory of Matrix Polynormials: 1. Matrix polynomials 2. Spectral triples for matrix polynomials 3. Monic matrix polynomials 4. Further results Part II. Manifolds and Vector Bundles: 5. Manifolds and vector bundles 6. Differential forms Part III. Pseudo-Differential Operators and Elliptic Boundary Value Problems: 7. Pseudo-differential operators on Rn 8. Pseudo-differential operators on a compact manifold 9. Elliptic systems on bounded domains in Rn Part IV. Reduction Of A Boundary Value Problem To An Elliptic System On The Boundary: 10. Understanding the L-condition 11. Applications to the index 12. BVPs for ordinary differential operators and the connection with spectral triples 13. Behaviour of a pseudo-differential operator near a boundary 14. The Main Theorem revisited Part V. An Index Formula For Elliptic Boundary Problems In The Plane: 15. Further results on the Lopatinskii Condition 16. The index in the plane 17. Elliptic systems with 2 x 2 real coefficients.
165 citations
TL;DR: In this article, the authors introduce a new free-boundary problem for the heat equation, which is obtained in the description of laminar flames as an asymptotic limit for high activation energy.
Abstract: We introduce a new free-boundary problem for the heat equation, of interest in combustion theory. It is obtained in the description of laminar flames as an asymptotic limit for high activation energy. The problem asks for the determination of a domain in space-time, Q C Rn x (0, T) , and a function u(x, t) > 0 defined in Q, such that ut = Au in Q, u takes certain initial conditions, u(x, 0) = uo(x) for x e Qo = OQfn {t = 0}, and two conditions are satisfied on the free boundary r = a2 n {t > 0}: u = O and u1, =-1, where u1, denotes the derivative of u along the spatial exterior normal to r. We approximate this problem by means of a certain regularization on the boundary and prove the existence of a weak solution under suitable assumptions on the initial data.
141 citations
TL;DR: In this article, the spontaneous boundary magnetization is found by solving the difference equations for the case of semi-infinite spin chain systems with integrable boundary conditions, and the axioms satisfied by the form factors in the boundary theory are formulated.
128 citations
TL;DR: The viscous Cahn-Hilliard equation as mentioned in this paper is a singular limit of the phase-field model of phase transitions and contains both the Cahn and Allen-Cahn equations as particular limits.
Abstract: The viscous Cahn-Hilliard equation arises as a singular limit of the phase-field model of phase transitions. It contains both the Cahn-Hilliard and Allen-Cahn equations as particular limits. The equation is in gradient form and possesses a compact global attractor A, comprising heteroclinic orbits between equilibria. Two classes of computation are described. First heteroclinic orbits on the global attractor are computed; by using the viscous Cahn-Hilliard equation to perform a homotopy, these results show that the orbits, and hence the geometry of the attractors, are remarkably insensitive to whether the Allen-Cahn or Cahn-Hilliard equation is studied. Second, initial-value computations are described; these computations emphasize three differing mechanisms by which interfaces in the equation propagate for the case of very small penalization of interfacial energy. Furthermore, convergence to an appropriate free boundary problem is demonstrated numerically.
94 citations
Journal Article•
TL;DR: In this paper, the authors considered the free boundary problem for the heat equations with a Stefan-like condition expressing conservation of energy at the interface and another condition relating the velocity of the interface to its curvature, the surface tension and the local temperature.
Abstract: We study certain approximate solutions of a system of equations formulated in an earlier paper (Physica D43 44{62 (1990)) which in dimensionless form are 2 t = 2 r 2 +F (;u); where u is (dimensionless) temperature, is an order parameter, w( )i s the temperature{independent part of the energy density, and F involves the {derivative of the free-energy density. The constants and are of order 1 or smaller, whereas could be as small as 10 8 . Assuming that a solution has two single{phase regions separated by a moving phase boundary ( t), we obtain the dierential equations and boundary conditions satised by the ‘outer’ solution valid in the sense of formal asymptotics away from and the ‘inner’ solution valid close to . Both rst and second order transitions are treated. In the former case, the ‘outer’ solution obeys a free boundary problem for the heat equations with a Stefan{like condition expressing conservation of energy at the interface and another condition relating the velocity of the interface to its curvature, the surface tension and the local temperature. There are O() eects not present in the standard phase{eld model, e.g. a correction to the Stefan condition due to stretching of the interface. For second{order transitions, the main new eect is a term proportional to the temperature gradient in the equation for the interfacial velocity. This eect is related to the dependence of surface tension on temperature. We also consider some cases in which the temperature u is very small, and possibly or as well; these lead to further free boundary problems,
93 citations
86 citations
TL;DR: In this paper, elementary boundary integral equations for the Helmholtz equation in the exterior domain, based on Green's formula or through representation of the solution by layer potentials, are considered.
TL;DR: In this article, a complete study of boundary bound states and related boundary S-matrices for the sine-Gordon model with Dirichlet boundary conditions is presented, based on the bootstrap procedure and partly on the explicit solution of the inhomogeneous XXZ model with boundary magnetic field and solution of boundary Thirring model.
Abstract: We present a complete study of boundary bound states and related boundary S-matrices for the sine-Gordon model with Dirichlet boundary conditions. Our approach is based partly on the bootstrap procedure and partly on the explicit solution of the inhomogeneous XXZ model with boundary magnetic field and solution of the boundary Thirring model. We identify boundary bound states with new `boundary strings` in the Bethe ansatz. The boundary energy is also computed.
TL;DR: In this paper, the numerical solution of the one-dimensional, unsteady heat conduction equation in which Dirichlet boundary conditions are specified at two space locations and the temperature distribution at a particular time, say T 0, is given.
TL;DR: In this article, a construction scheme of deriving transparent boundary conditions for the numerical solution of Fresnel's equation is presented, which leads to a sequence of coupled boundary value problems, and the resulting boundary conditions appear to be of a nonlocal Cauchy type.
Abstract: The paper presents a construction scheme of deriving transparent, i.e., reflection-free, boundary conditions for the numerical solution of Fresnel's equation (being formally equivalent to Schrodinger's equation). In contrast to previous suggestions, the method advocated here treasts the discrete problem after discretization of the time-like variable, i.e., in a Rothe method, which leads to a sequence of coupled boundary value problems. The thus obtained boundary conditions appear to be of a nonlocal Cauchy type. As it turns out, each kind of linear implicit discretization induces its own discrete transparent boundary conditions. Numerical experiments on technologically relevant examples from integrated optics are included.
01 Jan 1995
TL;DR: In this paper, it is shown that the problem is solvable in non-tangentially accessible domains, a class including the well-known vonKoch snowflake, and there are estimates of the behaviour of the solution at any point close to the boundary in terms of the distance from the point to the boundaries.
Abstract: A classical problem in differential geometry is the prescribed curvature problemin a bounded domain, where the task is to conformally deform the Euclidean metric to another complete Riemannian metric with a prescribed curvature function.If this function is negative, the problem is equivalent to solving a nonlinear elliptic partial differential equation with boundary blow-up. The problem is solvablein non-tangentially accessible domains, a class including the well-known vonKoch snowflake, and there are estimates of the behaviour of the solution at anypoint close to the boundary in terms of the distance from the point to the boundary. There is a characterization of the negative continuous functions for whichthe prescribed curvature problem is solvable. Another result is the existence ofa unique distribution solution of a particular boundary-blow-up problem relevant in probability theory, in any bounded domain. If the Laplace operator in the original problem is exchanged for the pseudo-Laplace operator or the real Monge-Ampere operator, it is still possible to state necessary and sufficient growth conditions on the nonlinear absorption term, which depends on the solution, for the corresponding problem to be solvable. Blow-up estimates andsome uniqueness results hold. The behaviour of a solution in a ball determineswhether there does or does not exist an entire solution of the equation, i.e. asolution defined throughout the Euclidean space. Existence results and growthestimates for solutions of a generalized mean curvature flow problem follow fromthe general technique which is the foundation of all the results in the thesis.
TL;DR: A mathematical model is presented for the dynamical strain-stress response of shape memory wires which incorporates bistable potentials together with a physical constant, independent of stress or temperature, which accounts for mechanically dissipative processes.
Abstract: A mathematical model is presented for the dynamical strain-stress response of shape memory wires. This model is based on a free boundary problem for the heat equation which incorporates bistable potentials together with a physical constant, independent of stress or temperature, which accounts for mechanically dissipative processes. An expression in closed form is derived for one of the most important observables, namely the width of the hysteresis loops as a function of the elongation rates. Excellent agreement with experimental results is observed.
TL;DR: In this paper, the authors analyzed the structure of the set of positive solutions of a class of semilinear boundary value problems and showed that the global continuum of positive solution emanating from the trivial equilibrium at the principal eigenvalue of the linearization is constituted by a regular curve if the slope of the kinetic at the trivial solution is large enough and Ω is convex.
01 Oct 1995
TL;DR: In this paper, the determinant of a matrix-valued elliptic differential operator of even order on a line segment [0, T] with boundary is given for the case where T is the boundary.
Abstract: In this paper we present a formula for the determinant of a matrix-valued elliptic differential operator of even order on a line segment [0, T] with boundary
TL;DR: In this paper, the use of characteristic boundary conditions for the Navier-Stokes equations with constant coefficients matrices is analyzed and new boundary conditions of characteristic type that yield fast convergence to steady state, a strongly well posed continuous problem and a strongly stable semi-discrete problem are derived.
TL;DR: In this paper, a modified classical penalty method for solving a Dirichlet boundary value problem is presented, based on the fact that one can expand the boundary measure under the chosen basis which leads to a fast, approximate calculation of boundary integral.
Abstract: A modified classical penalty method for solving a Dirichlet boundary value problem is presented. This new fictitious domain penalty method eliminates the traditional need of generating a complex computation grid in the case of irregular domains. It is based on the fact that one can expand the boundary measure under the chosen basis which leads to a fast, approximate calculation of boundary integral. The compact support and orthonormality of the basis are essential for representing the boundary measure numerically, and therefore for implementing this methodology.
TL;DR: In this article, the determinant of an elliptic self-adjoint boundary value problem for the Dirac operator over an odd-dimensional manifold was constructed, and it was shown that the ζ-function determinant coincides with the δ-function for 1-dimensional manifolds.
Abstract: We present a canonical construction of the determinant of an elliptic selfadjoint boundary value problem for the Dirac operatorD over an odd-dimensional manifold. For 1-dimensional manifolds we prove that this coincides with the ζ-function determinant. This is based on a result that elliptic self-adjoint boundary conditions forD are parameterized by a preferred class of unitary isomorphisms between the spaces of boundary chiral spinor fields. With respect to a decompositionS
1=X
0∪X
1, we explain how the determinant of a Dirac-type operator overS
1 is related to the determinants of the corresponding boundary value problems overX
0 andX
1.
TL;DR: In this paper, the Navier-Stokes equations were used to determine the domain occupied by the fluid (j) (j = 1, 2) at the moment t>0 together with the velocity vector field v(j,t)=(v1(j),v3(j))(x,t), the pressure p(j)(x and t) and with the absolute temperature Į(j).
Abstract: (1) and (2), respectively, at the initial moment. We assume that aS (1) ~(1) U T, ~(2) UT, ~(1) nr = q, ~(2) nr = ~, ~(1) f~(2) _ q („C is the initial interface between fluids (1) and (2), ƒ°(1), ƒ°(2) are fixed). Then our problem consists in determining the domain ƒ¶(j)(t), occupied by the fluid (j) (j=1, 2) at the moment t>0 together with the velocity vector field v(j)(x,t)=(v1(j),v3(j))(x,t), the pressure p(j) (x, t) and with the absolute temperature ƒÆ(j) (x, t) of the fluid (j), satisfying the system of Navier-Stokes equations:
TL;DR: In this article, the motion of an incompressible fluid in a rigid porous medium of infinite extent is studied, where the fluid is bounded below by a fixed, impermeable layer and above by a free surface moving under the influence of gravity.
Abstract: This paper is concerned with the motion of an incompressible fluid in a rigid porous medium of infinite extent. The fluid is bounded below by a fixed, impermeable layer and above by a free surface moving under the influence of gravity. The laminar flow is governed by Darcy's law.
TL;DR: A survey of boundary integral methods for free boundary value problems arising from fluid dynamics and materials science can be found in this article, where the level-set approach for water waves, general multi-fluid interfaces, Hele-Shaw cells, crystal growth and solidification are discussed.
Abstract: Many physically interesting problems involve propagation of free surfaces. Vortex-sheet roll-up in hydrodynamic instability, wave interactions on the ocean's free surface, the solidification problem for crystal growth and Hele-Shaw cells for pattern formation are some of the significant examples. These problems present a great challenge to physicists and applied mathematicians because the underlying problem is very singular. The physical solution is sensitive to small perturbations. Naive discretisations may lead to numerical instabilities. Other numerical difficulties include singularity formation and possible change of topology in the moving free surfaces, and the severe time-stepping stability constraint due to the stiffness of high-order regularisation effects, such as surface tension.This paper reviews some of the recent advances in developing stable and efficient numerical algorithms for solving free boundary-value problems arising from fluid dynamics and materials science. In particular, we will consider boundary integral methods and the level-set approach for water waves, general multi-fluid interfaces, Hele–Shaw cells, crystal growth and solidification. We will also consider the stabilising effect of surface tension and curvature regularisation. The issue of numerical stability and convergence will be discussed, and the related theoretical results for the continuum equations will be addressed. This paper is not intended to be a detailed survey and the discussion is limited by both the taste and expertise of the author.
TL;DR: In this paper, a direct approach to extending the formulation of the free boundary problem to a larger fixed domain without modifying the differential operator is described, based on an approximate controllability-type property.
Abstract: Both free boundary problems and optimal design problems involve unknown or variable domains. We describe a direct approach extending the formulation of the problem to a larger fixed domain without modifying the differential operator. This argument is based on an approximate controllability-type property.
Journal Article•
TL;DR: In this article, an algorithm is presented to numerically treat a free boundary problem arising in the theory of phase transition, where material is transported between particles and within particles by diusion, driven by curvature which expresses the eect of surface tension.
Abstract: An algorithm is presented to numerically treat a free boundary problem arising in the theory of phase transition. The problem is one in which a collection of simple closed curves (particles) evolves in such a way that the enclosed area remains constant while the total arclength decreases. Material is transported between particles and within particles by diusion, driven by curvature which expresses the eect of surface tension. The algorithm is based on a reformulation of the problem, using boundary integrals, which is then discretized and cast as a semi-implicit scheme. This scheme is implemented with a variety of congurations of initial curves showing that convexity or even topological type may not be preserved.
21 Jun 1995
TL;DR: In this article, the authors prove convergence of a trial free boundary method to a classical solution of a Bernoulli-type free boundary problem for the p-Laplace equation, 1 < p < ∞.
Abstract: We prove convergence of a trial free boundary method to a classical solution of a Bernoulli-type free boundary problem for the p-Laplace equation, 1 < p <∞. In addition, we prove the existence of a classical solution in N dimensions when p = 2 and, for 1 < p < ∞, results on uniqueness and starlikeness of the free boundary and continuous dependence on the fixed boundary and on the free boundary data. Finally, as an application of the trial free boundary method, we prove (also for 1 < p <∞) that the free boundary is convex when the fixed boundary is convex.
TL;DR: In this article, the boundary S-matrix of the sine-Gordon Lagrangian is matched to the solution of Ghoshal and Zamolodchikov.