Showing papers on "Free boundary problem published in 1997"
TL;DR: In this article, a variational iteration method is proposed to solve nonlinear partial differential equations without linearization or small perturbations, where a correction functional is constructed by a general Lagrange multiplier, which can be identified via variational theory.
571 citations
TL;DR: In this article, the following second order ordinary differential equation, x 0 t s f t, x t, x 9 t q e t, t g 0, 1, 1.
267 citations
TL;DR: In this paper, a level set method is used where the interface is the zero level set of a continuous function while the two fluids are solutions of the incompressible Navier-Stokes equation.
Abstract: We present a number of three-dimensional axisymmetric free boundary problems for two immiscible fluids, such as air and water. A level set method is used where the interface is the zero level set of a continuous function while the two fluids are solutions of the incompressible Navier–Stokes equation. We examine the rise and distortion of an initially spherical bubble into cap bubbles and toroidal bubbles. Steady solutions for gas bubbles rising in a liquid are computed, with favourable comparisons to experimental data. We also study the inviscid limit and compare our results with a boundary integral method. The problems of an air bubble bursting at a free surface and a liquid drop hitting a free surface are also computed.
242 citations
212 citations
TL;DR: In this article, the American put option valuation in a jump-diffusion model was considered and the optimal stopping problem was compared to a parabolic integro-differential free-boundary problem, with special attention to the behavior of the optimal-stopping boundary.
Abstract: This paper considers the American put option valuation in a jump-diffusion model and relates this optimal-stopping problem to a parabolic integro-differential free-boundary problem, with special attention to the behavior of the optimal-stopping boundary. We study the regularity of the American option value and obtain in particular a decomposition of the American put option price as the sum of its counterpart European price and the early exercise premium. Compared with the Black-Scholes (BS) [5] model, this premium has an additional term due to the presence of jumps. We prove the continuity of the free boundary and also give one estimate near maturity, generalizing a recent result of Barleset al. [3] for the BS model. Finally, we study the effect of the market price of jump risk and the intensity of jumps on the American put option price and its critical stock price.
200 citations
Journal Article•
TL;DR: In this paper, the existence of globally smooth convex solutions of MongeAmpere equations of the form detDw=/(x,M,Z)w) in Ω subject to the boundary condition Du(Q) = Ω* where Ω and Ω * are smooth uniformly convex domains in R\".
Abstract: We prove the existence of globally smooth convex Solutions of MongeAmpere equations of the form detDw=/(x,M,Z)w) in Ω subject to the boundary condition Du(Q) = Ω* where Ω and Ω* are smooth uniformly convex domains in R\".
194 citations
01 Jan 1997
TL;DR: The main features of the theory were formed in the 60's, beginning with the equivalence of the ellipticity conditions and the Fredholm property of the corresponding operator in the simplest Sobolev spaces.
Abstract: This paper, somewhat delayed in the series “Fundamental Directions,” is devoted to general linear elliptic boundary problems on a smooth compact manifold with boundary. The paper is intended for a large circle of readers. We hope the paper will be useful to many mathematicians with diverse scientific interests. The main features of the theory we will discuss were formed in the 60’s, beginning with the proof of the equivalence of the ellipticity conditions and the Fredholm property of the corresponding operator in the simplest Sobolev spaces. This was done on the basis of achievements of many mathematicians during the preceding decades. This basis was very extensive, but the results were incomplete. The elaboration of the general theory was stimulated by investigations of the index problem (see e.g. (Palais 1965) and (Fedosov 1990)) and went on under a strong influence of the microlocal analysis, beginning with the appearance of the calculus of pseudodifferential operators. During the last decades, new variants of the general theory appeared, and the old variants were enriched by new results.
183 citations
TL;DR: The problem of existence of a solution for the three-point boundary value problem was studied by Gupta, Ntouyas, and Tsamatos when α≤1 and when α>1 with αη as mentioned in this paper.
181 citations
TL;DR: In this article, the problem of determining quantitative information about corrosion occurring on an inaccesible part of a specimen is considered and a regularized numerical method for obtaining approximate solutions to the problem is presented.
Abstract: We consider the problem of determining quantitative information about corrosion occurring on an inaccesible part of a specimen. The data for the problem consist of prescribed current flux and voltage measurements on an accessible part of the specimen boundary. The problem is modelled by Laplace's equation with an unknown term in the boundary conditions. Our goal is recovering from the data. We prove uniqueness under certain regularity assumptions and construct a regularized numerical method for obtaining approximate solutions to the problem. The numerical method, which is based on the assumption that the specimen is a thin plate, is tested in numerical experiments using synthetic data.
179 citations
TL;DR: In this article, a four-step algorithm for solving nonlinear boundary-value problems on infinite or semi-infinite intervals is described, which can be implemented in as few as seven lines of Maple (sample program provided!).
Abstract: We describe a four-step algorithm for solving ordinary differential equation nonlinear boundary-value problems on infinite or semi-infinite intervals. The first step is to compute high-order Taylor series expansions using an algebraic manipulation language such as Maple or Mathematica. These expansions will contain one or more unknown parameters z which will be determined by the boundary condition at infinity. The second step is to convert the Taylor expansions into diagonal Pade approximants. The boundary condition that u(x) decays to zero at infinity becomes the condition that the coefficient of the highest power of x in the numerator polynomial must be zero. The third step is to solve this equation for the free parameter z. The final step is to evaluate each of the multiple solutions of this equation for physical plausibility and convergence (as N increases). This algorithm can be implemented in as few as seven lines of Maple (sample program provided!). We illustrate the method with three examples: the Flierl–Petviashvili vortex of geophysical fluid dynamics, the quartic oscillator of quantum mechanics, and the Blasius function for the boundary layer above a semi-infinite plate in fluid mechanics. Methods for nonlinear problems are almost always iterative and need a first guess to initialize the iteration. The Pade algorithm is unusual in that it is a direct method that requires no a priori information about the solution. © 1997 American Institute of Physics.
157 citations
Journal Article•
TL;DR: In this article, the implicit Neumann scheme for the interior and exterior Bernoulli's free-boundary problem has been proposed, and super linear convergence of a semi discrete variant is proved under a natural non-degeneracy condition.
Abstract: Bernoulli's free-boundary problem arises in ideal fluid dynamics, optimal insulation and electro chemistry. In electrostatic terms we design an annular condenser with a prescribed and an unknown boundary component such that the electrostatic field is constant in magnitude along the free boundary. Typically the interior Bernoulli problem has two Solutions, an elliptic one close to the fixed boundary and a hyperbolic one far from it. Previous results mainly deal with elliptic Solutions exploiting their monotonicity s discovered by A. Beurling. Hyperbolic Solutions are more delicate for analysis and numerical approximation. Nevertheless we derive a second order trial free-boundary method, the implicit Neumann scheme, with equally good performance for both types of Solutions. Super linear convergence of a semi discrete variant is proved under a natural non-degeneracy condition. Numerical examples computed by this method confirm analytic predictions including questions of uniqueness, connectedness, elliptic and hyperbolic limits. 1. Interior and exterior Bernoulli problem The interior Bernoulli problem is the following. Given a connected domain in (T and a constant Q > 0, find a subset A c Ω and a potential u: Ω\Α -* i such that -Δι/ = 0 in Ω\Α, u = Ο οη ΘΩ, u = l on dA , du „ Λ . — = on dA. ov The potential u lives on the domain Ω\Α, typically an annulus (Fig. 1). The exterior unit normal of this domain is denoted by v. In the classical setting the free-boundary condition means 166 (1) Flucher and Rumpf , Bernoulli's free-boundary problem
TL;DR: In this paper, the existence and approximation of solutions of asymptotic or periodic boundary value problems of mixed functional differential equations are discussed, via monotone iteration and non-standard ordering in the profile set.
TL;DR: In this paper, a mixed boundary value problem for elliptic second order equations was studied under weak assumptions on the data and the dependence of the solution with respect to perturbations of the boundary sets carrying the Dirichlet and the Neumann conditions.
Abstract: We study a mixed boundary value problem for elliptic second order equations obtaining optimal regularity results under weak assumptions on the data. We also consider the dependence of the solution with respect to perturbations of the boundary sets carrying the Dirichlet and the Neumann conditions.
TL;DR: In this article, the potential and boundary condition of the Sturm-Liouville problem were reconstructed from nodes of its eigenfunctions using the same method as McLaughlin.
Abstract: In this paper, we reconstruct the potential and the boundary condition of the Sturm - Liouville problem from nodes of its eigenfunctions. We also give the uniqueness for general boundary conditions using the same method as McLaughlin.
TL;DR: In this article, the authors prove domain perturbation theorems for linear and nonlinear elliptic equations under Robin boundary conditions, and show that the limiting problem is the Dirichlet problem.
TL;DR: The time-stepping BEM procedure produces instabilities and high numerical damping, when the time step size is chosen too small and too large, respectively as discussed by the authors, when the fundamental solution is known only in the frequency domain such that the time history of a response can only be obtained by an inverse transformation of the frequencydomain results.
Abstract: The usual time domain Boundary Element Method (BEM) contains fundamental solutions which are convoluted with time-dependent boundary data and integrated over the boundary surface. If the fundamental solution is known, e.g., in Elastodynamics, the temporal convolution can be performed analytically when the boundary data are approximated by polynomial shape functions in time and in the boundary elements. This formulation is well known, but the resulting time-stepping BEM procedure produces instabilities and high numerical damping, when the time step size is chosen too small and too large, respectively. Moreover, in case of viscoelastic or poroelastic domains, the fundamental solution is known only in the frequency domain such that the time history of a response can only be obtained by an inverse transformation of the frequency domain results.
TL;DR: In this paper, the quotient of the zeta functional determinants for certain elliptic boundary value problems on Riemannian 3 and 4-manifolds with smooth boundary was derived.
TL;DR: In this paper, a transformation of the free boundary problem together with an asymptotic analysis (performed about the solution when the transaction cost is zero) leads to solutions which are shown to be good approximations for cases which can be solved by numerical methods.
Abstract: It is known that the optimal trading strategy for a certain portfolio problem featuring fixed transaction costs is obtained from the solution of a free boundary problem. The latter can only be solved with numerical methods, and computations become formidable when the number of available securities is larger than three or four. This paper shows how a transformation of the free boundary problem together with an asymptotic analysis (performed about the solution when the transaction cost is zero) leads to solutions which are shown to be good approximations for cases which can be solved by numerical methods. These approximately optimal trading strategies are easy to compute, even when there are many risky securities, as is illustrated for the case of the 30 Dow Jones Industrials.
01 Jan 1997
TL;DR: In this article, a general theorem (principle of a priori boundedness) on solvability of the boundary value problem dx(t) dt = f (x)(t); h(x) = 0 is established, where are continuous operators.
Abstract: A general theorem (principle of a priori boundedness) on solvability of the boundary value problem dx(t) dt = f (x)(t); h(x) = 0 is established, where are continuous operators. As an application, a two-point boundary value problem for the system of ordinary diierential equations is considered. reziume. damt kicebu l ia zogadi Teor ema (apr ior ul i Semosa-zGvru l obis principi) dx(t) dt = f (x)(t); h(x) = 0 sasazG vro amocanis amoxsnad obis Sesaxeb, sadac f : C ((a; b]; R n) ! L((a; b]; R n) da h : C ((a; b]; R n) ! R n uCKvet i operator ebia. am Teor emis saPu Zvelze gamokvl eul ia orCer t il ovani sasazG vr o amocana hveul ebr iv diPerencial u r gant ol ebaTa sist emisaTvis.
TL;DR: In this paper, the authors prove unique existence of solution for the impedance (or third) boundary value problem for the Helmholtz equation in a half-plane with arbitrary L∞ boundary data.
Abstract: We prove unique existence of solution for the impedance (or third) boundary value problem for the Helmholtz equation in a half-plane with arbitrary L∞ boundary data. This problem is of interest as a model of outdoor sound propagation over inhomogeneous flat terrain and as a model of rough surface scattering. To formulate the problem and prove uniqueness of solution we introduce a novel radiation condition, a generalization of that used in plane wave scattering by one-dimensional diffraction gratings. To prove existence of solution and a limiting absorption principle we first reformulate the problem as an equivalent second kind boundary integral equation to which we apply a form of Fredholm alternative, utilizing recent results on the solvability of integral equations on the real line in [5].
TL;DR: In this paper, the problem of constructing boundary conditions for nonlinear equations compatible with higher symmetries is considered, in particular for the sine - Gordon, Jiber - Shabat, Liouville and KdV equations.
Abstract: The problem of constructing boundary conditions for nonlinear equations compatible with higher symmetries is considered. In particular, this problem is discussed for the sine - Gordon, Jiber - Shabat, Liouville and KdV equations. New results are obtained for the last two ones. The boundary condition for the KdV contains two arbitrary constants. The substitution maps it onto the boundary condition with linear dependence on t for the potentiated KdV.
TL;DR: In this paper, a detailed model is constructed in order to determine the full 3D weld pool and keyhole geometry by setting the appropriate energy and pressure balances, taking into account heat conduction, ablation losses and evaporation effects at the keyhole open surfaces, as well as the most relevant energyabsorption mechanisms, namely Fresnel and inverse Bremsstrahlung.
Abstract: A detailed model is constructed in order to determine the full 3D weld pool and keyhole geometry by setting the appropriate energy and pressure balances. The energy balance takes into account heat conduction, ablation losses and evaporation effects at the keyhole open surfaces, as well as the most relevant energy-absorption mechanisms, namely Fresnel and inverse Bremsstrahlung. The pressure balance ensures mechanical stability of the keyhole by including ablation pressure against surface tension pressure. The model provides a full description of the temperature field, electronic density, degree of ionization and absorption coefficient within the plasma, as well as setting the maximum penetration depth for a given set of laser parameters such as power, focusing radius and oscillation transversal mode. The keyhole boundary is initially taken to be an unknown free boundary and is obtained as a part of the solution of the problem. For low and medium welding speeds this boundary is successfully described with a family of ovoids. Good agreement with experimental results is achieved for a wide range of laser powers and plate thicknesses.
TL;DR: In this article, an explicit expression for vacuum expectation values of the boundary field in the boundary sine-Gordon model with zero bulk mass was proposed, which is consistent with known exact results for the boundary free energy and with perturbative calculations.
TL;DR: In this article, the boundary layer problem associated with the steady thermal conduction problem in a thin laminated plate is considered and two cases of boundary conditions, Dirichlet and Neumann, are treated.
Abstract: We consider the boundary layer problem associated with the steady thermal conduction problem in a thin laminated plate. Two cases of boundary conditions, Dirichlet and Neumann, are treated in the paper. Transmission conditions across the interfaces should be added since the plate is laminated. The study of the structure of the solution in the matching region of the layer with the basis solution in the plate leads to consideration of an eigenvalue problem for a second-order operator pencil with piecewise continuous coefficients and the corresponding boundary and transmission conditions. Twofold completeness of root functions of the latter problem is proved. The boundary layer term can then be expressed as a combination of these functions.
TL;DR: In this article, a finite element method for the solution of linear elliptic problems in infinite domains is proposed, where an artificial boundary B is first introduced, to make the computational domain Ω finite, and then the exact nonlocal Dirichlet-to-Neumann (DtN) boundary condition is derived on B. This condition is localized, and a sequence of local boundary conditions on B, of increasing order, is obtained.
TL;DR: In this article, a general technique for constructing nonlocal transparent boundary conditions for one-dimensional Schrodinger-type equations is presented, which avoids direct and inverse transforms between time and frequency domains and implements the boundary conditions in a direct manner.
TL;DR: It is proven that the distribution of particles tends for large time to a Maxwellian determined by the solution of the Poisson--Boltzmann equation with Dirichlet boundary condition.
Abstract: The asymptotic behavior for the Vlasov--Poisson--Fokker--Planck system in bounded domains is analyzed in this paper. Boundary conditions defined by a scattering kernel are considered. It is proven that the distribution of particles tends for large time to a Maxwellian determined by the solution of the Poisson--Boltzmann equation with Dirichlet boundary condition. In the proof of the main result, the conservation law of mass and the balance of energy and entropy identities are rigorously derived. An important argument in the proof is to use a Lyapunov-type functional related to these physical quantities.
TL;DR: In this paper, an n-dimensional (n = 2,3) inverse problem for the parabolic/diffusion equation,,, is considered, where the problem consists of determining the function a(x) inside of a bounded domain given the values of the solution u(x,t) for a single source location on a set of detectors where is the boundary of.
Abstract: An n-dimensional (n = 2,3) inverse problem for the parabolic/diffusion equation , , , is considered. The problem consists of determining the function a(x) inside of a bounded domain given the values of the solution u(x,t) for a single source location on a set of detectors , where is the boundary of . A novel numerical method is derived and tested. Numerical tests are conducted for n = 2 and for ranges of parameters which are realistic for applications to early breast cancer diagnosis and the search for mines in murky shallow water using ultrafast laser pulses. The main innovation of this method lies in a new approach for a novel linearized problem (LP). Such a LP is derived and reduced to a well-posed boundary-value problem for a coupled system of elliptic partial differential equations. A principal advantage of this technique is in its speed and accuracy, since it leads to the factorization of well conditioned, sparse matrices with non-zero entries clustered in a narrow band near the diagonal. The authors call this approach the elliptic systems method (ESM). The ESM can be extended to other imaging modalities.