Showing papers on "Free boundary problem published in 1991"
TL;DR: In this article, it was shown that the problem of pricing the American put is equivalent to solving an optimal stopping problem and that there is a unique solution to this problem which has a lower boundary.
Abstract: We show that the problem of pricing the American put is equivalent to solving an optimal stopping problem. the optimal stopping problem gives rise to a parabolic free-boundary problem. We show there is a unique solution to this problem which has a lower boundary. We identify an integral equation solved by the boundary and show that it is the unique solution to this equation satisfying certain natural additional conditions. the proofs also give a natural decomposition of the price of the American option as the sum of the price of the European option and an “American premium.”
570 citations
TL;DR: In this article, it was shown how to incorporate the magnetic boundary data and to apply the idea to transmission problems and prove coercivity without using symbols of pseudo-differential operators on the boundary.
230 citations
TL;DR: In this article, the boundary conditions for elastic waves are developed for elastic wave motions in unbounded spatial domains based on compositions of simple first-order differential operators, which can be applied without modification to problems in both two and three dimensions.
Abstract: Absorbing boundary conditions are needed for computing numerical models of wave motions in unbounded spatial domains. The boundary conditions developed here for elastic waves are generalizations of ones developed earlier for acoustic waves. These conditions are based on compositions of simple first-order differential operators. The formulas can be applied without modification to problems in both two and three dimensions. The boundary conditions are stable for all values of the ratio of P-wave velocity to S-wave velocity, and they are effective near a free surface and in a horizontally stratified medium. The boundary conditions are approximated with simple finite-difference equations that use values of the solution only along grid lines perpendicular to the boundary. This property facilitates implementation, especially near a free surface and at other corners of the computational domain.
229 citations
TL;DR: In this article, the vector parasite problem is studied in the context of finite-element solutions of Maxwell's equations for driven boundary-value problems, and an expanded weak form which combines the divergence equation with the conventional weak form of the double-curl equation is introduced.
Abstract: The vector parasite problem is studied in the context of finite-element solutions of Maxwell's equations for driven boundary-value problems. An expanded weak form which combines the divergence equation with the conventional weak form of the double-curl equation is introduced. This form is related to penalty methods where the penalty or weighting factor varies with the dielectric constant. The resulting algebraic system is identical to the Galerkin-Helmholtz operator on homogeneous subregions. Normal and tangential boundary conditions arise in terms of the divergence and curl of the field on the boundary which can be reexpressed as equivalent charges and currents. Computational results show the occurrence of two distinct types of parasitic modes in driven problems and their elimination with the formulation presented. Practical observations concerning the conditions which provoke spurious modes in these problems are reported. Spurious solutions arise from improper or unphysical boundary conditions, and the importance of careful specification of boundary-value problems is illustrated. Most conceptual difficulties with boundary conditions per se are removed when hybrid methods are used to couple the interior finite-element solution to the exterior problem. which focuses attention on the physics of the source distribution. >
113 citations
TL;DR: In this paper, three different finite difference schemes for solving the heat equation in one space dimension with boundary conditions containing integrals over the interior of the interval are considered, based on the forward Euler, the backward Euler and the Crank-Nicolson methods.
Abstract: Three different finite difference schemes for solving the heat equation in one space dimension with boundary conditions containing integrals over the interior of the interval are considered. The schemes are based on the forward Euler, the backward Euler and the Crank-Nicolson methods. Error estimates are derived in maximum norm. Results from a numerical experiment are presented.
105 citations
01 Jan 1991
TL;DR: In this paper, boundary integral equations and their application to the solution of boundary and initial-boundary value problems for partial differential equations are studied. But their application is limited to boundary integral problems.
Abstract: This article is devoted to boundary integral equations and their application to the solution of boundary and initial-boundary value problems for partial differential equations.
99 citations
TL;DR: In this paper, an application of a boundary integral equation method (BIEM) to an inverse problem of determining the shape and the location of cracks by boundary measurements is discussed, where the authors propose to solve this problem by minimizing the error of a certain boundary integral equations (BIE).
Abstract: This paper discusses an application of a boundary integral equation method (BIEM) to an inverse problem of determining the shape and the location of cracks by boundary measurements. Suppose that a given body contains an interior crack, the shape and the location of which are unknown. On the exterior boundary of this body one carries out measurements which are interpreted mathematically as prescribing Dirichlet data and measuring the corresponding Neumann data, or vice versa, for a field governed by Laplace's equation. The inverse problem considered here attempts to determine the geometry of the crack from these experimental data. We propose to solve this problem by minimizing the error of a certain boundary integral equation (BIE). The process of this minimization, however, is shown to require solutions of certain are proposed. Several 2D and 3D numerical examples are given in order to test the performance of the present method.
93 citations
TL;DR: In this article, the authors studied the behavior of the discrete-ordinate method in highly scattering regimes, where the leading behavior of its solution is determined by the solution of a diffusion equation.
Abstract: In highly scattering regimes, the transport equation has a limit in which the leading behavior of its solution is determined by the solution of a diffusion equation. A boundary layer with a thickness of a few mean free paths usually forms between the domain boundary and the interior region, through which the given transport boundary conditions are matched to the interior tablesolution by properly choosing the boundary conditions for the diffusion equation. In order for a numerical scheme to be effective in these regimes, it must have both a correct interior diffusion limit and a correct boundary condition limit. The behavior of the discrete-ordinate method is studied in these limits and formulas for the resulting diffusion equation and its boundary conditions are found. By imposing the condition that these limiting formulas be the same as those for the transport equation, a new constraint on the quadrature set is obtained and a way to discretize the boundary data based on the discrete W-function ...
83 citations
TL;DR: In this paper, the integrable initial boundary value problem on a semi-line for the nonlinear Schrodinger equation is considered, and it is shown that by means of the Backlund transformation this problem can be reduced to the well known Cauchy problem for the same equation on the line.
Abstract: The integrable initial boundary value problem on a semi-line for the nonlinear Schrodinger equation is considered. It is shown that by means of the Backlund transformation this problem can be reduced to the well known Cauchy problem for the same equation on the line.
78 citations
TL;DR: In this paper, the authors considered boundary-value problems for a parabolic equation with mixed derivatives, where the coefficient of the highest-order derivatives involves a parameter varying in the half-open interval (0, 1), and showed that the attempt to use adjustive methods to construct difference schemes that are uniformly convergent (with respect to the parameter) for such systems meets certain difficulties.
Abstract: Boundary-value problems for an equation of parabolic type, in which the coefficient of the highest-order derivatives involves a parameter varying in the half-open interval (0,1], are considered. As the parameter approaches zero, parabolic boundary layers develop near the boundary of the domain. It is shown that the attempt to use adjustive methods to construct difference schemes that are uniformly convergent (with respect to the parameter) for such systems meets certain difficulties; in fact, for uniform grids there is no such adjustive scheme. A study is presented of two problems for a parabolic equation with mixed derivatives: a periodic boundary-value problem in a strip and the Dirichlet problem in a two-dimensional domain whose boundary is a smooth curve. In both cases it is possible to construct difference schemes that converge uniformly in the parameter throughout the domain.
78 citations
TL;DR: In this article, the small-slope theory of Mullins is generalized to the entire range of dihedral angles by using a boundary integral formulation of the associated free boundary problem, and excellent agreement with experimental groove shapes is obtained by using the diffusivity measured by Jordon and Hunt.
TL;DR: The generalized multipole technique as mentioned in this paper is a new method for solving electromagnetic boundary value problems, which is used which may be thought of as equivalent sources which are displaced from the boundary of the scatterer.
TL;DR: In this article, a class of Hamilton-Jacobi equations arising in generalized time-optimal control problems and differential games is considered and the uniqueness of the continuous solution and of the free boundary is proved in the framework of viscosity solutions.
Abstract: A class of Hamilton-Jacobi equations arising in generalized timeoptimal control problems and differential games is considered. The natural global boundary value problem for these equations has a singular boundary condition on a free boundary. The uniqueness of the continuous solution and of the free boundary is proved in the framework of viscosity solutions. A local uniqueness theorem is also given, as well as some existence results and several applications to control and game theory. In particular a relaxation theorem (weak form of the bang-bang principle) is proved for a class of nonlinear differential games. 0. Introduction In this paper we study viscosity solutions of Hamilton-Jacobi (HJ) equations of the form (0.1) H(x,DU) = 0, for Hamiltonians admitting the representation (0.2) H (x, p) :=minma\\{-f(x, a, b)-p h(x, a, b)} for all jc ,/? € R , beB a£A with A,B compact, f,h sufficiently smooth, and h satisfying the condition (0.3) h(x,a,b)>h0>0 for all x e RN, a e A, b e B. We prove two types of uniqueness results which seem to be characteristic of this particular class of HJ equations. The first problem we consider is ' H(x, DU) = 0 infiVT\", (0.4) \\ U = g on&^, U(x) -* +ce as x —y x0 e dQ, where 3~ and g are given, f7~ c R^ is closed. Under very general assumptions we show that there exists at most one pair ( U, Q) such that U is continuous Received by the editors December 7, 1988 and, in revised form, April 17, 1989. This paper was presented at the conferences \"I Reunion Hispano-ltaliana de Analisis no Lineal y Matemática Aplicada\", El Escorial (Spain), June 5-9, 1989, and \"Workshop on Control Theory\", Pisa (Italy), July 17-18, 1989. 1980 Mathematics Subject Classification (1985 Revision). Primary 35F20, 49C20, 90D25. ©1991 American Mathematical Society 0002-9947/91 $1.00+ $.25 per page
TL;DR: A free boundary problem related to singular stochastic control: the parabolic case was studied in this article. But the problem was not addressed in this paper, since it is a special case of the problem.
Abstract: (1991). A free boundary problem related to singular stochastic control: the parabolic case. Communications in Partial Differential Equations: Vol. 16, No. 2-3, pp. 373-424.
TL;DR: In this article, the initial and boundary layer analyses of an asymptotic expansion that yields a diffusive description of linear particle transport are carried out in some generality, including boundary curvature, the variation of the transport boundary condition along the bounding surface, and spatial and temporal variations of the interaction coefficients.
Abstract: The initial and boundary layer analyses of an asymptotic expansion that yields a diffusive description of linear particle transport are carried out in some generality. This yields initial and boundary conditions that apply to the diffusion equation previously reported in the literature. Effects treated include boundary curvature, the variation of the transport boundary condition along the bounding surface, and spatial and temporal variations of the interaction coefficients (cross sections) in the initial and boundary layers.
TL;DR: In this article, the authors studied stochastic differential equations with boundary conditions at the endpoints of a time interval (instead of the customary initial condition), and proved that the solution is a Markov field and the drift is affine.
Abstract: In this paper, we study stochastic differential equations with boundary conditions at the endpoints of a time interval (instead of the customary initial condition). We present existence and uniqueness results and study the Markov property of the solution. In the one-dimensional case, we prove that the solution is a Markov field $\operatorname{iff}$ the drift is affine.
TL;DR: In this paper, the free boundary is determined by an algebraic relation and an approximate solution derived, and a suitable modification of the approximate solution gives the exact solution, which implies the expression determined by the algebraic relations is the true critical price, at which the option should be exercised.
Abstract: The solution of the American option valuation problem is the solution of a parabolic partial differential equation satisfying free boundary conditions. The free boundary represents the critical price, at which the option should be exercised. In this paper the free boundary is determined by an algebraic relation and an approximate solution derived. A suitable modification of the approximate solution gives the exact solution. The uniqueness of the free boundary implies the expression determined by the algebraic relation is the true critical price
TL;DR: In this paper, an integral equation technique for solving stochastic boundary value problems in groundwater flow is presented, where the mean and covariance of head and flux are derived using the distribution of fictitious sources and dipoles.
Abstract: This paper presents an integral equation technique for solving stochastic boundary value problems in groundwater flow The aquifer considered has deterministic hydraulic conductivity but is subject to random boundary condition and domain recharge Using the distribution of fictitious sources and dipoles, stochastic integral equations for the mean and covariance of head and flux are derived An iterative boundary element technique is applied for numerical solution Two one-dimensional examples are examined and compared with exact solution A two-dimensional problem is then presented
TL;DR: In this article, various layer potential operators are constructed for general elliptic systems of partial differential equations with constant coefficients, and some regularity results for the Dirichlet problem are obtained.
TL;DR: In this paper, the augmented Lagrangian method is applied to impose a friction boundary condition on walls sustaining very large shear-stresses, and numerical implementation of this boundary condition involves variational inequalities that are briefly described in the context of Bingham fluid.
Abstract: The augmented Lagrangian method is applied to impose a friction boundary condition on walls sustaining very large shear-stresses. If the flow field is such that the wall shear-stress is smaller than the threshold value, the fluid adherence to the boundary is imposed and we recover the classical no-slip condition. When the wall shear stress reaches the critical value, the fluid is allowed to slip along the boundary with a given friction coefficient. This can be viewed as a free boundary problem since the regions of the boundary where the fluid will adhere or slip are not known a priori. The numerical implementation of this boundary condition involves variational inequalities that are briefly described in the context of a Bingham fluid. Numerical examples will also be presented.
01 Jun 1991
01 Jan 1991
TL;DR: This paper deals with interface operators in boundary value problems: how to define them, which is their meaning in both mathematical and physical sense, how to use them to derive numerical approximations based on domain decomposition approaches.
Abstract: This paper deals with interface operators in boundary value problems: how to define them, which is their meaning in both mathematical and physical sense, how to use them to derive numerical approximations based on domain decomposition approaches. When matching partial differential equations set in adjacent subregions of a domain Ω of ℝn, the interface operators ensure the fulfillement of transmission conditions between the different solutions. From the mathematical side, they make it possible to reduce the overall boundary value problem into a subproblem depending solely on the trace of the solution upon the interface. Once the solution of such a problem is available, the original solution can be reconstructed through the solution of independent boundary value problems within each subregion. The above independency feature is very likely behind the increasing interest for the use in the recent years of interface operators in scientific computing. Indeed, the subdomain approach yields a problem to be solved for the interface gridvalues only, then a family of reduced problems are left to solve simultaneously, hopefully by a multiprocessor architecture.
TL;DR: In this paper, a novel approach is presented, where hyper-singular kernels for stresses on the boundary are made numerically tractable through the imposition of certain equilibrated displacement modes.
Abstract: The boundary element method (BEM) for linear elasticity in its curent usage is based on the boundary integral equation for displacements. The stress field in the interior of the body is computed by differentiating the displacement field at the source point in the BEM formulation, via the strain field. However, at the boundary, this method gives rise to a hypersingular integral relation which becomes numerically intractable. A novel approach is presented here, where hyper-singular kernels for stresses on the boundary are made numerically tractable through the imposition of certain equilibrated displacement modes. Numerical results are also presented for benchmark problems, to illustrate the efficacy of the present approach. Solutions are compared to the commonly used boundary stress algorithm wherein the boundary stresses are computed from known boundary tractions, and derivatives of known displacements tangential to the boundary. An extension of this approach to solve linear elasticity problems using the traction boundary integral equation (TBIE) is also discussed.
TL;DR: In this paper, a boundary element method is described for the solution of two-dimensional steady-state convective heat diffusion problems in homogeneous and isotropic media with both linear and nonlinear boundary conditions.
TL;DR: In this article, the problem of the wave field continuation and recovering of coefficients for the wave equation in a bounded domain in R n, n > 1 was considered, where the inverse data is a response operator mapping Neumann boundary data into Dirichlet ones.
Abstract: We consider the problem of the wave field continuation and recovering of coefficients for the wave equation in a bounded domain in R n , n > 1 . The inverse data is a response operator mapping Neumann boundary data into Dirichlet ones. The reconstruction procedure is local. This means that, observing boundary response for larger times, we may recover coefficients deeper in the domain. The approach is based upon ideas and results of the boundary control theory, yielding some natural multidimensional analogs of the classical Gel'fand-Levitan-Krein equations.