Showing papers on "Free boundary problem published in 1990"
TL;DR: It is shown that the optimal buying and selling policies are the local times of the two-dimensional process of bank and stock holdings at the boundaries of a wedge-shaped region which is determined by the solution of a nonlinear free boundary problem.
Abstract: In this paper, optimal consumption and investment decisions are studied for an investor who has available a bank account paying a fixed rate of interest and a stock whose price is a log-normal diffusion. This problem was solved by Merton and others when transactions between bank and stock are costless. Here we suppose that there are charges on all transactions equal to a fixed percentage of the amount transacted. It is shown that the optimal buying and selling policies are the local times of the two-dimensional process of bank and stock holdings at the boundaries of a wedge-shaped region which is determined by the solution of a nonlinear free boundary problem. An algorithm for solving the free boundary problem is given.
1,320 citations
Journal Article•
TL;DR: In this article, the boundary stabilizability of the solutions of the wave equation y''−Δy=0 in a bounded domain Ω⊂R n with smooth boundary Γ, subject to mixed boundary conditions y=0 on Γ 1 and δy/δv=F(x,y'), was studied.
323 citations
TL;DR: In this article, a boundary element method is developed to compute the electrostatic potential inside and around molecules in an electrolyte solution, and a set of boundary integral equations are derived based on the integral formulations of the Poisson equation and the linearized Poisson Boltzmann equation.
Abstract: A boundary element method is developed to compute the electrostatic potential inside and around molecules in an electrolyte solution. A set of boundary integral equations are derived based on the integral formulations of the Poisson equation and the linearized Poisson-Boltzmann equation. The boundary integral equations are then solved numerically after discretizing the molecular surface into a number of flat triangular elements. The method is applied to a spherical molecule for which analytical solutions are available. Use is made of both constant and linearly varying unknowns over the boundary elements, and the method is tested for various values of parameters such as the dielectric constant of the molecule, ionic strength, and the location of the interior point charge. The use of the boundary integral method incorporating the nonlinear Poisson-Boltzmann equation is also briefly discussed.
218 citations
TL;DR: In this article, a class of radiation boundary conditions for two-and three-dimensional elastic wave propagation is developed based on compositions of simple first-order operators, which are defined by relatively simple formulas that apply without modification to problems in both two and three dimensions.
Abstract: In this paper, a class of radiation boundary conditions for two-and three-dimensional elastic wave propagation is developed. The boundary conditions are based on compositions of simple first-order operators.Some recent empirical investigations have suggested that certain earlier radiation boundary conditions may be unstable if the ratio of P-wave velocity to S-wave velocity is sufficiently large. The boundary conditions developed in the present paper are stable for all values of the velocity ratio. The boundary operators used here are defined by relatively simple formulas that apply without modification to problems in both two and three dimensions. Given an arbitrary angle of incidence, some parameters in the boundary conditions can be adjusted to make the conditions perfectly absorbing for P- waves and/or S-waves traveling at that angle of incidence to the boundary.
170 citations
TL;DR: The concept of desingularization in three-dimensional boundary integral computations is re-examined in this article, where the singularity of the fundamental solution is placed away from the boundary and outside the domain of the problem.
Abstract: SUMMARY The concept of desingularization in three-dimensional boundary integral computations is re-examined. The boundary integral equation is desingularized by moving the singular points away from the boundary and outside the problem domain. We show that the desingularization gives better solutions to several problems. As a result of desingularization, the surface integrals can be evaluated by simpler techniques, speeding up the computation. The effects of the desingularization distance on the solution and the condition of the resulting system of algebraic equations are studied for both direct and indirect versions of the boundary integral method. Computations show that a broad range of desingularization distances gives accurate solutions with significant savings in the computation time. The desingularization distance must be carefully linked to the mesh size to avoid problems with uniqueness and ill-conditioning. As an example, the desingularized indirect approach is tested on unsteady non-linear three-dimensional gravity waves generated by a moving submerged disturbance; minimal computational difficulties are encountered at the truncated boundary. Boundary integral methods provide a powerful technique for the solution of linear, homogeneous boundary value problems. The method employs a fundamental solution, which satisfies the differential equation (and possibly part of the boundary conditions), to reformulate the problem as an integral equation on the boundary. In conventional boundary integral formulations, singularities of the fundamental solution are placed on the domain boundary. This requires special evaluation of singular integrands, which can result in costly numerical calculations. In time-dependent non-linear free surface problems' - a boundary integral problem is solved at each time step. Since most of the computation time is devoted to the boundary integral problem, an effective solution method is critical in the time-marching procedure. When the singularity of the fundamental solution is placed away from the boundary and outside the domain of the problem, a desingularized boundary integral equation is obtained. We will show two advantages to this desingularization: a more accurate solution can be obtained for a given truncation, and a numerical quadrature can be used to reduce the computational time to obtain the algebraic system representing the discretized boundary integral problem. There are two types of non-singular boundary integral formulations: direct and indirect. In the direct method, Green's second identity is used to derive the boundary integral equation, and the solution of the problem is obtained directly by solving the boundary integral equation. In the
164 citations
TL;DR: In this article, a singular perturbation analysis of the forward Fokker-Planck equation is presented, based on Kramers' approach, for a process with absorption at the boundary and a source at the attractor.
Abstract: This paper considers the problem of exit for a dynamical system driven by small white noise, from the domain of attraction of a stable state. A direct singular perturbation analysis of the forward equation is presented, based on Kramers’ approach, in which the solution to the stationary Fokker–Planck equation is constructed, for a process with absorption at the boundary and a source at the attractor. In this formulation the boundary and matching conditions fully determine the uniform expansion of the solution, without resorting to “external” selection criteria for the expansion coefficients, such as variational principles or the Lagrange identity, as in our previous theory. The exit density and the mean first passage time to the boundary are calculated from the solution of the stationary Fokker–Planck equation as the probability current density and as the inverse of the total flux on the boundary, respectively. As an application, a uniform expansion is constructed for the escape rate in Kramers’ problem o...
151 citations
TL;DR: In this paper, the convergence of the Galerkin approximation is shown to hold for the exact as well as the discretized problems, and the typical saddle point structure of such problems is analyzed.
Abstract: A class of transmission problems is considered in which a nonlinear variational problem in one domain is coupled with a linear elliptic problem in a second domain. A typical example is a problem from three-dimensional elasticity theory where an elastoplastic material is embedded into a linear elastic material. The nonlinear problem is given in variational form with a strictly convex functional. The linear elliptic problem is described by boundary integral equations on the coupling boundary. The typical saddle point structure of such problems is analyzed. Galerkin approximations are studied which consist of a finite element approximation in the first domain coupled with a boundary element method on the coupling boundary. The convergence of the Galerkin approximation is based on the saddle-point structure which is shown to hold for the exact as well as the discretized problems.
151 citations
TL;DR: In this article, the Gibbs-Thomson law relating the melting temperature to the mean curvature of the phase interface is considered and solutions, global in time, are constructed which satisfy the natural a priori estimates.
Abstract: The coupling of the Stefan equation for the heat flow with the Gibbs-Thomson law relating the melting temperature to the mean curvature of the phase interface is considered. Solutions, global in time, are constructed which satisfy the natural a priori estimates. Mathematically the main difficulty is to prove a certain regularity in time for the temperature and the indicator function of the phase separately. A capacity type estimate is used to give an Lx bound for fractional time derivatives.
150 citations
TL;DR: In this article, an open boundary condition is constructed for three dimensional primitive equation ocean circulation models, which utilises dominant balances in the governing equations to assist calculations of variables at the boundary.
Abstract: An open boundary condition is constructed for three dimensional primitive equation ocean circulation models. The boundary condition utilises dominant balances in the governing equations to assist calculations of variables at the boundary. The boundary condition can be used in two forms. Firstly as a passive one in which there is no forcing at the boundary and phenomena generated within the domain of interest can propagate outwards without distorting the interior. Secondly as an active condition where a model is forced by the boundary condition. Three simple idealised tests are performed to verify the open boundary condition, (1) a passive condition to test the outflow of free Kelvin waves, (2) an active condition during the spin up phase of an ocean, (3) finally an example of the use of the condition in a tropical ocean.
147 citations
TL;DR: In this article, the authors consider the solution of partial differential equations for initial/boundary conditions using the decomposition method and show that the partial solutions obtained from the seperate equations for the highest-ordered linear operator terms are identical when the boundary conditions are general, and asymptotically equal when boundary conditions in one independent variable are independet of other variables.
Abstract: We consider the solution of partial differential equations for initial/boundary conditions using the decomposition method. The partial solutions obtained from the seperate equations for the highest-ordered linear operator terms are shown to be identical when the boundary conditions are general, and asymptotically equal when the boundary conditions in one independent variable are independet of other variables.
TL;DR: In this paper, the stationary Vlasov-Poisson boundary value problem in a spatially one-dimensional domain is studied and uniqueness is proved for (physically realistic) boundary conditions which are decreasing functions of the velocity variable.
Abstract: The stationary Vlasov-Poisson boundary value problem in a spatially one-dimensional domain is studied The equations describe the flow of electrons in a plane diode Existence is proved when the boundary condition (the cathode emission distribution) is a bounded function which decays super-linearly or a Dirac mass Uniqueness is proved for (physically realistic) boundary conditions which are decreasing functions of the velocity variable It is shown that uniqueness does not always hold for the Dirac mass boundary conditions
TL;DR: In this paper, the Lagrangian boundary layer equations decouple into a momentum equation for the motion parallel to the boundary, and a hyperbolic continuity equation (essentially a conserved Jacobian) for the normal motion normal to the boundaries.
Abstract: Although unsteady, high-Reynolds number, laminar boundary layers have conventionally been studied in terms of Eulerian coordinates, a Lagrangian approach may have significant analytical and computational advantages. In Lagrangian coordinates the classical boundary layer equations decouple into a momentum equation for the motion parallel to the boundary, and a hyperbolic continuity equation (essentially a conserved Jacobian) for the motion normal to the boundary. The momentum equations, plus the energy equation if the flow is compressible, can be solved independently of the continuity equation. Unsteady separation occurs when the continuity equation becomes singular as a result of touching characteristics, the condition for which can be expressed in terms of the solution of the momentum equations. The solutions to the momentum and energy equations remain regular. Asymptotic structures for a number of unsteady 3-D separating flows follow and depend on the symmetry properties of the flow. In the absence of any symmetry, the singularity structure just prior to separation is found to be quasi 2-D with a displacement thickness in the form of a crescent shaped ridge. Physically the singularities can be understood in terms of the behavior of a fluid element inside the boundary layer which contracts in a direction parallel to the boundary and expands normal to it, thus forcing the fluid above it to be ejected from the boundary layer.
01 Jan 1990
TL;DR: In this paper, the first boundary-value problem is considered in relation to an equation of parabolic type in which the coefficient of the highest derivative is a small parameter, and a difference scheme which converges uniformly with respect to the parameter is constructed.
Abstract: The first boundary-value problem is considered in relation to an equation of parabolic type in which the coefficient of the highest derivative is a small parameter. The boundary function, which is specified at the initial time and at the endpoints of the interval has a discontinuity of the first kind at a finite number of points. A difference scheme, which converges uniformly with respect to the parameter, is constructed for this problem.
Journal Article•
TL;DR: An application of a third-order generalized boundary condition to scattering by a filled rectangular groove is presented in this article, where errors of such higher-order boundary conditions are addressed, and a correction for the present case is proposed.
Abstract: An application of a third-order generalized boundary condition to scattering by a filled rectangular groove is presented Deficiencies of such higher-order boundary conditions are addressed, and a correction for the present case is proposed As part of the process of examining and correcting the accuracy of the proposed generalized boundary conditions, an exact solution is developed and a comparison is provided with a solution based on the standard impedance boundary condition >
TL;DR: In this paper, the Navier-Stokes equations for solenoidal fields with scalar potentials have been derived and the equivalence with the original problem has been proved.
Abstract: The representation of solenoidal fields by means of two scalar potentials can be a very useful method for a wide range of problems, in particular for the incompressible Navier–Stokes equations, though in finite containers boundary conditions may not be easily handled. The differential equations for the potentials are of an order higher than the original Navier–Stokes ones. As a consequence additional boundary conditions are needed to solve them. These differential equations and the corresponding boundary conditions for any geometry have been derived and the equivalence with the original problem has been proved. Special emphasis has been laid on domains with nontrivial geometry in which integral boundary conditions appear. As an example, the results have been applied to the periodic Couette flow. In this case the integral boundary conditions can be avoided by an appropriate change of variables, hence reducing the order of the equations obtained.
TL;DR: In this paper, a fully-Galerkin approach to the coefficient recovery (parameter identification) problem for a linear parabolic partial differential equation is introduced, where the forward problem is discretised with a sinc basis in the temporal domain and a finite element basis in a spatial domain.
Abstract: A fully-Galerkin approach to the coefficient recovery (parameter identification) problem for a linear parabolic partial differential equation is introduced. The forward problem is discretised with a sinc basis in the temporal domain and a finite element basis in the spatial domain. Tikhonov regularisation is applied to deal with the ill-posedness of the inverse problem. In the solution of the resulting nonlinear optimisation problem, advantage is taken of the diagonalisation solution procedure used for the discretised forward problem. An example with noisy data is included.
TL;DR: In this paper, the similarity properties of nonlinear ordinary free boundary value problems are considered and a method for finding the location of the free boundary s through the first numerical integration and the numerical solution by means of a second integration is presented.
Abstract: We consider the similarity properties of nonlinear ordinary free boundary value problems, i.e., u″ = f(x,u,u′) x∊(0,s) s>0 u(0) = α; u(s) = u′(s) = 0; α≠0. By making use of group properties we show that for the two classes of problems it is possible to define a method that allows us to find the location of the free boundary s through the first numerical integration and the numerical solution by means of a second integration. Moreover, by requiring invariance of some parameter, we give an important extension of the method to solve a problem that does not belong to the two classes in point. Finally we remark that the method is self-validating.
TL;DR: In this article, a set of asymptotic boundary conditions for three-dimensional quasi-static problems for a spherical outer boundary is derived, and the second-order boundary condition is generalized to a box-shaped outer boundary to solve the potential problem of a rectangular microstrip patch.
Abstract: The finite element method is used to analyze open three-dimensional transmission line structures in the quasi-TEM regime. Starting from the general solution of the Laplace equation in spherical coordinates, a set of asymptotic boundary conditions are derived for three-dimensional quasi-static problems for a spherical outer boundary. The second-order boundary condition is generalized to a box-shaped outer boundary and implemented in the finite element method to solve the potential problem of a rectangular microstrip patch. Numerical results show that the asymptotic boundary conditions yield more accurate results than those obtainable with a perfectly conducting shield placed at the same location. >
TL;DR: In this article, the free boundary problem has a unique solution which is radially symmetric using a best possible maximum principle, and the authors show that this solution has radially asymmetric solution.
Abstract: Let Ωi ⊂ ℝN, i = 0, 1, be two bounded separately star-shaped domains such that We consider the electrostatic potential u defined in :
The geometry of the two boundary components Γ0 and Γ1 is not given, but instead the electrostatic potential u is supposed to satisfy the further boundary conditions
Using a best possible maximum principle, we show that this free boundary problem has a unique solution which is radially symmetric
TL;DR: In this article, the existence, uniqueness, and continuous dependence of solutions of the Helmholtz equation, which are subject to either resistive or conductive conditions at the boundary of a closed smooth scatterer, are considered.
Abstract: The existence, uniqueness, and continuous dependence of solutions of the Helmholtz equation, which are subject to either resistive or conductive conditions at the boundary of a closed smooth scatterer, are considered. A boundary integral equation of the second kind for each problem whose unique solution is the trace on the boundary of the unique solution of that problem is devised. Continuous dependence results are proven using the integral equations.
TL;DR: In this paper, a nonlinear boundary value problem for Laplace's equation is solved numerically by using a reformulation as a non-linear boundary integral equation, both using product integration to approximate the singu- lar integrals in the equation.
Abstract: A nonlinear boundary value problem for Laplace's equation is solved numerically by using a reformulation as a nonlinear boundary integral equation. Two numerical methods are proposed and analyzed for discretizing the integral equation, both using product integration to approximate the singu- lar integrals in the equation. The first method uses the product Simpson's rule, and the second is based on trigonometric interpolation. Iterative methods (in- cluding two-grid methods) for solving the resulting nonlinear systems are also discussed extensively. Numerical examples are included.
TL;DR: In this article, the problem of boundary layer flow on a flat plate with injection and a constant velocity opposite in direction to that of the uniform mainstream is analyzed, and the critical values of λ andC are obtained numerically and their significance in drag reduction is discussed.
Abstract: The problem of boundary layer flow on a flat plate with injection and a constant velocity opposite in direction to that of the uniform mainstream is analyzed. It is shown that the solution of this boundary layer problem not only depends on the ratio of the velocity of the plate to the velocity of the free stream (λ), but also on the injection velocity parameter (C). It is also shown that there exists a range of values of λ andC for which the differential equations associated with the boundary layer problems admit analytic solutions. The critical values of λ andC are obtained numerically and their significance in drag reduction is discussed.
TL;DR: In this article, a hierarchy of absorbing boundary conditions is derived for transonic small perturbation equations of unsteady flows, and the best approximations must be in the canonical forms which can be factorized into first-order operators.
TL;DR: In this paper, the second-order necessary conditions of the Kuhn-Tucker type for optimality in a domain optimization problem are studied, and the boundary data are given in terms of the solution and the first variation.
Abstract: Second-order necessary conditions of the Kuhn-Tucker type for optimality in a domain optimization problem are studied. The second variation, corresponding to a boundary variation, of the solution to a boundary-value problem is shown to exist and is given as the solution of a boundary-value problem of the same type. The boundary data are shown to be given in terms of the solution and the first variation of the solution. From these results, the second variation of the objective function is calculated to derive second-order necessary conditions of the Kuhn-Tucker type.