Showing papers on "Free boundary problem published in 2007"
TL;DR: In this article, a new method for treating free boundary problems in perfect fluids was proposed, and proved local-in-time wellposedness in Sobolev spaces for the free-surface incompressible 3D Euler equations with or without surface tension for arbitrary initial data, and without any irrotationality assumption on the fluid.
Abstract: We provide a new method for treating free boundary problems in perfect fluids, and prove local-in-time well-posedness in Sobolev spaces for the free-surface incompressible 3D Euler equations with or without surface tension for arbitrary initial data, and without any irrotationality assumption on the fluid. This is a free boundary problem for the motion of an incompressible perfect liquid in vacuum, wherein the motion of the fluid interacts with the motion of the free-surface at highest-order
433 citations
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17 Apr 2007TL;DR: In this article, the authors approximate the approximate bounding matrix of boundary element matrices using boundary integral integral equations and approximate boundary element matrix approximations, based on the approximation of boundary element matrix.
Abstract: Boundary Integral Equations.- Boundary Element Methods.- Approximation of Boundary Element Matrices.- Implementation and Numerical Examples.
242 citations
TL;DR: Numerical results show that the formulation that incorporates the law of the wall yields an improvement over the original method, which presents a robust technique for flows of industrial interest.
224 citations
TL;DR: In this article, the problem of solving the one-dimensional parabolic partial differential equation subject to given initial and non-local boundary conditions is considered, and several approaches for the numerical solution of this boundary value problem which have been considered in the literature, are reported.
Abstract: Various processes in the natural sciences and engineering lead to the nonclassical parabolic initial boundary value problems which involve nonlocal integral terms over the spatial domain. The integral term may appear in the boundary conditions. It is the reason for which such problems gained much attention in recent years, not only in engineering but also in the mathematics community. In this paper the problem of solving the one-dimensional parabolic partial differential equation subject to given initial and nonlocal boundary conditions is considered. Several approaches for the numerical solution of this boundary value problem which have been considered in the literature, are reported. New finite difference techniques are proposed for the numerical solution of the one-dimensional heat equation subject to the specification of mass. Numerical examples are given at the end of this paper to compare the efficiency of the new techniques. Some specific applications in various engineering models are introduced.
156 citations
TL;DR: In this article, a system of semilinear parabolic equations with a free boundary was studied in a predator-prey ecological model. And the conditions for the existence and uniqueness of a classical solution were obtained.
Abstract: This article is concerned with a system of semilinear parabolic equations with a free boundary, which arises in a predator–prey ecological model The conditions for the existence and uniqueness of a classical solution are obtained The evolution of the free boundary problem is studied It is proved that the problem addressed is well posed, and that the predator species disperses to all domains in finite time
149 citations
TL;DR: This work derive and rigorously prove high order asymptotic expansions for the early exercise boundary near expiry, and provides an ode iterative scheme which can reach its numerical fixed point in five iterations for all time to expiry.
Abstract: We study a free boundary problem arising from American put options. In particular we prove existence and uniqueness for this problem, and we derive and rigorously prove high order asymptotic expansions for the early exercise boundary near expiry. We provide four approximations for the boundary: one is explicit and is valid near expiry (weeks); two others are implicit involving inverse functions and are accurate for longer time to expiry (months); the fourth is an ODE initial value problem which is very accurate for all times to expiry, is extremely stable, and hence can be solved instantaneously on any computer. We further provide an ode iterative scheme which can reach its numerical fixed point in five iterations for all time to expiry. We also provide a large time (equivalent to regular expiration times but large interest rate and/or volatility) behavior of the exercise boundary. To demonstrate the accuracy of our approximations, we present the results of a numerical simulation.
132 citations
TL;DR: In this paper, a nonlocal diffusion operator in a bounded smooth domain prescribing the flux through the boundary is studied, and the authors prove existence, uniqueness and a comparison principle for the problem.
128 citations
TL;DR: This Poisson solver is primarily intended for real space methods where the charge density and the potential are given on a uniform grid and has an O(N log N) computational cost.
Abstract: We present a method that gives highly accurate electrostatic potentials for systems where we have periodic boundary conditions in two spatial directions but free boundary conditions in the third direction. These boundary conditions are needed for all kinds of surface problems. Our method has an O(N log N) computational cost, where N is the number of grid points, with a very small prefactor. This Poisson solver is primarily intended for real space methods where the charge density and the potential are given on a uniform grid.
128 citations
TL;DR: By means of He's homotopy perturbation method (HPM), an approximate solution of a boundary layer equation in unbounded domain is obtained and the method is very effective and simple.
Abstract: By means of He's homotopy perturbation method (HPM) an approximate solution of a boundary layer equation in unbounded domain is obtained. Comparison is made between the obtained results and those in open literature. The results show that the method is very effective and simple.
115 citations
TL;DR: Two widely used kinetic boundary conditions in the lattice Boltzmann equation: the combination of the bounce-back and specular-reflection scheme and the discrete Maxwell's scheme are studied.
Abstract: The lattice Boltzmann equation (LBE) has shown its promise in the simulation of microscale gas flows. One of the critical issues with this advanced method is to specify suitable slip boundary conditions to ensure simulation accuracy. In this paper we study two widely used kinetic boundary conditions in the LBE: the combination of the bounce-back and specular-reflection scheme and the discrete Maxwell's scheme. We show that (i) both schemes are virtually equivalent in principle, and (ii) there exist discrete effects in both schemes. A strategy is then proposed to adjust the parameters in the two kinetic boundary conditions such that an accurate slip boundary condition can be implemented. The numerical results demonstrate that the corrected boundary conditions are robust and reliable.
114 citations
TL;DR: In this paper, the problem of imaging the conductivity from knowledge of one current and corresponding voltage on a part of the boundary of an inhomogeneous isotropic object and of the magnitude |J(x)| of the current density inside was reduced to a boundary value problem with partial data.
Abstract: We consider the problem of imaging the conductivity from knowledge of one current and corresponding voltage on a part of the boundary of an inhomogeneous isotropic object and of the magnitude |J(x)| of the current density inside. The internal data are obtained from magnetic resonance measurements. The problem is reduced to a boundary value problem with partial data for the equation ∇ |J(x)||∇u|−1∇u = 0. We show that equipotential surfaces are minimal surfaces in the conformal metric |J|2/(n−1)I. In two dimensions, we solve the Cauchy problem with partial data and show that the conductivity is uniquely determined in the region spanned by the characteristics originating from the part of the boundary where measurements are available. We formulate sufficient conditions on the Dirichlet data to guarantee the unique recovery of the conductivity throughout the domain. The proof of uniqueness is constructive and yields an efficient algorithm for conductivity imaging. The computational feasibility of this algorithm is demonstrated in numerical experiments.
TL;DR: This paper considers thermoacoustic tomography as the inverse problem of determining from lateral Cauchy data the unknown initial conditions in a wave equation with spatially varying coefficients and derives a numerical scheme for the solution of the quasi-reversibility problem by a $B$-spline Galerkin method.
Abstract: In this paper we consider thermoacoustic tomography as the inverse problem of determining from lateral Cauchy data the unknown initial conditions in a wave equation with spatially varying coefficients. This problem also occurs in several applications in the area of medical imaging and nondestructive testing. Using the method of quasi-reversibility, the original ill-posed problem is replaced with a boundary value problem for a fourth order partial differential equation. We find a weak $H^2$ solution of this problem and show that it is a well-posed elliptic problem. Error estimates and convergence of the approximation follow from observability estimates for the wave equation, which are proved using a Carleman estimate. We derive a numerical scheme for the solution of the quasi-reversibility problem by a $B$-spline Galerkin method, for which we give error estimates. Finally, we present numerical results supporting the robustness of this method for the reconstruction of initial conditions from full and limited boundary data.
TL;DR: In this paper, the well-posedness problem of an evolutionary weighted p -Laplacian with boundary degeneracy was studied, and it was shown that the degenerate portion of the boundary should be decomposed into two parts.
TL;DR: In this paper, the outer boundary of a Schwarzschild black hole with an outgoing gravitational wave perturbation is compared to a reference solution obtained by placing the boundary at a large radius, and the full solutions including constraint violations and extracted gravitational waves are compared to those of the reference solution.
Abstract: Various methods of treating outer boundaries in numerical relativity are compared using a simple test problem: a Schwarzschild black hole with an outgoing gravitational wave perturbation. Numerical solutions computed using different boundary treatments are compared to a 'reference' numerical solution obtained by placing the outer boundary at a very large radius. For each boundary treatment, the full solutions including constraint violations and extracted gravitational waves are compared to those of the reference solution, thereby assessing the reflections caused by the artificial boundary. These tests are based on a first-order generalized harmonic formulation of the Einstein equations and are implemented using a pseudo-spectral collocation method. Constraint-preserving boundary conditions for this system are reviewed, and an improved boundary condition on the gauge degrees of freedom is presented. Alternate boundary conditions evaluated here include freezing the incoming characteristic fields, Sommerfeld boundary conditions, and the constraint-preserving boundary conditions of Kreiss and Winicour. Rather different approaches to boundary treatments, such as sponge layers and spatial compactification, are also tested. Overall the best treatment found here combines boundary conditions that preserve the constraints, freeze the Newman–Penrose scalar Ψ0, and control gauge reflections.
TL;DR: In this paper, the existence and nonexistence of positive solutions for the nonlinear fractional boundary value problem (D0) was studied. But the existence of positive solution was not investigated.
Abstract: We are concerned with the existence and nonexistence of positive solutions for the nonlinear fractional boundary value problem: D0
TL;DR: An optimal control approach for the solidification process of a melt in a container using a sharp interface model for the control of the free boundary based on a two phase Stefan problem modelled as a graph.
TL;DR: Variational iteration technique can be viewed as an efficient and reliable method for solving a wide class of linear and nonlinear boundary value problems.
Abstract: In this paper, we apply the variational iteration method for solving fourth order boundary value problems. The analytical results are in terms of convergent series with easily computable components. Several examples are given to illustrate the efficiency and implementation of the method. Comparison is made to confirm the reliability of this technique. Variational iteration technique can be viewed as an efficient and reliable method for solving a wide class of linear and nonlinear boundary value problems.
TL;DR: In this article, the scaling limits of three different aggregation models on Z^d were studied: internal DLA, where particles perform random walks until reaching an unoccupied site; the rotor-router model, in which particles perform deterministic analogues of random walks; and the divisible sandpile, where each site distributes its excess mass equally among its neighbors.
Abstract: We study the scaling limits of three different aggregation models on Z^d: internal DLA, in which particles perform random walks until reaching an unoccupied site; the rotor-router model, in which particles perform deterministic analogues of random walks; and the divisible sandpile, in which each site distributes its excess mass equally among its neighbors. As the lattice spacing tends to zero, all three models are found to have the same scaling limit, which we describe as the solution to a certain PDE free boundary problem in R^d. In particular, internal DLA has a deterministic scaling limit. We find that the scaling limits are quadrature domains, which have arisen independently in many fields such as potential theory and fluid dynamics. Our results apply both to the case of multiple point sources and to the Diaconis-Fulton smash sum of domains.
TL;DR: In this article, the inverse problem is used to recover the interior boundary curve of a doubly connected planar domain from a pair of Cauchy data on the exterior boundary of the planar curve.
Abstract: We consider the inverse problem to recover a part $\Gamma_c$
of the boundary of a simply connected planar
domain
$D$ from a pair of Cauchy data of a harmonic function $u$ in $D$ on the remaining part
$\partial D\setminus \Gamma_c$ when $u$ satisfies a homogeneous impedance boundary
condition on $\Gamma_c$. Our approach extends a method that has been suggested
by Kress and Rundell [17] for recovering the interior boundary curve of
a doubly connected planar domain from a pair of Cauchy data on the exterior boundary
curve and is based on a system of nonlinear integral equations. As a byproduct,
these integral equations can also be used for the problem to extend incomplete Cauchy data and to solve
the inverse problem to recover an impedance profile on a known boundary curve.
We present
the mathematical foundation of the method and illustrate
its feasibility by numerical examples.
TL;DR: In this article, boundary integral equation (BIE) methods for elastic wave propagation problems are discussed and compared. But they can be grouped into two families: direct and indirect BIE methods.
Abstract: We review the application of boundary integral equation (BIE) methods to elastic wave propagation problems. BIE methods express the wavefield as an integral equation defined over the boundary of the domain studied. They can be grouped into two families. “Direct” BIE relate the wavefield (generally the displacements and tractions) to the values of the wavefield at the boundary of the domain, while “Indirect” BIE rely on an intermediate unknown, which is usually a distribution of fictitious sources along the boundary. We present the mathematical bases of the methods and discuss and review their various numerical implementations. We illustrate some of the applications for seismic wave propagation.
TL;DR: In this paper, a mixed boundary value problem for the Stokes system in a polyhedral domain is considered, where different boundary conditions (in particular, Dirichlet, Neumann, free surface conditions) are prescr...
Abstract: A mixed boundary value problem for the Stokes system in a polyhedral domain is considered. Here different boundary conditions (in particular, Dirichlet, Neumann, free surface conditions) are prescr ...
TL;DR: In this paper, an approximate analytical solution of the boundary value problem (BVP) for the nonlinear shallow waters equations is proposed, based on the Carrier and Greenspan [1] hodograph transformation.
Abstract: We propose an approximate analytical solution of the boundary value problem (BVP) for the nonlinear shallow waters equations. Our work, based on the Carrier and Greenspan [1] hodograph transformation, focuses on the propagation of nonlinear nonbreaking waves over a uniformly plane beach. Available results are briefly discussed with specific emphasis on the comparison between the Initial Value Problem and the BVP; the latter more completely representing the physical phenomenon of wave propagation on a beach. The solution of the BVP is achieved through a perturbation approach solely using the assumption of small waves incoming at the seaward boundary of the domain. The most significant results, i.e., the shoreline position estimation, the actual wave height and velocity at the seaward boundary, the reflected wave height and velocity at the seaward boundary are given for three specific input waves and compared with available solutions.
TL;DR: The inverse problems consisting in the simultaneous estimation of unknown thermophysical and/or geometrical parameters of the tumor region are solved and the evolutionary algorithm coupled with the multiple reciprocity boundary element method has been applied.
01 Jan 2007
TL;DR: In this article, the authors describe the shooting method and its Maple implementation, shoot, which is used to analyze three common two-point boundary value problems from chemical engineering: the Blasius solution for laminar boundary-layer flow past a at plate, the reactivity behavior of porous catalyst particles subject to both internal mass concentration gradients and temperature gradients, and the steady-state flow near an in finite rotating disk.
Abstract: One of the strengths of Maple is its ability to provide a wide variety of information about di erential equations. Explicit, implicit, parametric, series, Laplace transform, numerical, and graphical solutions can all be obtained via the dsolve command. Numerical solutions are of particular interest due to the fact that exact solutions do not exist, in closed form, for most engineering and scienti c applications. The numerical solution methods available within dsolve are applicable only to initial value problems. Thus, at rst glance, Maple appears to be very limited in its ability to analyze the multitude of two-point boundary value problems that occur frequently in engineering analysis. A commonly used numerical method for the solution of two-point boundary value problems is the shooting method. This well-known technique is an iterative algorithm which attempts to identify appropriate initial conditions for a related initial value problem (IVP) that provides the solution to the original boundary value problem (BVP). The rst objective of this paper is to describe the shooting method and its Maple implementation, shoot. Then, shoot is used to analyze three common two-point BVPs from chemical engineering: the Blasius solution for laminar boundary-layer ow past a at plate, the reactivity behavior of porous catalyst particles subject to both internal mass concentration gradients and temperature gradients, and the steady-state ow near an in nite rotating disk.
TL;DR: In this paper, it was shown that the optimal exercise boundary for the American put option with non-dividend-paying asset is convex and a simple rigorous argument providing an accurate asymptotic behavior for the exercise boundary near expiry.
Abstract: We show that the optimal exercise boundary for the American put option with non-dividend-paying asset is convex. With this convexity result, we then give a simple rigorous argument providing an accurate asymptotic behavior for the exercise boundary near expiry.
TL;DR: In this paper, boundary asymptotics are proved for solutions of weighted p-Laplacian equations that take infinite value on the boundary of a bounded domain. But their results are restricted to the case where weights are unbounded in the domain.
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TL;DR: In this paper, the authors considered the Cahn-Hilliard equation with Wentzell boundary condition and proved the convergence of a global solution to an equilibrium as time goes to infinity by means of a suitable L ojasiewicz-Simon type inequality with boundary term.
Abstract: In this paper we consider the Cahn-Hilliard equation endowed with Wentzell boundary condition which is a model of phase separation in a binary mixture contained in a bounded domain with permeable wall. Under the assumption that the nonlinearity is analytic with respect to unknown dependent function, we prove the convergence of a global solution to an equilibrium as time goes to infinity by means of a suitable \L ojasiewicz-Simon type inequality with boundary term. Estimates of convergence rate are also provided.
TL;DR: In this paper, the authors established the existence, stability, and asymptotic behavior of transonic flows with a transonic shock for the steady, full Euler equations in two-dimensional infinite nozzles of slowly varying cross-sections.
TL;DR: In this article, a system of equations describing a motion of a self-gravitating one-dimensional gaseous medium in the presence of radiation and reacting process is considered, and the free-boundary problem is reduced to the problem in a fixed domain with an explicit gravitational term.
TL;DR: In this article, the Ohta-Kawasaki density functional theory of diblock copolymers gives rise to a nonlocal free boundary problem, where the boundary of the droplet satisfies an equation that involves the curvature of the boundary and a quantity that depends nonlocally on the whole pattern.
Abstract: The Ohta–Kawasaki density functional theory of diblock copolymers gives rise to a nonlocal free boundary problem. Under a proper condition between the block composition fraction and the nonlocal interaction parameter, a pattern of a single droplet is proved to exist in a general planar domain. A smaller parameter range is identified where the droplet solution is stable. The droplet is a set that is close to a round disc. The boundary of the droplet satisfies an equation that involves the curvature of the boundary and a quantity that depends nonlocally on the whole pattern. The location of the droplet is determined by the regular part of a Green’s function of the domain. This droplet pattern describes one cylinder in space in the cylindrical phase of diblock copolymer morphology.