Showing papers on "Free boundary problem published in 1993"
01 Jan 1993
TL;DR: In this article, the authors describe some of the recent developments in the mathematical theory of linear and quasilinear elliptic and parabolic systems with nonhomogeneous boundary conditions.
Abstract: It is the purpose of this paper to describe some of the recent developments in the mathematical theory of linear and quasilinear elliptic and parabolic systems with nonhomogeneous boundary conditions. For illustration we use the relatively simple set-up of reaction-diffusion systems which are — on the one h and — typical for the whole class of systems to which the general theory applies and — on the other h and — still simple enough to be easily described without too many technicalities. In addition, quasilinear reaction-diffusion equations are of great importance in applications and of actual mathematical and physical interest, as is witnessed by the examples we include.
704 citations
01 Jan 1993
306 citations
196 citations
TL;DR: The existence and instability of a third stationary solution with the same mass is proved: A spike-like solution called a canonical nucleus within the class of solutions which are even with respect to the center of the spike, it has a one-dimensional unstable manifold.
Abstract: When a constant metastable solution of the Cahn–Hilliard equation is subjected to a spatially localized large-amplitude perturbation, a transition process may be triggered leading to a globally stable stationary solution. In one space dimension, the existence and instability of a third stationary solution with the same mass is proved: A spike-like solution called a canonical nucleus. Within the class of solutions which are even with respect to the center of the spike, it has a one-dimensional unstable manifold. In addition, the process of nucleation by formal arguments using two space scales and two timescales is described. The last stage in the process can be approximated by a nonlinear Stefan free boundary problem.
171 citations
108 citations
TL;DR: In this paper, a mathematical model for the description of heat conduction in a conductor in the presence of Joule heating is considered, and the existence of a weak solution is proved using Schauder's fixed point theorem together with elliptic estimates.
106 citations
TL;DR: In this paper, the uniform stabilization of the wave equation by means of a nonlinear dissipative boundary feedback was studied, and a Neumann condition on the whole boundary was considered, and the observation is the boundary displacement and velocity.
Abstract: We study the uniform stabilization of the wave equation by means of a nonlinear dissipative boundary feedback. We consider a Neumann condition on the whole boundary, and the observation is the boundary displacement and velocity. Extending a result of E. Zuazua, we obtain, in a nonlinear framework, estimates of the decay, for any displacement. We establish a similar result for the one-dimensional wave equation with a variable coefficient.
81 citations
01 Jan 1993
TL;DR: In this paper, the authors present an approach for the regularization of the Displacement and Traction BIE for 3D Elastodynamic using indirect methods, which is based on the Symmetric Galerkin Boundary Element Method and Coupled BEM/FEM.
Abstract: 1. Regularization of the Displacement and Traction BIE for 3D Elastodynamic Using Indirect Methods.- 2. Stochastic Boundary Element Methods.- 3. A BEM Approach for Transient Conduction-Convection in Machining Processes.- 4. Improved Integration Methods for P-Adaptive Boundary Element Techniques.- 5. Hybrid BE-FE Stress Analysis of the Excavation of a Tunnel Bifurcation on the Basis of a Substructuring Technique.- 6. Computational Analysis of Singular Integral Equations for Crack Problems.- 7. Analytical Contributions for Viscoelastic BEM Formulations in Time and Frequency Domain.- 8. Symbolic Computation of Hypersingular Boundary Integrals.- 9. On Volterra Boundary Integral Equations of the First Kind for Nonstationary Stokes Equations.- 10. On the Engineering Analysis of 2D Problems by the Symmetric Galerkin Boundary Element Method and Coupled BEM/FEM.- 11. Sparse Blocked Equation Solving Techniques in Boundary Element Analysis.- 12. Three-Dimensional Transient Coupled Analysis of Groundwater Flow and Nuclide Migration by the Boundary Element Method.- 13. Fortran Codes for the Evaluation of the Discrete Helmholtz Integral Operators.- 14. On a New Formulation for the Boundary Integration Equation Method ofElastostatics.- 15. A Galeridn Symmetric Boundary-Element Method in Plasticity: Formulation and Implementation.- 16. Potential Compressible Flows Around Helicopter Rotors in Arbitrary Motion.- 17. A. Consistent Boundary/Interior Element Method for Evolutive Elastic Plastic Structural Analysis.- 18. Integration of Boundary Element Analysis with Computer Aided Design.- 19. Some Boundary Methods for Analysis of Elastic-Wave Propagation.- 20. Symmetric Coupling of Finite Elements and Boundary Elements.- 21. Boundary Element Analysis of Nonlinear Free Surface Flow in Containers.- 22. Non-Conforming Boundary Elements for 3D Steady-State Electromagnetic Fields.- 23. Expert System for Boundary Element Elastostatic Analysis.- 24. A Panel Method for the Simulation of Nonlinear Gravity Waves and Ship Motions.
78 citations
TL;DR: In this article, the spherically symmetric motion of viscous barotropic gas surrounding a solid ball was studied and the existence of a global weak solution with some regular properties was shown.
Abstract: We study the spherically symmetric motion of viscous barotropic gas surrounding a solid ball. We are interested in the density distribution which contacts with the vacuum at a finite radius. This is a free boundary problem. We obtained the existence of a global weak solution with some regular properties. We can show that such a solution is unique.
77 citations
TL;DR: In this paper, a family of finite-difference methods was developed for the solution of special nonlinear 8-order boundary-value problems, including methods with second-, fourth-, sixth-and eighth-order covergence.
Abstract: A family of finite-difference methods is developed for the solution of special nonlinear eighth-order boundary-value problems. Methods with second-, fourth-, sixth- and eighth-order covergence are contained in the family. The problem is also solved by writing the eighth-order differential equation as a system of four second-order differential equations. A second-order method is then used to obtain the solution.
75 citations
TL;DR: In this paper, the authors examined the Black-Scholes partial differential equation for the pricing of a traded option (an American call option on an asset paying a continuous dividend) and made comparisons with other well known free boundary diffusion problems, such as the oxygen consumption problem.
Abstract: We examine the Black–Scholes partial differential equation for the pricing of a traded option (an American call option on an asset paying a continuous dividend) and make comparisons with other well known free boundary diffusion problems, such as the oxygen consumption problem. The pricing of American options can be viewed as a free boundary problem and is, therefore, inherently nonlinear. We consider the short and long time behaviour of the free boundary, present analytic results for the option value in such limits, and consider the formulation of the problem as a variational inequality, and its numerical solution.
TL;DR: In this paper, the shape of growing crystals is determined by an interplay of complex processes that include transport of energy and matter through bulk phases, capillarity-related processes that determine local equilibrium conditions at the crystal-nutrient interface, and non-equilibrium kinetic processes that take place locally to that interface.
TL;DR: In this article, heat equation methods and invariance theory were used to compute the index of the classical elliptic complexes with pseudo-differential boundary conditions of Atiyah-Patodi-Singer type where the structure are not product near the boundary.
TL;DR: In this article, sufficient second-order optimality conditions are established for parabolic boundary control problems with nonlinear boundary condition and constraints on the control and the state, by means of a semigroup approach and a two-norm technique.
Abstract: In this paper sufficient second-order optimality conditions are established for parabolic boundary control problems with nonlinear boundary condition and constraints on the control and the state. The main idea is to extend the known theory for systems governed by ordinary differential equations to the case of partial differential equations. This is performed by means of a semigroup approach and a two-norm technique. The verification of the second-order conditions is discussed.
TL;DR: In this paper, a viscosity solution for the boundary control of a parabolic equation with Neumann boundary conditions is defined and the existence and uniqueness results for the viscoity solution are obtained.
TL;DR: In this article, Richardson derived a mathematical model for describing Hele-Shaw flows with a free boundary produced by the injection of fluid into a narrow channel, which can be represented in the following form (see also [3]): given fo(z), f0(0)=0, analytic and univalent in a neighbourhood of Izl_
Abstract: In [7] Richardson derived a mathematical model for describing Hele-Shaw flows with a free boundary produced by the injection of fluid into a narrow channel. This model can be represented in the following form (see also [3]): Given fo(z), f0(0)=0, analytic and univalent in a neighbourhood of Izl_
TL;DR: In this paper, a boundary element formulation for a general dynamic crack problem in a linear elastic material is presented, where the displacement equation is applied to one of the crack surfaces and the traction equation to the other.
Abstract: A boundary element formulation, which does not require domain integration, is presented for a general dynamic crack problem in a linear elastic material. The problem is solved using the dual boundary element method, that is, the displacement equation is applied to one of the crack surfaces and the traction equation to the other. Domain integrals in the elastodynamic equation are transformed into boundary integrals using the dual reciprocity method. As the result of the discretization a set of ordinary differential equations in time is obtained. Several numerical examples are considered, showing good agreement with the results given by other methods.
TL;DR: These boundary conditions have substantially smaller finite-size effects than periodic or open boundary conditions and can be applied to nearly any short-ranged Hamiltonian system in any dimensionality and within almost any type of numerical approach.
Abstract: We introduce a new type of boundary conditions, smooth boundary conditions, for numerical studies of quantum lattice systems. In a number of circumstances, these boundary conditions have substantially smaller finite-size effects than periodic or open boundary conditions. They can be applied to nearly any short-ranged Hamiltonian system in any dimensionality and within almost any type of numerical approach.
TL;DR: For the Laplace equation with Signorini boundary conditions, two equivalent boundary variational inequality formulations are deduced in this article, and an algorithm based on decomposition-coordination is used to solve the discretized problems.
Abstract: For the Laplace equation with Signorini boundary conditions two equivalent boundary variational inequality formulations are deduced. We investigate the discretization by a boundary element Galerkin method and obtain quasi-optimal asymptotic error estimates in the underlying Sobolev spaces. An algorithm based on the decomposition-coordination method is used to solve the discretized problems. Numerical examples confirm the predicted rate of convergence.
TL;DR: In this article, the temporarily global solution for the two phase free boundary problem is discussed and the global solution is obtained near the equilibrium state under sufficiently small initial data and external forces.
Abstract: We shall discuss the temporarily global solution for the two phase free boundary problem. Both fluids are regarded as immiscible, nonhomogeneous, viscous and incompressible and subject to surface tention on the interface. The global solution is obtained near the equilibrium state under the sufficiently small initial data and external forces.
TL;DR: In this paper, it was shown that the ordinary Poisson brackets in field theory do not fulfill the Jacobi identity if boundary values are not reasonably fixed by special boundary conditions, and that these brackets can be modified by adding some surface terms to lift this restriction.
Abstract: The ordinary Poisson brackets in field theory do not fulfill the Jacobi identity if boundary values are not reasonably fixed by special boundary conditions. It is shown that these brackets can be modified by adding some surface terms to lift this restriction. The new brackets generalize a canonical bracket considered by Lewis, Marsden, Montgomery, and Ratiu for the free boundary problem in hydrodynamics. The definition of Poisson brackets used herein permits the treating of to treat boundary values of a field on equal footing with its internal values and the direct estimation of estimate the brackets between both surface and volume integrals. This construction is applied to any local form of Poisson brackets.
TL;DR: In this article, a generalization of the canonical Poisson bracket for the free boundary problem in hydrodynamics is proposed, which treats boundary values of a field on equal footing with its internal values and directly estimates the brackets between both surface and volume integrals.
Abstract: The ordinary Poisson brackets in field theory do not fulfil the Jacobi identity if boundary values are not reasonably fixed by special boundary conditions. We show that these brackets can be modified by adding some surface terms to lift this restriction. The new brackets generalize a canonical bracket considered by Lewis, Marsden, Montgomery and Ratiu for the free boundary problem in hydrodynamics. Our definition of Poisson brackets permits to treat boundary values of a field on equal footing with its internal values and directly estimate the brackets between both surface and volume integrals. This construction is applied to any local form of Poisson brackets. A prescription for delta-function on closed domains and a definition of the {\it full} variational derivative are proposed.
TL;DR: In this article, a computationally convenient and simple procedure is presented to match solution to boundary conditions in two-point boundary value problems for linear differential and partial differential equations for the nonlinear case.
06 Jul 1993
01 Sep 1993
TL;DR: In this article, the authors discuss the construction of wavelet bases with preassigned boundary value conditions on the unit interval, which arise naturally from boundary value differential equations, and the wavelets are Riesz bases of the Sobolev spaces and of the Holder spaces where the solutions of these equations live.
Abstract: . We discuss the construction of wavelet bases with preassigned boundary value conditions on the unit interval. These boundary conditions arise naturally from boundary value differential equations, and the wavelets are Riesz bases of the Sobolev spaces and of the Holder spaces where the solutions of these equations live. Our construction rests on the theory of Hermite interpolation and on the theory of multiresolution analyses.
TL;DR: In this article, a regularized boundary integral equation relating the boundary displacement gradients to the tractions is proposed to calculate the boundary stress components, which can be evaluated numerically by using the standard Gaussian quadrature formula.
Abstract: Accurate computation of boundary stress components is important in stress-base shape optimization. The boundary element method usually gives us more accurate solutions of them compared with those obtained by the finite element method. In the present paper we propose a new approach to calculate the boundary stress components by using the regularized boundary integral equation relating the boundary displacement gradients to the tractions. Although the original integral equation must be evaluated in the sense of the Cauchy principal value, it can be regularized by subtracting and adding the displacement gradients at the source point from and to the original integral equation. When the displacement gradients satisfy the Holder continuity at the source point, the regularized integral equation becomes non-singular and can be evaluated numerically by using the standard Gaussian quadrature formula. The numerical implementation of the proposed integral equation is presented and the effectiveness of the approach is also discussed through some numerical demonstrations.
TL;DR: In this paper, the boundary conditions for the incompressible, time-dependent Navier-Stokes equation were investigated and new boundary conditions were proposed so that these boundary layers are suppressed.
TL;DR: In this paper, a probabilistic interpretation of a system of second order quasilinear elliptic partial differential equations under a Neumann boundary condition is obtained by introducing a kind of backward stochastic differential equations in the infinite horizon case.
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TL;DR: An overview of the free boundary problems connected to the problem of Stefan type or, more generally, to problems describing a change of phase can be found in this paper, where a sketch of the historical development of the research in the area is presented.
Abstract: This paper presents an overview of the free boundary problems connected to the problem of Stefan type or, more generally, to problems describing a change of phase. After a sketch of the historical development of the research in the area, some specific questions are addressed, such as classical solvability of the problem in several space dimensions, regularization of supercooling, dynamical contact angle.