Showing papers on "Free boundary problem published in 1986"

An analysis of a phase field model of a free boundary

Abstract: A mathematical analysis of a new approach to solidification problems is presented. A free boundary arising from a phase transition is assumed to have finite thickness. The physics leads to a system of nonlinear parabolic differential equations. Existence and regularity of solutions are proved. Invariant regions of the solution space lead to physical interpretations of the interface. A rigorous asymptotic analysis leads to the Gibbs-Thompson condition which relates the temperature at the interface to the surface tension and curvature.

Exact boundary conditions at an artificial boundary for partial differential equations in cylinders

Abstract: The numerical solution of partial differential equations in unbounded domains requires a finite computational domain. Often one obtains a finite domain by introducing an artificial boundary and imposing boundary conditions there. This paper derives exact boundary conditions at an artificial boundary for partial differential equations in cylinders. An abstract theory is developed to analyze the general linear problem. Solvability requirements and estimates of the solution of the resulting finite problem are obtained by use of the notions of exponential and ordinary dichotomies. Useful representations of the boundary conditions are derived using separation of variables for problems with constant tails. The constant tail results are extended to problems whose coefficients obtain limits at infinity by use of an abstract perturbation theory. The perturbation theory approach is also applied to a class of nonlinear problems. General asymptotic formulas for the boundary conditions are derived and displayed in detail.

A free boundary problem for semilinear elliptic equations.

Abstract: Etude de la nature de la frontiere libre Ω∩∂{u>0} pour un probleme de Dirichlet semilineaire

Symmetry, groups and boundary value problems. A progressive introduction to noncommutative harmonic analysis of partial differential equations in domains with geometrical symmetry

Abstract: Let Ω be a plane or spatial domain and G a group of isometries which leave Ω invariant. Suppose one has to solve on Ω a boundary value problem Au = f or an eigenvalue problem Au = μu . It is assumed (‘equivariance’) that the coefficients of A are also invariant by the action of G (but this is not required of the right-hand side ƒ). Then, instead of solving the original problem on the whole domain, one can solve a set of related problems (whose number does not exceed the order n of G ) on a reduced domain, the ‘symmetry cell’, n times smaller than the original Ω. The process, which obviously promises interesting savings in structural analysis and other fields, generalizes Fourier analysis and may be referred to as “non-commutative harmonic analysis” [1]. The theoretical foundations are essentially those of group representation theory. The paper, which is mainly expository, aims at introducing the minimal amount of this theory necessary to understand how the process works and how it can be implemented in finite element codes. It also addresses the question (which seems to have been neglected in the literature) of the derivation of proper boundary conditions for the subproblems to be solved on the symmetry cell.

Artificial boundary conditions for the linear advection diffusion equation

Abstract: A family of artificial boundary conditions for the linear advection diffusion equation with small viscosity is developed Well-posedness for the associated initial boundary value problem is analyzed The error produced by truncating the domain is estimated Numerical results are presented 1 Introduction When computing the solution of a partial differential equation in an unbounded domain, one often introduces artificial boundaries In order to limit the computational cost, these boundaries must be chosen not too far from the domain of interest Therefore, the boundary conditions must be good approxima- tions to the so-called " transparent"i boundary condition (ie,, such Ithat the solution of the problem in the bounded domain is equal to the solution in the original domain) The transparent boundary condition is usually an integral relation in time and space between u and its normal derivative on the boundary, which makes it impractical from a numerical point of view One must approximate this relation to get local boundary conditions: they are often called absorbing or artificial boundary conditions This question is of crucial interest in such different areas as geophysics, plasma physics, fluid dynamics (1), (2), (3), and the use of such conditions is now classical in geophysics Our interest for the linear advection diffusion equation comes from the Navier- Stokes equation, but it arises also in other fields as, for example, meteorology (6) The incompressible Navier-Stokes equation can be written as

Boundary Conditions at Spatial Infinity

Abstract: There are many, both conceptually and technically different ways to obtain the Adm expression for the energy of the gravitational field, some of the published methods containing inconsistencies, most of them raising doubts about uniqueness of the final result. The author wishes to present here a simple way of obtaining this expression in a geometrical hamiltonian setting allowing for an exact analysis of all ambiguities present. One of the results of this study is a considerable weakening of the boundary conditions at spatial infinity, for which the energy-momentum of an initial data set is finite and well defined. The derivation of the Adm hamiltonian presented here is the simplest one known to the author, as far as calculations are concerned.

An Integral Solution for the Diffusion-Advection Equation

Abstract: A new integral formulation, based on Green's second identity, is used to solve the unsteady transport (diffusion-advection) equation which governs the storage and movement of pollutants in porous media. It uses the fundamental solution for the terms of the differential equation with the highest derivatives (the order of the equation) and, by applying Green's second identity, those terms are cast into a boundary integral while the remaining terms with lower derivatives are weighted with that fundamental solution and integrated over the solution domain. The resulting integral representation is no longer a boundary integral but comprises both boundary and domain integral portions. Because the terms with the highest derivatives constitute the Laplacian, the fundamental solution employed in the formulation is the logarithmic function. The boundary integral portion of the formulation is similar to that encountered when the boundary element method is applied to elliptic equations but the domain portion, which contains the temporal derivative and advection terms, is evaluated by discretizing the solution domain into elements, as is frequently done in finite element formulations. The classic one-dimensional semi-infinite and a two-dimensional semi-infinite diffusion-advection problems are solved for a wide range of local Peclet numbers in order to verify and demonstrate the usefulness of the present formulation. The results are satisfactory.

On a Problem in the Polymer Industry: Theoretical and Numerical Investigation of Swelling

Abstract: A recent model for the penetration of solvents into polymers leads to a parabolic free boundary problem with unusual boundary conditions. It is shown that the model equations are well posed, and some qualitative features of the free boundary are established. A numerical method for the free boundary problem is suggested and its convergence is proved. A numerical calculation is included to illustrate the theoretical results.

Equations of Motion of Compressible Viscous Fluids

Abstract: We survey the global solutions of equations for one-dimensional motion of compressible, viscous and heat-conductive fluids. Initial value problems with fixed and free boundaries are treated about solutions global in time and about the asymptotic behaviors as time tends to infinity.

On the boundary behavior of orientation-preserving harmonic mappings

Abstract: Nonconstant soluliuns of the partial differential equation where a is analytic in the open unit disk and |a| <1, are orientation-preserving harmonic mappings. We consider the case where ais a finite Blaschke product, so that ‖a‖= 1, and describe the behavior at the boundary. Some applications include conditions under which the image is a polygon and a situation where a univalent solution does not exist that maps onto a prescribed domain with a normalization at three boundary points.

Necessary conditions for a domain optimization problem in elliptic boundary value problems

Abstract: This paper deals with a domain optimization problem suggested by a real physical problem, the shape optimization problem for high-beta plasmas in nuclear fusion research. The function of the space dependent variable, which is the solution of an elliptic boundary value problem defined on a variable domain, should be optimized. The fundamental equation which evaluates the variation of the solution according to the boundary variation is given. From this equation, a variational equation and its equivalent which relate the variation of the solution to the variation of the boundary are derived. This variational equation is exploited to derive a necessary condition for the optimality of the domain. The necessary condition is composed of the Euler–Lagrange equation in the wider sense and the transversality condition. Another form of the necessary condition is also obtained from the equivlaent variational equation.

Numerical solutions for Bayes sequential decision problems

Abstract: Certain sequential decision problems involving normal random variables reduce to optimal stopping problems which can be related to the solution of corresponding free boundary problems for the heat equation. The numerical solution of these free boundary problems can then be approximated by calculating the solution of simpler optimal stopping problems by backward induction. This approach is not well adapted for very precise results but is surprisingly effective for rough pproximations. An estimate of the difference between the solutions of the related problems permits one to make continuity corrections which provide considerably improved accuracy. Further reductions in the necessary computational effort are possible by considering truncated procedures for one-sided boundaries and by exploiting monotone and symmetric boundaries.

Approximate solution of a non-linear boundary value problem with a small parameter for the highest-order differential

Abstract: A numerical method, which has an estimate of the rate of convergence which is uniform with respect to a small parameter, is proposed for a non-linear boundary value problem with a small parameter for the highest-order derivative.

Simply supported plates by the boundary integral equation method

Abstract: Plates governed by Kirchhoff's equation have been analysed by the boundary integral equation method using the fundamental solution of the biharmonic equation. In the case of supported plates, the boundary conditions permit the uncoupling of the field equation into two harmonic equations that originate, due to the nature of the fundamental solution, easier integration kernels and a simpler system of equations. The calculation of bending and twisting moments and transverse shear force can be formed, combining derivatives of the integral equation which defines the expression of the deflection on any point of the plate. The uncoupling of the biharmonic equation into two Poisson's equations involves the discretization of the domain of the studied problems. Nevertheless, the unknown quantity of the problem does not appear in the domain integrations for which a refined discretization is unnecessary. In the paper, however, a numerical alternative is considered to express the domain integral by means of boundary integrals. In this way, we need only discretize the boundary of the plate, making it necessary to solve a supplementary system of equations in order to calculate the coefficients of the approximation carried out.

Regularity results for an elliptic-parabolic free boundary problem

Abstract: On etudie un probleme aux limites libre elliptique-parabolique a une dimension d'espace. On donne plusieurs resultats de regularite pour la solution faible et la frontiere libre. On donne des conditions pour que la frontiere libre soit une courbe C 1

A front tracking method for one dimensional moving boundary problems

Abstract: We introduce a front tracking method for the numerical solution of one-dimensional Stefan problems. It consists in the formulation of the Stefan problem as an ordinary differential initial-value problem for the moving boundary coupled with a parabolic partial differential equation for the distribution of temperatures. We present a variable time step procedure in which the initial-value problem is solved with a predictor-corrector scheme; in the corrector step the function evaluation is done, iteratively, through an implicit time discretization of the parabolic equation. Numerical results for one-dimensional, one-phase Stefan problems with straight and curved moving boundary trajectories are presented. For these cases the front tracking method presented gives greatly improved results.

A geometrical inverse problem

Abstract: A simple solution is given to the problem of finding the unknown boundary from the extra boundary condition.

An elliptic-parabolic free boundary problem

Abstract: in which c is a continuous nondecreas~Rg function defined on R such that C(U) is strictly increasing when u 0, Ui = 1 if bi = 0 and bi = 1 if ai = 0 for i = 0 and i = 1 and fO, fi and t’. are prescribed functions. Problem I arises in the theory of fluid Aow through parGaIly saturated media. Then t6 denotes the potential due to capiitary suction and c the moisture content. The dependence of c on IE is found empirically to be as in Fig. I, c being bounded above by the saturation value c, = 1. Thus, in regions where the medium is saturated, the flow is of potential type, and described by an ehiptic equation, and in regions where the medium is unsaturated, the flow is of diffusive type and described by a parabolic equation. At the boundary between these regions-the interface-one expects U’= 0 and u, continuous.