Showing papers on "Free boundary problem published in 1972"

Difference approximations for boundary and eigenvalue problems for ordinary differential equations

Abstract: The boundary value problem for ordinary differential equations is considered and a general theory for difference approximation is developed. In particular,theinfluenceof extra boundary conditions is investigated and the eigenvalue problem is considered in detail.

Boundary Layers in Linear Elliptic Singular Perturbation Problems

Abstract: This paper presents a survey of recent results obtained in the theory of singular perturbations for linear elliptic partial differential equations in two independent variables. The emphasis is on the constructive procedures and the various aspects of the boundary layer phenomenon.

Uniqueness and existence for the geodetic boundary value problem using the known surface of the earth

Abstract: The geodetic boundary value problem using the known surface of the earth is defined and shown to have at most one solution Furthermore it is proved that the solution exists and that its harmonic part can be represented by the potential of a simple layer under the sufficient condition that at the surface of the earth directions are known which lie differentially close to the gradients of the gravity field The advantages of this boundary value problem are outlined in comparison to the boundary value problem formulated by Molodensky

On dirichlet problems using subspaces with nearly zero boundary conditions

Abstract: Publisher Summary This chapter focuses on Dirichlet problems using subspaces with nearly zero boundary conditions. The classical Rayleigh-Ritz methods for elliptic boundary value problems with Dirichlet data assume that the approximating functions fulfill the boundary conditions exactly. In most cases, that is, when the boundary is curved, the construction of such subspaces is delicate and implies many difficulties in practice. To overcome this problem there are now known modifications of the variational principles obtained by adding appropriate boundary terms. In these cases the approximating subspaces which are used need not fulfill any boundary conditions. The chapter presents versions of the Ritz and least squares methods and shows them to be quasi-optimal. Besides the concept of nearly zero boundary values and a certain inverse assumption, a sharper condition of approximability than normally used is needed. The cubic spline functions are also discussed in the chapter.

A variational inequality with mixed boundary conditions

Abstract: In this paper we study a variational inequality for a second order uniformly elliptic operator on a bounded domain, the solution of which is required to lie above a given obstacle and to assume assigned values on a part of the boundary of the domain. We are mainly concerned with the regularity of the solution in relation to the regularity of the data.

Expansion of the Master Equation for One‐Dimensional Random Walks with Boundary

Abstract: In order to understand the behavior of coarse‐grained equations in the presence of a boundary, the following model is investigated. A homogeneous one‐dimensional random walk is bounded on one side by some boundary conditions of rather arbitrary form. The corresponding master equation is approximated by the Fokker‐Planck equation plus partial differential equations for the higher orders. The boundary condition for the Fokker‐Planck approximation is well known; but those for the higher order terms are here derived. To the second order they amount to a virtual displacement of the boundary. The case of a two‐step random walk, however, gives rise to an unexpected complication, inasmuch as nonpropagating solutions of the master equation cannot be ignored in the boundary condition, although they do not contribute to the differential equations themselves.

The method of orthogonal projections and the dirichlet problem in domains with a fine-grained boundary

Abstract: The Dirichlet problem is considered for an elliptic selfadjoint operator in a domain , where is a bounded domain in and the are nonintersecting closed sets (grains). It is shown that, if the grain diameters tend to zero, and the number of grains tends to infinity, the solution of the problem reduces, under certain conditions, to the solution of another boundary value problem in the simpler domain .Bibliography: 8 items.

Semidiscrete least-squares methods for a parabolic boundary value problem

Abstract: In this paper some approximate methods for solving the initial-boundary value problem for the heat equation in a cylinder under homogeneous boundary condi- tions are analyzed. The methods consist in discretizing with respect to time and solving approximately the resulting elliptic problem for fixed time by least squares methods. The approximate solutions will belong to a finite-dimensional subspace of functions in space which will not be required to satisfy the homogeneous boundary conditions. 1. Introduction. The purpose of this paper is to analyze some approximate methods for solving the initial-boundary value problem for the heat equation in a cylinder under homogeneous boundary conditions. The methods consist in discretizing with respect to time and solving approximately the resulting elliptic problem for fixed time by least squares methods. The approximate solutions will belong to a finite- dimensional subspace of functions in space which will not be required to satisfy the homogeneous boundary conditions. Let Ql be a bounded domain in Euclidean N-space RN with smooth boundary au. We shall consider the approximate solution of the following mixed initial-boundary value problem for u = u(x, t), namely,

Generalized Bleustein Modes

Abstract: : This article presents a general formulation of the boundary value problem for a Bleustein mode piezoelectric surface wave. The problem is solved explicitly for those cases where the secular equation is biquadratic and Bleustein modes are shown to exist upon certain cuts of crystal in Classes 4, 6, 4mm, 6mm bar 4, bar 6 m 2, 23 bar 4 3 m and bar 4 2 m.

Numerical Treatment of a Singularity in a Free Boundary Problem

Abstract: The numerical solution of free boundary problems gives rise to many computational difficulties. One such difficulty is due to the singularity at the separation point between the fixed and free boundaries. A method is suggested which uses complex variable techniques to determine the shape of the free boundary near to the separation point. This complex variable solution is also used to improve the accuracy of the finite-difference solution in the neighbourhood of the singularity. The analytical study was incorporated into an algorithm for the numerical solution of a particular free boundary problem concerning the percolation of a fluid through a porous dam. Some numerical results for this problem are presented.

The Eigenvalue Problem for a Pair of Coupled Integral Equations Arising in the Numerical Solution of Laplace’s Equation

Abstract: Using Green’s third formula one can reduce the mixed boundary value problem for Laplace’s equation in two dimensions to a pair of coupled integral equations. This pair of integral equations can be solved numerically, and in many cases of interest this is an efficient way to solve the boundary value problem. Here we investigate the spectrum for the associated eigenvalue problem. It follows from our results that it is always feasible to solve numerically the boundary value problem by the above method.

On nonexistence of boundary arcs in control problems with bounded state variables

Abstract: When solving optimal control problems with bounded state variables, one must determine whether the optimal trajectory intersects the boundary only at isolated points in time (boundary point) or remains on the boundary for a nonzero length of time (boundary arc). Previously, this determination has been made by trial and error. The task is complicated by the fact that the necessary conditions in common use for these problems assume that the solution has a boundary arc, and can thus yield a boundary arc when the solution has no boundary arc. In this paper the necessary conditions of [1] are used to derive conditions under which the optimal trajectory cannot have a boundary arc. These conditions include the condition for no boundary arcs developed in [1] as a special case. The application of these conditions is illustrated via several examples.

Partial differential equations and boundary-value problems

Abstract: Partial differential equations are classified as to order and linearity in the same way as ordinary differential equations. The order of an equation is the order of the highest-order partial derivatives of the unknown function that appear in the equation. A boundary value problem possessing a unique solution that depends on the prescribed values in the boundary conditions is said to be a well-posed problem. Boundary conditions that yield a well-posed problem with a hyperbolic equation do not in general yield a well-posed problem with an equation of elliptic type. For some partial differential equations, it is possible to find expressions that represent all solutions, that is, represent the general solution. Such expressions contain arbitrary functions instead of arbitrary constants, as in the case of ordinary differential equations. The expression for the Laplacian of a function in spherical coordinates can be derived in the same way as was done for cylindrical coordinates, although the algebra is more complicated. In problems where the number of independent variables is greater than two, the method of separation of variables leads to the notion of a multiple Fourier series.

Asymptotic Solutions of Inhomogeneous Initial Boundary Value Problems for Weakly Nonlinear Partial Differential Equations

Abstract: The two-time perturbation procedure is applied to a class of inhomogeneous initial boundary value problems for weakly nonlinear partial differential equations of wave type. An operator calculus approach is adopted to treat the problem as an initial value problem in Hilbert space. The first order equation is applied to two problems in the forced vibration of a nonlinear string. They are governed by the generalized Duffing and van der Pol equations respectively. In the case of the Duffing equation, the results show that there exists a stable, periodic vibration when small damping is present and how the system approaches this final state in the course of time. In the absence of damping, the system does not in general possess a periodic motion. For the van der Pol equation, our results generalize that of Kelley and Kogelman [1]. Here we find that there exist self excited motions in the form of many periodic motions superimposed upon an externally excited periodic motion. The dependence of the final motions on...

Unsteady-state temperature distribution in a convecting fin of constant area

Abstract: A solution for the unsteady-state temperature distribution in a fin of constant area dissipating heat only by convection to an environment of constant temperature, is obtained. The partial differential equation is separated into an ordinary differential equation with position as the independent variable, and a partial differential equation with position and time as the independent variables. The problem is solved for either a step function in temperature or a step function in heat flow rate, for zero time, at one boundary while the other boundary is insulated. The initial condition is taken as an arbitrary constant. The unspecified boundary values (temperature or heat flow rate) are presented for both cases by utilizing dimensionless plots. Experimental verification is presented for the case of constant heat flow rate boundary condition.

Boundary Free Energy in the Lattice Model. I. General Formulation

Abstract: In order to provide the basis for solving the paradox in the theory of the boundary structure, the general expression for the boundary free energy between two phases is derived by applying the cluster variation method rigorously. The surface free energy nσ for the phase boundary is expressed as exp(−nσ/kT)=h(II)·g(I), where h and g are the row and column vectors used in the eigenvalue formulation of the cooperative problem, and (II) and (I) signify the two phases meeting at the boundary. This expression is the same as the one previously derived by Clayton and Woodbury. In the process of the derivation, the local free energy density is identified which will play the key role in solving the paradox in a subsequent paper. The symmetric boundary, the asymmetric boundary, and the case of long‐range interaction are discussed.

Two point boundary value problems for nonlinear functional differential equations

Abstract: This paper is concerned with the existence of solutions of two point boundary value problems for functional differential equations. Specifically, we consider /(f) = L(t, yt) +f(t, y,), Mya + Nyb =

Control of functional differential equations with function space boundary conditions

Abstract: The results of various authors dealing with problems involving functional differential equations with terminal conditions in function space are reviewed. The review includes not only very recent results, but also some little known results of Soviet mathematicians prior to 1970. Particular attention is given to results concerning controllability, existence of optimal controls, and necessary and sufficient conditions for optimality.