Showing papers on "Free boundary problem published in 1971"

On Diffusion in a Fractured Medium

Abstract: A “narrow fracture” approximation is applied to the diffusion equation on an inhomogeneous region in order to convert the parabolic interface problem into a boundary value problem with tangential derivatives on part of the boundary. This boundary value problem is shown to have a unique weak solution. Finally, a particular steady state model is solved, essentially in closed form, with a “Galerkin by lines” technique.

The method of Christopherson for solving free boundary problems for infinite journal bearings by means of finite differences

Abstract: A method for solving free boundary problems for journal bearings by means of finite differences has been proposed by Christopherson We analyse Christopherson's method in detail for the case of an infinite journal bearing where the free boundary problem is as follows: Given T > 0 and /(t) find r E (0, T) and p(t) such that (i) (h3p')' = h' for t E (0, r), (ii) p(O) = 0, (iii) p(t) = 0 for t (T, Ir7, and (iv) p'(r - 0) = 0 First, it is shown that the discrete approximation is accurate to O((At)2) where At is the step size Next, it is shown that the discrete problem is equivalent to a quadratic pro- gramming problem Then, the iterative method for computing the discrete approximation is analysed Finally, some numerical results are given 1 Introduction A journal bearing consists of a rotating cylinder which is separated from a "bearing surface" by a thin film of lubricating fluid (see Fig 1) The fluid is fed in at A and flows out at B The width of the film is smallest at C, and we set 1 = O/0, where 0 is as shown in Fig 1 Between C and B, the width of the film increases so that the pressure in the lubricating fluid may be expected to decrease We assume that for t - r the pressure becomes so low that the fluid vaporizes The point t T the interface between the two phases of the fluid, is called the free boundary The mathematical problem can now be formulated (see Pinkus and Sternlicht

A two phase Stefan problem with temperature boundary conditions

Abstract: We studied a two phase Stefan problem in a infinite plane slab, when the temperatures are prescribed on the two limiting planes.

General One-Phase Stefan Problems and Free Boundary Problems for the Heat Equation with Cauchy Data Prescribed on the Free Boundary

Abstract: where u(x, t) and the free boundary s(t) are to be determined. Here f, g, h, A, p, and a are the data of problem (1.1), with f, g, h, A defined for t > 0 and (p(x) defined for 0 ? x 0 and a > 0 and compatibility and regularity conditions described in Theorem 1 of ? 2. In Theorem 1 we prove existence and uniqueness of the solution of (1.1). If a is the supremum of those t for which (1.1) has a solution, then we prove, in Theorem 1, that either a = oo, or a is finite with s(a-) 0 and s(t) > 0 for t < a, or a is finite, s(t) does not tend to 0 as t T a, and lim inf s'(t) = lim inf ux(s(t), t) = oo as t T a. We describe the Stefan problem (1.1) as general because there are no sign restrictions on f, g, h and (p. We may interpret (1.1) as a problem of heat conduction with melting when

A two phase Stefan problem with flux boundary conditions

Abstract: We studied a two phase Stefan problem in a infinite plane slab, when the thermal fluxes are assigned on the two limiting planes.

On Boundary Value Problems for Elliptic Equations Degenerating on the Boundary

Abstract: in SCR", where & is an interior or, an exterior of a smooth 'and compact hypersurface, A is a complex number and p(x) is a realvalued function satisfying (1) p(#) is continuous and 0

An Implicit Free Boundary Problem for the Heat Equation

Abstract: An existence and uniqueness theorem is proved for an implicit free boundary problem for the heat equation—denoted as “implicit” due to the absence of the derivative of the free boundary in a free boundary condition.

A numerical method for two-phase stefan problems*

Abstract: A numerical technique for the solution of one-dimensional parabolic free interface (Stefan) problems is discussed. The method of lines is employed to approximate the partial differential equations at discrete time levels by free interface problems for ordinary differential equations which are solved by conversion to initial value problems. Comments on multiphase problems and a numerical example round out the discussion.

Integral characteristics of the growth of spectral functions for generalized second order boundary problems with boundary conditions at a regular end

Abstract: For the spectral function of the generalized second order boundary problem and for the function , which may belong to an extremely large class of positive functions that are nonincreasing on , the problem of characterizing the growth of the function as and of the convergence of the integral is connected with the behavior as of the function . The results that are proved in this article were announced by the author in [5] (MR 37 #5456).

Use of Finite Difference Approximations to Partial Differential Equations for Problems Having Boundaries at Infinity (Correspondence)

Abstract: A computationally simple technique is presented for solving finite difference equations arising from potential problems, part of whose boundary is at infinity. The procedure makes use of an arbitrary "fictitious" boundary drawn close to the regions of physical interest. An initial guess is made of the potential on this boundary as well as at all interior points. Well-known iterative techniques are used to correct the values of the interior potentials. Meanwhile the potentials on the boundary are corrected iteratively by recalculating them from the sources or charges in the entire region, which in turn are calculated from the current iteration of the interior potential. The technique is valid even if parts of the physical structure, such as an air-dielectric interface in microstrip, extend toward infinity. The fictitious boundary need not include all of the structure, providing the rate of falloff of the sources outside the boundary is known.

Asymptotic treatment of the stability of a rotating layer of fluid with rigid boundaries

Abstract: Chandrasekhar (1961) has summarized the stability results of Benard convection in a rotating fluid for the cases where the boundary surfaces are both rigid and free, and for both exchange of stabilities and overstability. His analysis provides very accurate results for a limited range of Taylor number J. Bisshopp and Niiler (1965) presented an asymptotic analysis of the rigid boundary problem for exchange of stabilities which is valid for very large Taylor number. The present paper makes use of modern rotating fluid theory to develop an approximate scheme for evaluating the Rayleigh number and other parameters and variables. Known asymptotic results for the free boundary problem at large J are used and an expansion in powers of E1/6 (the Ekman number, E = 2J −½) yields a sequence of equations and appropriate boundary conditions for the rigid boundary problem. After the algorithm for the calculation is developed, results are given for the problem to second order in the expansion parameter for the ...

Boundary behavior of solutions of parabolic equations with discontinuous coefficients

Abstract: where we have employed the convention of summation over repeated indices. Here x = (#1, ) denotes a point in E with n^l and t denotes a point on the real line. Assume that the coefficients of (1) are bounded measurable functions of (xr t) in S = E X( — 1, 2T) for some T>0 and that there is a constant A>0 such that a»y(x, t)ZiZj^\\\\z\\ almost everywhere in 5 for all zÇzE. Let G be a bounded domain in E n X(0 , T). We prove that to each f unction ƒ £ C(dG) there corresponds a weak solution u of the boundary value problem

Stability of Boundary Layer Flow Subject to Rotation

Abstract: The stability of three‐dimensional infinitesimal waves moving in a Blasius boundary layer in a rotating flow is investigated for the situation where the rotation axis coincides with the leading edge of a semiinfinite flat plate. Primary interest is in the effect of rotation on the critical point of the neutral stability curve. The problem is formulated in terms of a sixth‐order differential equation, which reduces to the Orr‐Sommerfeld equation for zero rotation, with homogeneous boundary conditions. The eigenvalue problem thus formed is solved numerically. Results indicate a destabilizing or stabilizing effect depending on the sense of the rotation vector. This rotation effect is influenced by the spanwise wavenumber so that the effect disappears as the spanwise wavenumber vanishes. The results compare favorably with known experimental results.

Asymptotic behavior of solutions of pseudo-parabolic partial differential equations

Abstract: Consider a solution to a second-order pseudo-parabolic equation with sufficiently smooth time-independent coefficients in a cylindrical domain. If it vanishes on the cylindrical surface for all times and if its restriction to a fixed instant belongs toC2+a, then its pointwise values decay exponentially as t→∞ while its Dirichlet norm grows expontially as t→−∞. Similar conclusion still hold for solutions to non-homogeneous equations under non-homogeneous boundary conditions provided the free term and the boundary data posses these asymptotic behaviors.

Existence and uniqueness of solutions of boundary value problems for two dimensional systems of nonlinear differential equations

Abstract: The paper considers the nonlinear system x'=f(t,x,y), y'=g(t, x, y) with linear and nonlinear two point boundary conditions. With a Lipschitz condition, an interval of uniqueness for linear boundary conditions is determined using a comparison theorem. A corresponding existence theorem is established. Under the assumption of uniqueness, a general existence theorem is established for quite general nonlinearities in the functions and in the boundary conditions. Examples are provided. The results extend previous work on second order scalar differential equations. Introduction. Questions of the existence and uniqueness of solutions of the two point boundary value problem (i.i) y"+f(t,y,y') = o, (1.2) yia) = A, (1.3) yib) = 77, where f(t, y, y') satisfies a Lipschitz condition, have a long history, going back to Picard [9], 1893. The problem is to determine, in terms of the Lipschitz constants, the best possible interval [a, b] on which there exists a unique solution of (1.1), (1.2), (1.3). Recent results and references on this problem can be found in [2], [3]. A second question of interest is, given that solutions of (1.1), (1.2), (1.3) are unique, i.e. given that there exists at most one solution, when does this imply that there exists at least one solution. The first results along this line are due to Lasota and Opial [6] and Jackson [3]. This paper investigates these two types of questions for the second order system (1-4) x'=f(t,x,y), y'=f2(t,x,y) subject to more general boundary conditions (1.5) gi(x(a), y(a)) = cu (1.6) g2(x(b),y(b)) = c2. Received by the editors March 10, 1969 and, in revised form, February 11, 1970. AMS 1970 subject classifications. Primary 34B15.