Showing papers on "Neumann boundary condition published in 1983"
235 citations
TL;DR: The problem of determining both the existence and regularity properties of the solution u of au = a, where u is orthogonal to the null space of a on (p, q l)-forms, is known as the a-Neumann problem as discussed by the authors.
Abstract: Let a be a a-closed form of type (p, q) with L2-coefficients on a smoothly bounded domain Q in C'. The problem of determining both the existence and regularity properties of the solution u of au = a, where u is orthogonal to the null space of a on (p, q l)-forms, is known as the a-Neumann problem. One of the principal methods used in the investigation of this problem is the proof of certain a priori subelliptic estimates. Let U be a neighborhood of a point z0 in the boundary of Q. A subelliptic estimate is said to hold in U if the estimate
176 citations
TL;DR: In this paper, the critical behavior of a semi-infiniten-vector model with a surface term (c/2) ∫dSφ2 is studied in 4-e dimensions near the special transition.
Abstract: The critical behaviour of a semi-infiniten-vector model with a surface term (c/2) ∫dSφ2 is studied in 4-e dimensions near the special transition. It is shown that all critical surface exponents derive from bulk exponents and η∥, the anomalous dimension of the order parameter at the surface. The surface exponents and the crossover exponent Φ for the variablec are calculated to second order in e. It is found that Φ does not satisfy the relation Φ=1-ν predicted by Bray and Moore. The order-parameter profilem(z)= is calculated to first order in e. In contrast to mean-field theory,m(z) is not flat nor does it satisfy a Neumann boundary condition. General aspects of the field-theoretic renormalization program for systems with surfaces are discussed with particular attention paid to the explanation of the unfamiliar new features caused by the presence of surfaces.
122 citations
01 Jan 1983
TL;DR: In this paper, a flow with a free boundary is represented by a jet of fluid travelling through a region of constant pressure, and two typical situations are shown in Figure 1 : a jet impinging on a fixed wall and a jet emerging from a hole in the wall of a large reservoir.
Abstract: One example of a flow with a free boundary is that of a jet of fluid travelling through a region of constant pressure. There are two typical situations which are shown in Figure. The first is a jet impinging on a fixed wall and the second is a jet emerging from a hole in the wall of a large reservoir. These situations may either be two or three-dimensional, but we can make more analytical progress in the two-dimensional case.
82 citations
79 citations
TL;DR: In this article, error estimates for continuous and discrete time finite element procedures to approximate the solution of the degenerate parabolic equation (1.1)-(1.3) are proved.
Abstract: The degenerate parabolic equation aut = v ( Oulvvu), v >1 has been used to model the flow of gas through a porous medium. Error estimates for continuous and discrete time finite element procedures to approximate the solution of this equation are proved, and several new regularity results are given. 1. A Porous Medium Equation. Introduction. We shall study the porous medium equation (1.1) au/at = V (I u I"vu) on 2 X (0, T], (1.2) au/an = 0 on aa X [O, T], (1.3) u(x, 0)= uO(x) on 2, where v > 1 is a parameter and Q is a bounded domain in RN, N ? 3, with a smooth boundary. The initial function u0 is assumed to be nonnegative and four times continuously differentiable on U. Notice that the compatibility condition auo/an = 0 holds on asa. Our main result is the derivation of error estimates for numerical approximations to the problem (1. 1)-(1.3), which we shall refer to as "the porous medium equation" or "IPME". The PME does not, in general, admit classical solutions. Existence and uniqueness of weak solutions was proved in one space dimension by Oleinik, Kalashnikov, and Czou [15], [16] and in several space dimensions by Lions [12]. These proofs concern the PME with different boundary conditions, but the arguments carry over to the PME (1.1)-(1.3). The maximum principle implies that, since u0 is nonnegative on 92, u(x, t) is nonnegative for all (x, t) E Q X [0, T]; see [15], [16]. If u0 is nonzero, the Neumann boundary condition implies that u will eventually become strictly positive and (1.1) will become nondegenerate for all time t > To, To sufficiently large. We can rewrite (1.1) in the form (1.4) au/at=AK(u) on QX(0,T], Received May 11, 1979; revised July 24, 1981. 1980 Mathematics Subject Classification. Primary 35K65, 65N30, 65N15. *Current address: Reservoir Engineering Branch, Petroleum Engineering Division, BP Exploration (DOS), Britannic House, Moore Lane, London EC2Y 9BU, United Kingdom. (D1983 American Mathematical Society 0025-5718/82/0000-1 063/$08.25 435 This content downloaded from 157.55.39.55 on Tue, 23 Aug 2016 04:12:54 UTC All use subject to http://about.jstor.org/terms
59 citations
TL;DR: In this article, a general formulation based on space-time Green's functions is developed using the complete heat equation, followed by a simpler formulation using the Laplace equation, and the latter is pursued and applied in detail.
Abstract: Boundary integral equation methods are presented for the solution of some two-dimensional phase change problems. Convection may enter through boundary conditions, but cannot be considered within phase boundaries. A general formulation based on space-time Green's functions is developed using the complete heat equation, followed by a simpler formulation using the Laplace equation. The latter is pursued and applied in detail. An elementary, noniterative system is constructed, featuring linear interpolation over elements on a polygonal boundary. Nodal values of the temperature gradient normal to a phase change boundary are produced directly in the numerical solution. The system performs well against basic analytical solutions, using these values in the interphase jump condition, with the simplest formulation of the surface normal at boundary vertices. Because the discretized surface changes automatically to fit the scale of the problem, the method appears to offer many of the advantages of moving mesh finite element methods. However, it only requires the manipulation of a surface mesh and solution for surface variables. In some applications, coarse meshes and very large time steps may be used, relative to those which would be required by fixed grid domain methods. Computations are also compared to original lab data, describing two-dimensional soil freezing with a time-dependent boundary condition. Agreement between simulated and measured histories is good.
59 citations
TL;DR: In this article, the performance of the Open Boundary Condition (OC) with a simple finite difference representation has been tested for laminar wakes in a nonstratified fluid and for internal wave problems in a linearly stratified flow.
57 citations
TL;DR: In this article, some typical free boundary problems, such as the obstacle problem, the seepage surface problem, Stefan problem, and the Elenbaas equation, are all picked up into the scheme.
52 citations
TL;DR: In this paper, a nonlinear elliptic eigenvalue problem with neumann boundary conditions, with an application to population genetics, is studied. But the authors focus on population genetics only.
Abstract: (1983). On a nonlinear elliptic eigenvalue problem with neumann boundary conditions, with an application to population genetics. Communications in Partial Differential Equations: Vol. 8, No. 11, pp. 1199-1228.
49 citations
TL;DR: In this paper, the authors derive a spatial-interaction model in which places are related to each other through a set of simultaneous linear equations and the associated Lagrangians can be interpreted as pushes and pulls, or as shadow prices.
Abstract: Elementary geometric assumptions are used to derive a spatial-interaction model in which places are related to each other through a set of simultaneous linear equations. The system has simple properties with respect to aggregation and turnover, yet incorporates spatial competition, adjacency, and effects of geographic shadowing. The objective function satisfied by the model reduces congestion and minimizes the per capita work; solves a quadratic transportation-problem and fulfils in-sum and out-sum constraints. The associated Lagrangians can be interpreted as pushes and pulls, or as shadow prices. A spatially continuous version of the model consists of coupled elliptical partial differential equations with Neumann boundary conditions, solvable by numerical methods. With migration data from the United States of America the model yields an amazingly good fit, better than existing models and with fewer free parameters. Inversion of the model yields an estimate of distances between regions. In-movement rates ...
TL;DR: In this article, the Neumann problem is analyzed as a bifurcation problem with two parameters, i.e., λ and ϵ, and the qualitative properties of these parameters are obtained for different values of a.
TL;DR: In this paper, a parabolic equation defined on a bounded domain is considered, with input acting in the Neumann (or mixed) boundary condition, and expressed as a specified finite dimensional, nondynamical feedback of the Dirichlet trace of the solution ("boundary observation").
Abstract: A parabolic equation defined on a bounded domain is considered, with input acting in the Neumann (or mixed) boundary condition, and expressed as a specified finite dimensional, nondynamical feedback of the Dirichlet trace of the solution ("boundary observation"). The free system is assumed unstable. Conditions are given at the unstable eigenvalues, under which one can select boundary vectors of the feedback operator, so that the corresponding feedback solutions decay exponentially to zero, in the uniform operator norm ast ź + ź. These conditions consist of (i) verifiable algebraic (full rank) conditions, plus (ii) an Invertibility Condition. The latter depends crucially on properties of the Dirichlet traces of the (`free system') eigenfunctions, whose direct knowledge is available only in special cases. We then specialize--in the appendix--to canonical situations, involving the Laplacian (translated) on spheres and parallelepipeds. In these cases, we indicate how to construct (in infinitely many ways, in fact) boundary vectors of the feedback operator, which satisfy both the algebraic conditions and the Invertibility Condition, thereby yielding stabilization.
TL;DR: In this article, axisymmetric static thermal problems for a half-space when one boundary condition corresponds to partial insulation, either inside or outside the circle r = a, z = 0, are considered and the problems are reduced to the solution of integro-diffe rential equations of Abel type.
Abstract: SUMMARY Contact or crack problems in thermoelastic ity are usually analysed with the idealised boundary conditions of perfect conduction or perfect insulation. These boundary conditions, while simplifying the mathematics, sometimes lead to unrealistic, singular thermoelastic fields. This paper formulates axisymmetric static thermal problems for a half-space when one boundary condition corresponds to partial insulation, either inside or outside the circle r = a, z = 0. Four important cases are considered and the problems are reduced to the solution of integro-diffe rential equations of Abel type. In each case it is shown that the equation can be solved by using two simultaneous Fourier expansions of the unknown function.
TL;DR: In this article, a least-squares approximation of the symbol of the pseudodifferential operator is proposed to obtain differential operators as boundary conditions for hyperbolic systems, where the expectation of the reflected energy is lower than in the case of Taylor-and Pade-approximation.
Abstract: Engquist and Majda [3] proposed a pseudodifferential operator as asymptotically valid absorbing boundary condition for hyperbolic equations. (In the case of the wave equation this boundary condition is valid at all frequencies.) Here, least-squares approximation of the symbol of the pseudodifferential operator is proposed to obtain differential operators as boundary conditions. It is shown that for the wave equation this approach leads to Kreiss well-posed initial boundary value problems and that the expectation of the reflected energy is lower than in the case of Taylor- and Pade-approximations [3, 4]. Numerical examples indicate that this method works even more effectively for hyperbolic systems. The least-squares approach may be used to generate the boundary conditions automatically.
TL;DR: In this paper, a halfplane under plane wave excitation obeys a Dirichlet boundary condition on one side and a Neumann boundary conditions on the other, and the present problem leads to a system of integral equations of the Wiener-Hopf type which may be solved by a matrix factoring method.
Abstract: A half-plane under plane wave excitation obeys a Dirichlet boundary condition on one side and a Neumann boundary condition on the other. These boundary conditions contrast the ones used by A. Sommerfeld in his classical paper. The present problem leads to a system of integral equations of the Wiener-Hopf type which may be solved by a matrix factoring method suggested by A. E. Heins in 1950.
TL;DR: In this article, a simple model problem in optimal boundary heating of solids is analyzed by numerical methods, where the physical objective of the present "steady-state optimal control" problem is to achieve a desired temperature profile along a segment of the solid boundary with a minimum amount of a boundary heat flux which acts as the controlling function.
TL;DR: In this article, a necessary and sufficient condition for a system governed by Dirichlet and Neumann problems for a self-adjoint elliptic operator with an infinite number of variables to have an optimal control of the distributed type which is characterized by a set of inequalities is given.
Abstract: In this paper, we obtain a necessary and sufficient condition for a system governed by Dirichlet and Neumann problems for a self-adjoint elliptic operator with an infinite number of variables to have an optimal control of the distributed type which is characterized by a set of inequalities.
TL;DR: In this article, the problem of scattering from a rough interface separating two semi-infinite homogenous media was considered and a single coordinate-space integral equation of the first kind for the generalized reflection coefficient R was derived.
TL;DR: In this paper, it was shown that if one takes the first two moments of the Fokker-Planck equation, one obtains, in the lowest order, the telegraph equation irrespective of whether one uses a full-range or a half-range decomposition of the distribution function; it was pointed out that the exact boundary condition, according to which the distribution for emerging particles vanishes at the surface of an absorbing (black) sphere of radius R, may be replaced, in either case, by Marshak's boundary condition j+(R, t) = 0
Abstract: It is shown that, if one takes the first two moments of the Fokker–Planck equation, one obtains, in the lowest order, the telegraph equation irrespective of whether one uses a full‐range or a half‐range decomposition of the distribution function; it is pointed out that the exact boundary condition, according to which the distribution function for emerging particles vanishes at the surface of an absorbing (black) sphere of radius R, may be replaced, in either case, by Marshak’s boundary condition j+(R, t) =0, where j+ is the outward radial current. Drawing on Wilemski’s work, the second moment equation is finally replaced by Fick’s law, obtaining thereby the diffusion equation and the radiation boundary condition.
TL;DR: In this article, a weak formulation of two-phase free boundary evolution problems with free boundary jump conditions generalizing that of Stefan and with possibly vanishing coefficient of time derivative is given.
TL;DR: In this paper, it is shown that an external noise shifts deterministic bifurcation maps or generates new Bifurcations in a special bistable reaction-diffusion model, and the functional Fokker-planck equation is solved approximately.
Abstract: It is shown for a special bistable reaction-diffusion model that an external noise shifts deterministic bifurcation maps or generates new bifurcations. Numerical results and an analytical approximation for the bifurcation map of the stationary probability density distribution of the mean concentration in a discrete model are given. The functional Fokker-Planck equation is solved approximately.
Stochastische Bifurkationen in einem bistabilen Reaktions-Diffusionssystem mit Neumann-Randbedingungen
An einem speziellen bistabilen Reaktions-Diffusionsmodell wird gezeigt, das die Einbeziehung eines auseren Rauschens zur Verschiebung der deterministischen Bifurkations-netze bzw. zur Erzeugung neuer Bifurkationen fuhrt. Fur die stationare Wahrscheinlichkeitsdichte-verteilung der mittleren Konzentration in einem diskreten Modell werden numerische Ergebnisse und eine analytische Naherung fur das Bifurkationsnetz angegeben. Die funktionale Fokker-Planck-Gleichung kann naherungsweise gelost werden.
TL;DR: The purpose of this paper is the treatment of three numerical algorithms for solving large but mildly behaved diffusion problems on a vector computer with memory- to-memory architecture.
TL;DR: In this paper, the authors extended the results of Bailey et al. to boundary value problems similar to (l), (3), (4), (5), (6), (7), (8), (9), (10), (11), (12), (13), (14), (15), (16), (17), (18), (19), (20), (21), (22), (23), (24), (25), (26), (27), (28), (30), (31), (33), (34), (35
Abstract: Motivational results for solution matching were proven first in the text by Bailey et al. [l] where they dealt with solutions of two-point problems for y” = f(x, y, y’) by matching solutions of initial value problems. Then in 1973, Barr & Sherman [2] adapted the arguments of Bailey et al. to match solutions of two-point boundary value problems for third order equations to obtain solutions of three-point problems. In the same paper, they showed that this result could be generalized to equations of arbitrary order yielding solutions of (l), (2). Barr & Miletta [3] matched solutions of k-point problems for nth order linear equations and obtained solutions of n-point conjugate type boundary value problems. In 1978, Moorti & Garner [4] extended the results of [2] to boundary value problems similar to (l), (3), except p, Y were restricted
01 Jan 1983
TL;DR: In this second year of effort, truly large-scale computing aspects of PDE's (partial differential equations) have been addressed and singularities which are frequently troublesome in cylindrical and spherical co-ordinates are eliminated naturally in MFE inner product formulations.
Abstract: The mathematical background regarding the moving finite element (MFE) method of Miller and Miller (1981) is discussed, taking into account a general system of partial differential equations (PDE) and the amenability of the MFE method in two dimensions to code modularization and to semiautomatic user-construction of numerous PDE systems for both Dirichlet and zero-Neumann boundary conditions. A description of test problem results is presented, giving attention to aspects of single square wave propagation, and a solution of the heat equation.
TL;DR: In this article, the authors studied the numerical solution of nonlinear reaction-diffusion systems with homogeneous Neumann boundary conditions, via the known o-method, and showed that if conditions for the positivity of solutions are imposed, then the study of the asymptotic behavior of numerical solution can be done by means of the theory of stochastic matrices.
Abstract: In this paper we study the numerical solution of nonlinear reaction-diffusion systems with homogeneous Neumann boundary conditions, via the known o-method. We show that if conditions for the positivity of solutions are imposed, then the study of the asymptotic behavior of the numerical solution can be done by means of the theory of stochastic matrices. In this way it is shown that the numerical solution reproduces the asymptotic behavior of the corresponding theoretical one. In particular, we obtain the decay of the solution to its mean value. An analysis of the asymptotic stability of the equilibrium points and the convergence of the numerical scheme is given based on the use of M-matrices. Finally we consider the case in which the nonlinear term satisfies a condition of quasi- monotonicity.
TL;DR: In this paper, a boundary value problem is solved for the nonlinear first order complex differential equation in the plane, where the complex function H is measurable on, Lipschitz continuous with respect to the last...
Abstract: A boundary value problem is solved for the nonlinear first order complex differential equation in the plane. The complex function H is measurable on , Lipschitz continuous with respect to the last ...
TL;DR: For boundary value problems for linear differential equations in which the boundary conditions are linear, the uniqueness of solutions implies the existence of solutions as mentioned in this paper, which is not the case in general for nonlinear differential equations.
TL;DR: In this article, the limiting behavior of a pseudoparabolic partial differential equation with a total flux boundary condition on a sphere in the interior of a body is studied as the radius is allowed to approach zero.
Abstract: The limiting behaviour of a pseudoparabolic partial differential equation with a total flux boundary condition on a sphere in the interior of a body is studied as the radius is allowed to approach Zero. It is shown that the solutions, parametrized by the radii, converge to the solution of a corresponding problem with a Dirac measure with mass at the center of the sphere.