Showing papers on "Boundary value problem published in 1987"

Stochastic differential equations : an introduction with applications

Abstract: Some Mathematical Preliminaries.- Ito Integrals.- The Ito Formula and the Martingale Representation Theorem.- Stochastic Differential Equations.- The Filtering Problem.- Diffusions: Basic Properties.- Other Topics in Diffusion Theory.- Applications to Boundary Value Problems.- Application to Optimal Stopping.- Application to Stochastic Control.- Application to Mathematical Finance.

An introduction to the mathematics and methods of astrodynamics

Abstract: Part 1 Hypergeometric Functions and Elliptic Integrals: Some Basic Topics In Analytical Dynamics The Problem of Two Bodies Two-Body Orbits and the Initial-Value Problem Solving Kepler's Equation Two-Body Orbital Boundary Value Problem Solving Lambert's Problem Appendices Part 2 Non-Keplerian Motion: Patched-Conic Orbits and Perturbation Methods Variation of Parameters Two-Body Orbital Transfer Numerical Integration of Differential Equations The Celestial Position Fix Space Navigation Appendices

A global uniqueness theorem for an inverse boundary value problem

Abstract: In this paper, we show that the single smooth coefficient of the elliptic operator LY = v yv can be determined from knowledge of its Dirichlet integrals for arbitrary boundary values on a fixed region 2 C R', n ? 3. From a physical point of view, we show that an isotropic conductivity can be determined by steady state measurements at the boundary.

Wave Function of the Universe

Abstract: The quantum state of a spatially closed universe can be described by a wave function which is a functional on the geometries of compact three-manifolds and on the values of the matter fields on these manifolds. The wave function obeys the Wheeler-DeWitt second-order functional differential equation. We put forward a proposal for the wave function of the "ground state" or state of minimum excitation: the ground-state amplitude for a three-geometry is given by a path integral over all compact positive-definite four-geometries which have the three-geometry as a boundary. The requirement that the Hamiltonian be Hermitian then defines the boundary conditions for the Wheeler-DeWitt equation and the spectrum of possible excited states. To illustrate the above, we calculate the ground and excited states in a simple minisuperspace model in which the scale factor is the only gravitational degree of freedom, a conformally invariant scalar field is the only matter degree of freedom and $\ensuremath{\Lambda}g0$. The ground state corresponds to de Sitter space in the classical limit. There are excited states which represent universes which expand from zero volume, reach a maximum size, and then recollapse but which have a finite (though very small) probability of tunneling through a potential barrier to a de Sitter-type state of continual expansion. The path-integral approach allows us to handle situations in which the topology of the three-manifold changes. We estimate the probability that the ground state in our minisuperspace model contains more than one connected component of the spacelike surface.

Frictional–collisional constitutive relations for granular materials, with application to plane shearing

Abstract: Within a granular material stress is transmitted by forces exerted at points of mutual contact between particles. When the particles are close together and deformation of the assembly is slow, contacts are sustained for long times, and these forces consist of normal reactions and the associated tangential forces due to friction. When the particles are widely spaced and deformation is rapid, on the other hand, contacts are brief and may be regarded as collisions, during which momentum is transferred. While constitutive relations are available which model both these situations, in many cases the average contact times lie between the two extremes. The purpose of the present work is to propose constitutive relations and boundary conditions for this intermediate case and to solve the corresponding equations of motion for plane shear of a cohesionless granular material between infinite horizontal plates. It is shown that, in general, not all the material between the plates participates in shearing, and the solutions for the shearing material are coupled to a yield condition for the non-shearing material to give a complete solution of the problem.

Black-hole normal modes: A WKB approach. I. Foundations and application of a higher-order WKB analysis of potential-barrier scattering.

Abstract: We present a semianalytic technique for determining the complex normal-mode frequencies of black holes. The method makes use of the WKB approximation, carried to third order beyond the eikonal approximation. Mathematically, the problem is similar to studying one-dimensional quantum-mechanical scattering near the peak of a potential barrier, and determining the scattering resonances. Under such conditions, a modification of the usual WKB approach must be used. We obtain the connection formulas that relate the amplitudes of incident, reflected, and transmitted waves, to the third WKB order. By imposing the normal-mode (resonance) boundary condition of a zero incident amplitude with nonzero transmitted and reflected amplitudes, we find a simple formula that determines the real and imaginary parts of the normal-mode frequency of perturbation (or of the quantum-mechanical energy of the resonance) in terms of the derivatives (up to and including sixth order) of the barrier function evaluated at the peak, and in terms of the quantity (n+(1/2)), where n is an integer and labels the fundamental mode (resonance), first overtone, and so on. This higher-order approach may find uses in barrier-tunneling problems in atomic and nuclear physics.

Density-functional theory for time-dependent systems.

Abstract: The time-dependent density-functional theory of Runge and Gross [Phys. Rev. Lett. 52, 997 (1984)] is reexamined on the basis of its limitations, and the criticisms raised by Xu and Rajagopal [Phys. Rev. A 31, 2682 (1985)] are addressed, within the imposition of natural boundary conditions of vanishing density and potential at infinity. Also, for a single-particle system characterized by an arbitrary time-dependent potential, the uniqueness of the density-to-potential mapping is established explicitly for both bound and scattering states.

Boundary stabilization of thin plates

Abstract: Preface 1. Introduction: orientation Background Connection with exact controllability 2. Thin plate models: Kirchhoff model Mindlin-Timoshenko model von Karman model A viscoelastic plate model A linear termoelastic plate model 3. Boundary feedback stabilization of Mindlin-Timoshenko plates: Orientation: existence, uniqueness, and properties of solutions Uniform asymptotic stability of solutions 4. Limits of the Mindlin-Timoshenko system and asymptotic stability of the limit systems: Orientation The limit of the M-T system as KE 0+ The limit of the M-T system as K Study of the Kirchhoff system Uniform asymptotic stability of solutions Limit of the Kirchhoff system as 0+ 5. Uniform stabilization in some nonlinear plate problems: Uniform stabilization of the Kirchhoff system by nonlinear feedback Uniform asymptotic energy estimates for a von Karman plate 6. Boundary feedback stabilization of Kirchhoff plates subject to weak viscoelastic Damping: formulation of the boundary value problem Existence, uniqueness, and properties of solutions Asymptotic energy estimates 7. Uniform asymptotic energy estimates for thermoelastic plates: Orientation Existence, uniqueness, regularity, and strong stability Uniform asymptotic energy estimates Bibliography Index.

On pressure boundary conditions for the incompressible Navier‐Stokes equations

Abstract: The pressure is a somewhat mysterious quantity in incompressible flows. It is not a thermodynamic variable as there is no ‘equation of state’ for an incompressible fluid. It is in one sense a mathematical artefact—a Lagrange multiplier that constrains the velocity field to remain divergence-free; i.e., incompressible—yet its gradient is a relevant physical quantity: a force per unit volume. It propagates at infinite speed in order to keep the flow always and everywhere incompressible; i.e., it is always in equilibrium with a time-varying divergence-free velocity field. It is also often difficult and/or expensive to compute. While the pressure is perfectly well-defined (at least up to an arbitrary additive constant) by the governing equations describing the conservation of mass and momentum, it is (ironically) less so when more directly expressed in terms of a Poisson equation that is both derivable from the original conservation equations and used (or misused) to replace the mass conservation equation. This is because in this latter form it is also necessary to address directly the subject of pressure boundary conditions, whose proper specification is crucial (in many ways) and forms the basis of this work. Herein we show that the same principles of mass and momentum conservation, combined with a continuity argument, lead to the correct boundary conditions for the pressure Poisson equation: viz., a Neumann condition that is derived simply by applying the normal component of the momentum equation at the boundary. It usually follows, but is not so crucial, that the tangential momentum equation is also satisfied at the boundary.

Analysis of undrained soil deformation due to ground loss

Abstract: Closed form solutions are presented for obtaining the strain field in an initially isotropic and homogeneous incompressible soil due to near-surface ground loss. This problem fits into a category of cases in which the imposed boundary conditions are only or mainly in terms of displacements (strain-controlled problems). In these cases there is a possibility of eliminating the stresses and obtaining the strains by using only the incompressibility condition. The presence of the top free surface is considered by means of a virtual image technique and some results for the elastic half-space. The results are simple, especially for the movements of the soil surface. The application to some typical problems, such as soft ground tunnelling and pile driving or extraction, shows that the calculated movements agree well with the experimental observations and compare favourably with commonly used numerical methods. Some hints are given for extending the method to other cases, such as to compressible materials. Des sol...

Numerical absorbing boundary conditions for the wave equation

Abstract: On developpe une theorie des approximations par differences des conditions aux limites absorbantes pour l'equation d'ondes scalaire a plusieurs dimensions d'espace

Path-integral computation of superfluid densities

Abstract: The normal and superfluid densities are defined by the response of a liquid to sample boundary motion. The free-energy change due to uniform boundary motion can be calculated by path-integral methods from the distribution of the winding number of the paths around a periodic cell. This provides a conceptually and computationally simple way of calculating the superfluid density for any Bose system. The linear-response formulation relates the superfluid density to the momentum-density correlation function, which has a short-ranged part related to the normal density and, in the case of a superfluid, a long-ranged part whose strength is proportional to the superfluid density. These facts are discussed in the context of path-integral computations and demonstrated for liquid $^{4}\mathrm{He}$ along the saturated vapor-pressure curve. Below the experimental superfluid transition temperature the computed superfluid fractions agree with the experimental values to within the statistical uncertainties of a few percent in the computations. The computed transition is broadened by finite-sample-size effects.

Some errors estimates for the box method

Abstract: We define and analyze several variants of the box method for discretizing elliptic boundary value problems in the plane. Our estimates show the error to be comparable to a standard Galerkin finite element method using piecewise linear polynomials.

A new finite formulation for computational fluid dynamics: VII. The Stokes problem with various well-posed boundary conditions: symmetric formulations that converge for all velocity/pressure spaces

Abstract: Symmetric finite element formulations are proposed for the primitive-variables form of the Stokes equations and shown to be convergent for any combination of pressure and velocity interpolations. Various boundary conditions, such as pressure, are accommodated.

On the solution of integral equations with strongly singular kernels

Abstract: Some useful formulas are developed to evaluate integrals having a singularity of the form (t-x) sup-m ,m greater than or equal 1. Interpreting the integrals with strong singularities in Hadamard sense, the results are used to obtain approximate solutions of singular integral equations. A mixed boundary value problem from the theory of elasticity is considered as an example. Particularly for integral equations where the kernel contains, in addition to the dominant term (t-x) sup -m , terms which become unbounded at the end points, the present technique appears to be extremely effective to obtain rapidly converging numerical results.

A simple formula for diffusion calculations involving wall reflection and low density

Abstract: Diffusion theory is often employed to calculate the effects of wall destruction on the local concentration of an active species immersed in a scattering gas. In many situations the spatial dependence of the concentration is given to a good approximation by the fundamental diffusion mode, and the local loss frequency can be calculated using the container’s fundamental mode diffusion length Λ. The additional assumption that the density of the active species may be taken to be zero at the container boundaries gives a value of Λ=Λ0 which depends only on the container dimensions, but use of Λ0 can be seriously in error if the diffusion mean free path λm is comparable to the dimensions, or if the particle reflection coefficient R becomes of significance. An improved boundary condition may be written simply in terms of the linear extrapolation length λ, whose inverse is the logarithmic gradient of the particle density at the boundary. The equation λ=2(1+R)λm/3(1−R) allows the representation of the full range of ...

A finite-element approach to control the end-point motion of a single-link flexible robot

Abstract: A structural finite-element technique based on Bernoulli-Euler beam theory is presented which will permit the finding of the torques (or forces) that are necessary to apply at one end of a flexible link to produce a desired motion at the other end. This technique is suitable for the open loop control of the tip motion. It may also provide a good control law for feedback control. The finite-element method is used to discretize the equations of motion. This method has a major advantage in the fact that different material properties and boundary conditions like hubs, tip loads, changes in cross sections, etc., can be handled in a very simple and straightforward manner. The resulting differential equations are integrated via the frequency domain. This allows for the expansion of the desired end motion into its harmonic components and helps to visualize the complex wave propagation nature of the problem. The performance of the proposed technique is illustrated in the solution of a practical example. Results point out the potential that this technique has in the study of the dynamics and control not only of flexible robots, but also of any other flexible mechanisms like those used in biomechanics, where high precision at the tip of very light flexible arms is required.

Hardy spaces and the Neumann problem in L^p for laplace's equation in Lipschitz domains

Abstract: On donne des resultats optimaux pour la resolubilite du probleme de Neumann dans des domaines de Lipschitz a donnees dans L p . On obtient des resultats du point extreme correspondant pour des espaces de Hardy

Deflation of conjugate gradients with applications to boundary value problems

Abstract: A method for improving the convergence of the standard conjugate gradient method is given. This method involves the use of auxiliary subspaces It is shown how such subspaces may be constructed for boundary value problems and an analysis of convergence for second order problems is presented.

Young's modulus of porous brittle solids

Abstract: A new equationE =E0 (1 −aP)n whereE andE0 are the Young's moduli at porosity,P, and zero, respectively, a andn are material constants, has been derived semi-empirically for describing the porosity dependence of Young's modulus of brittle solids. The equation satisfies quite well the exact theoretical solution for the values of Young's moduli at different porosities for model systems with ideal and non-ideal packing geometry. The equation shows excellent agreement with the data Onα- andβ-alumina over a wide range of porosity. Unlike the existing porosity-elastic modulus equations, the proposed equation satisfies the boundary conditions and is inherently capable of treating isometric closed pores as well as non-isometric interconnected pores. The parameters a and n provide information about the packing geometry and pore structure of the material.

Boundary Value Problems in Abstract Kinetic Theory

Abstract: This monograph is intended to be a reasonably self -contained and fairly complete exposition of rigorous results in abstract kinetic theory. Throughout, abstract kinetic equations refer to (an abstract formulation of) equations which describe transport of particles, momentum, energy, or, indeed, any transportable physical quantity. These include the equations of traditional (neutron) transport theory, radiative transfer, and rarefied gas dynamics, as well as a plethora of additional applications in various areas of physics, chemistry, biology and engineering. The mathematical problems addressed within the monograph deal with existence and uniqueness of solutions of initial-boundary value problems, as well as questions of positivity, continuity, growth, stability, explicit representation of solutions, and equivalence of various formulations of the transport equations under consideration. The reader is assumed to have a certain familiarity with elementary aspects of functional analysis, especially basic semigroup theory, and an effort is made to outline any more specialized topics as they are introduced. Over the past several years there has been substantial progress in developing an abstract mathematical framework for treating linear transport problems. The benefits of such an abstract theory are twofold: (i) a mathematically rigorous basis has been established for a variety of problems which were traditionally treated by somewhat heuristic distribution theory methods; and (ii) the results obtained are applicable to a great variety of disparate kinetic processes. Thus, numerous different systems of integrodifferential equations which model a variety of kinetic processes are themselves modelled by an abstract operator equation on a Hilbert (or Banach) space.

Stochastic differential equations for multi-dimensional domain with reflecting boundary

Abstract: In this paper we prove that there exists a unique solution of the Skorohod equation for a domain inR d with a reflecting boundary condition. We remove the admissibility condition of the domain which is assumed in the work [4] of Lions and Sznitman. We first consider a deterministic case and then discuss a stochastic case.

Computation of vortex sheet roll-up in the Trefftz plane

Abstract: Two vortex-sheet evolution problems arising in aerodynamics are studied numerically. The approach is based on desingularizing the Cauchy principal value integral which defines the sheet's velocity. Numerical evidence is presented which indicates that the approach converges with respect to refinement in the mesh-size and the smoothing parameter. For elliptic loading, the computed roll-up is in good agreement with Kaden's asymptoic spiral at early times. Some aspects of the solution's instability to short-wavelength perturbations, for a small value of the smoothing parameter, are inferred by comparing calculations performed with different levels of computer round-off error. The tip vortices' deformation, due to their mutual interaction, is shown in a long-time calculation. Computations for a simulated fuselage-flap configuration show a complicated process of roll-up, deformation and interaction involving the tip vortex and the inboard neighboring vortices.

Second kind integral equation formulation of Stokes' flows past a particle of arbitary shape

Abstract: The problem of determining the slow viscous flow of an unbounded fluid past a single solid particle is formulated exactly as a system of linear Fredholm integral equations of the second kind for a ...

Documentation of computer program VS2D to solve the equations of fluid flow in variably saturated porous media

Abstract: This repprt documents a computer code for solving problems of variably saturated, single-phase flow in porous media. The mathematical model of this physical process is developed by combining the law of conservation of fluid mass with a nonlinear form of Darcy's law. The resultant mathematical model, or flow equation, is written with total hydraulic potential as the dependent variable. This allows straightforward treatment of both saturated and unsaturated conditions. The spatial derivatives in the flow equation are approximated by central differences written about grid-block boundaries. Time derivatives are approximated by a fully implicit backward scheme. Nonlinear storage terms are linearized by an implicit Newton-Raphson method. Nonlinear conductance terms, boundary conditions, and sink terms are linearized implicitly. Relative hydraulic conductivity is evaluated at cell boundaries by using full upstream weighting, the arithmetic mean, or the geometric mean of values from adjacent cells. Saturated hydraulic conductivities are evaluated at cell boundaries by using distance-weighted harmonic means. The linearized matrix equations are solved using the strongly implicit procedure. Nonlinear conductance and storage coefficients are assumed to be represented by one of three closed-form algebraic equations. Alternatively, these values may be interpolated from tabulated data. Nonlinear boundary conditions treated by the code include infiltration, evaporation, and seepage faces. Extraction by plant roots is included as a nonlinear sink term. The code is written in standard ANSI Fortran. Extensive use of subroutines and function subprograms provides a modular code that is easily modified. A complete listing of data-input requirements and input and output for a one-dimensional infiltration problem and for a two-dimensional problem involving infiltration, evaporation, and evapotranspiration (plant-root extraction) are included.

Functional determinants and geometry

Abstract: In this paper we study the zeta-function determinant in the context of elliptic boundary value problems. Our main technique is to relate the determinant of an operator, or a ratio of determinants, to the boundary values of the solutions of the operator. This has the advantage of restricting attention to the solutions of the operator, which do not depend on the boundary conditions and can often be written down explicitly, rather than the eigenvalues, which are usually difficult to work with. In addition, the problem is reduced to a calculation over the boundary of the manifold which is a closed manifold of dimension one less than the original manifold. This has special significance in the case that the manifold is a finite interval. In this case the boundary is a pair of points and the determinant of an ordinary differential operator is expressed in terms of the determinant of a finite matrix.