Showing papers on "Boundary value problem published in 1981"

Absorbing Boundary Conditions for the Finite-Difference Approximation of the Time-Domain Electromagnetic-Field Equations

Abstract: When time-domain electromagnetic-field equations are solved using finite-difference techniques in unbounded space, there must be a method limiting the domain in which the field is computed. This is achieved by truncating the mesh and using absorbing boundary conditions at its artificial boundaries to simulate the unbounded surroundings. This paper presents highly absorbing boundary conditions for electromagnetic-field equations that can be used for both two-and three-dimensional configurations. Numerical results are given that clearly exhibit the accuracy and limits of applicability of highly absorbing boundary conditions. A simplified, but equally accurate, absorbing condition is derived for two- dimensional time-domain electromagnetic-field problems.

Superlattice band structure in the envelope-function approximation

Abstract: The band structure of GaAs-GaAlAs and InAs-GaSb superlattices is calculated by matching propagating or evanescent envelope functions at the boundary of consecutive layers. For GaAs-GaAlAs materials, the envelope functions are the solutions of an effective Hamiltonian in which both band edges and effective masses are position dependent. The effective-mass jumps modify the boundary conditions which are imposed to the eigenstates of the effective-mass Hamiltonian. In InAs-GaSb superlattices, the dispersion relations, although quite similar to those obtained in GaAs-GaAlAs materials, reflect the genuine symmetry mismatch of InAs (electrons) and GaSb (light-holes) levels. The evolution of the InAs-GaSb band structure with increasing periodicity is calculated and found to be in excellent agreement with previous LCAO results. The dispersion relations of heavy-hole bands are obtained.

Collocation Software for Boundary-Value ODEs

Abstract: The use of a general-purpose code, COLSYS, is described. The code is capable of solving mixed-order systems of boundary-value problems in ordinary differential equations. The method of spline collocation at Gaussian points is implemented using a B-spline basis. Approximate solutions are computed on a sequence of automatically selected meshes until a user-specified set of tolerances is satisfied. A damped Newton's method is used for the nonlinear iteration. The code has been found to be particularly effective for difficult problems. It is intended that a user be able to use COLSYS easily after reading its algorithm description. The use of the code is then illustrated by examples demonstrating its effectiveness and capabilities.

Thermally optimum spacing of vertical, natural convection cooled, parallel plates

Abstract: While component dissipation patterns and system operating modes vary widely, many electronic packaging configurations can be modeled by symmetrically or asymmetrically isothermal or isoflux plates. The idealized configurations are amenable to analytic optimization based on maximizing total heat transfer per unit volume or unit primary area. To achieve this anlaytic optimization, however, it is necessary to develop composite relations for the variation of the heat transfer coefficient along the plate surfaces. The mathematical development and verification of such composite relations as well as the formulation and solution of the optimizing equations for the various boundary conditions of interest constitute the core of this presentation.

Nonlinear differential equations in abstract spaces

Abstract: Preface Directional derivatives Mean value theorems The Cauchy problem Successive approximations Types of approximate solutions Dissipative type conditions Dissipative operators The exponential formula Difference approximations Convergence of difference approximations Global existence Fundamental properties Differential inequalities in cones Existence of solutions in weak topology Equations with delay Boundary value problems Monotone iterative methods

An optimal order process for solving finite element equations

Abstract: A k-level iterative procedure for solving the algebraic equations which arise from the finite element approximation of elliptic boundary value problems is presented and analyzed. The work estimate for this procedure is proportional to the number of unknowns, an optimal order result. General geometry is permitted for the underlying domain, but the shape plays a role in the rate of convergence through elliptic regularity. Finally, a short discussion of the use of this method for parabolic problems is presented.

Global invertibility of Sobolev functions and the interpenetration of matter

Abstract: A global inverse function theorem is established for mappings u: Ω → ℝn, Ω ⊂ ℝn bounded and open, belonging to the Sobolev space W1.p(Ω), p > n. The theorem is applied to the pure displacement boundary value problem of nonlinear elastostatics, the conclusion being that there is no interpenetration of matter for the energy-minimizing displacement field.

Newton-Kantorovitch algorithm applied to an electromagnetic inverse problem

Abstract: An efficient algorithm is presented to compute the shape of perfectly conducting cylinders via knowledge of scattering cross sections. This choice of scattering data avoids the difficulties linked with the phase measurements or reconstructions. The mathematical analysis is performed in terms of operator and functions, and the fundamental instability of the problem is demonstrated. Then the stability is restored by means of a Tikhonov-Miller regularization. The efficiency of the method is outlined by numerical examples. References are given which show that the same algorithm applies to other electromagnetic inverse problems, especially to gratings.

Theory of electronic transitions in slow atomic collisions

Abstract: This review deals with quantitative descriptions of electronic transitions in atom-atom and ion-atom collisions. In one type of description, the nuclear motion is treated classically or semiclassically, and a wave function for the electrons satisfies a time-dependent Schr\"odinger equation. Expansion of this wave function in a suitable basis leads to time-dependent coupled equations. The role played by electron-translation factors in this expansion is noted, and their effects upon transition amplitudes are discussed. In a fully quantum-mechanical framework there is a wave function describing the motion of electrons and nuclei. Expansion of this wave function in a basis which spans the space of electron variables leads to quantum-mechanical close-coupled equations. In the conventional formulation, known as perturbed-stationary-states theory, certain difficulties arise because scattering boundary conditions cannot be exactly satisfied within a finite basis. These difficulties are examined, and a theory is developed which surmounts them. This theory is based upon an intersecting-curved-wave picture. The use of rotating or space-fixed electronic basis sets is discussed. Various bases are classified by Hund's cases (a)-(e). For rotating basis sets, the angular motion of the nuclei is best described using symmetric-top eigenfunctions, and an example of partial-wave analysis in such functions is developed. Definitions of adiabatic and diabatic representations are given, and rules for choosing a good representation are presented. Finally, representations and excitation mechanisms for specific systems are reviewed. Processes discussed include spin-flip transitions, rotational coupling transitions, inner-shell excitations, covalent-ionic transitions, resonant and near-resonant charge exchange, fine-structure transitions, and collisional autoionization and electron detachment.

Solvability of Nonlinear Equations and Boundary Value Problems

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Some twisted self-dual solutions for the Yang-Mills equations on a hypertorus

Abstract: TheSU(N) Yang-Mills equations are considered in a four-dimensional Euclidean box with periodic boundary conditions (hypertorus). Gauge-invariant twists can be introduced in these boundary conditions, to be labeled with integersn μν (= −n μν ), defined moduloN. The Pontryagin number in this space is often fractional. Whenever this number is zero there are solutions to the equationsG μν =0 HereG μν is the covariant curl. When this number is not zero we find a set of solutions to the equations $$G_{\mu u } = \tilde G_{\mu u } $$ , provided that the periodsa μ of the box satisfy certain relations.

Nonlinear two-point boundary value problems

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Nonlinear Parabolic Equations Involving Measures as Initial Conditions

Abstract: : The Cauchy problem is considered for certain equations with a boundary condition and an initial condition. A solution of the equations exists if and only if O p n+2/n. This paper deals with the question of existence (and uniqueness) when the initial data is a measure, for example a Dirac mass. Physically this corresponds to the important case when the initial temperature (or initial density etc. ..) is extremely high near one point. The main novelty of this paper is to show that a solution exists only under some severe restrictions on the parameter P (or m); namely P must be less than n+2/n (mn+2/n). For example, one striking conclusion reached is the fact that an equation possesses no solution in any dimension n or = 1 and on any time interval (O,T). This result pinpoints the sharp contrast between linear and nonlinear equations from the point of view of existence.

Critical conditions for magnetic instabilities in force-free coronal loops

Abstract: The remarkable magnetohydrodynamic stability of solar coronal loops has been attributed to the anchoring of the ends of loops in the dense photosphere. However, all the previous analyses of such line-tying have been approximate, in the sense that they give only upper or lower bounds on the critical amount of twist (or the critical loop-length) required for the breakdown of stability. The object of the present paper is to remove these approximations and determine the exact value for the critical twist. When it is exceeded the magnetic field becomes kink unstable and a flare may be initiated. A simple analytic stability calculation is described for an idealised loop. This is followed by the development of a general numerical technique for any loop profile, which involves solving the partial differential equations of motion. It is found, for example, that a force-free field of uniform twist possesses a critical twist of 2.49 π, by comparison with the previous bounds of 2π, for stability, and 3.3π, f...

On the regular and the asymptotic characteristic initial value problem for Einstein's vacuum field equations

Abstract: The regular characteristic initial value problem for Einstein’s vacuum field equations where data are given on two intersecting null hypersurfaces is reduced to a characteristic initial value problem for a symmetric hyperbolic system of differential equations. This is achieved by making use of the spin-frame formalism instead of the harmonic gauge condition. The method is applied to the asymptotic characteristic initial value problem for Einstein’s vacuum field equations, where data are given on part of past null infinity and on an incoming null-hypersurface. A uniqueness theorem for this problem is proved by showing that a solution of the problem must satisfy a regular symmetric hyperbolic system of differential equations in a neighbourhood of past null infinity.

Lubricated Squeezing Flow: A New Biaxial Extensional Rheometer

Abstract: Squeezing flow between two disks with lubricated surfaces was found to generate a homogeneous compression or equal biaxial extension in a high viscosity polydimethylsiloxane sample The apparatus is extremely simple: two steel disks with a central rod to provide alignment and prevent sample slip, an LVDT to measure displacement, and a silicone oil bath The mass and area of the upper disk provide for a constant force boundary condition The biaxial viscosity was found to be approximately six times the shear viscosity over biaxial extension rates er from 0003 to 10 s−1 Lubrication could be achieved up to Hencky strains of about 25 Some data were also taken on the same polyisobutylene sample used by Stephenson and Meissner in their biaxial stretching study Agreement was very good

Sturmian Theory for Ordinary Differential Equations

Abstract: I. Historical Prologue.- 1. Introduction.- 2. Methods Based Upon Variational Principles.- 3. Historical Comments on Terminology.- II. Sturmian Theory for Real Linear Homogeneous Second Order Ordinary Differential Equations on a Compact Interval.- 1. Introduction.- 2. Preliminary Properties of Solutions of (1.1).- 3. The Classical Oscillation and Comparison Theorems of Sturm.- 4. Related Oscillation and Comparison Theorems.- 5. Sturmian Differential Systems.- 6. Polar Coordinate Transformations.- 7. Transformations for Differential Equations and Systems.- 8. Variational Properties of Solutions of (1.1).- 9. Comparison Theorems.- 10. Morse Fundamental Quadratic Forms for Conjugate and Focal Points.- 11. Survey of Recent Literature.- 12. Topics and Exercises.- III. Self-Adjoint Boundary Problems Associated with Second Order Linear Differential Equations.- 1. A Canonical Form for Boundary Conditions.- 2 Extremum Problems for Self-Adjoint Systems.- 3. Comparison Theorems.- 4. Comments on Recent Literature.- 5. Topics and Exercises.- IV. Oscillation Theory on a Non-Compact Interval.- 1. Introduction.- 2. Integral Criteria for Oscillation and Non-Oscillation.- 3. Principal Solutions.- 4. Theory of Singular Quadratic Functionals.- 5. Interrelations Between Oscillation Criteria and Boundary Problems.- 6. Strong and Conditional Oscillation.- 7. A Class of Sturmian Problems on a Non-Compact Interval.- 8. Topics and Exercises.- V. Sturmian Theory for Differential Systems.- 1. Introduction.- 2. Special Examples.- 3. Preliminary Properties of Solutions of (2.5).- 4. Associated Riccati Matrix Differential Equations.- 5. Normality and Abnormality.- 6. Variational Properties of Solutions of (3.1).- 7. Comparison Theorems.- 8. Morse Fundamental Hermitian Forms.- 9. Generalized Polar Coordinate Transformations for Matrix Differential Systems.- 10. Matrix Oscillation Theory.- 11. Principal Solutions.- 12. Comments on Systems (3.1) Which are Not Identically Normal.- 13. Comments on the Literature on Oscillation Theory for Hamiltonian Systems (3.1).- 14. Higher Order Differential Equations.- 15. Topics and Exercises.- VI. Self-Adjoint Boundary Problems.- 1. Introduction.- 2. Normality and Abnormality of Boundary Problems.- 3. Self-Adjoint Boundary Problems Associated with (B).- 4. Comparison Theorems.- 5. Treatment of Self-Adjoint Boundary Problems by Matrix Oscillation Theory.- 6. Notes and Comments on the Literature.- 7. Topics and Exercises.- VII. A Class of Definite Boundary Problems.- 1. Introduction.- 2. Definitely Self-Adjoint Boundary Problems.- 3. Comments on Related Literature.- 4. Topics and Exercises.- VIII. Generalizations of Sturmian Theory.- 1. Introduction.- 2. Integro-Differential Boundary Problems.- 3. A Class of Generalized Differential Equations.- 4. Hestenes Quadratic Form Theory in a Hilbert Space.- 5. The Weinstein Method of Intermediate Problems.- 6. Oscillation Phenomena for Hamiltonian Systems in a B*-Algebra.- 7. Topological Interpretations of the Sturmian Theorems.- Abbreviations for Mathematical Publications Most Frequently Used.- Special Symbols.- Author Index.

Solution of Poisson’s equation: Beyond Ewald‐type methods

Abstract: A general method for solving Poisson’s equation without shape approximation for an arbitrary periodic charge distribution is presented The method is based on the concept of multipole potentials and the boundary value problem for a sphere In contrast to the usual Ewald‐type methods, this method has only absolutely and uniformly convergent reciprocal space sums, and treats all components of the charge density equivalently Applications to band structure calculations and lattice summations are also discussed

An evaluation of a model equation for water waves

Abstract: This study assesses a particular model for the unidirectional propagation of water waves, comparing its predictions with the results of a set of laboratory experiments. The equation to be tested is a one-dimensional representation of weakly nonlinear, dispersive waves in shallow water. A model for such flows was proposed by Korteweg & de Vries (1895) and this has provided the theoretical basis for a number of laboratory experiments. Some recent studies that have been made in the area are those of Zabusky & Galvin (1971), Hammack (1973) and Hammack & Segur (1974). In each case the theoretical model gave a good qualitative account of the experiments, but the quantitative comparisons were not very extensive. One of the purposes of this paper is to provide a more detailed quantitative assessment of a particular model than has been given to date. An important aspect of the formulation of the theoretical model is the specification of the initial conditions and the boundary conditions for the equation. Zabusky & Galvin (1971) considered an initial-value problem having spatial periodicity, whereas Hammack (1973) and Hammack & Segur (1974) considered an initial-value problem posed on the real line. In contrast, we shall consider an initial-value problem posed on the half line with boundary data specified at the origin. This problem was chosen to correspond with an experiment in which waves were generated at one end of a long channel, and obviates certain difficulties inherent in the other formulations.

Scaling theory for the criticality of fluids between plates

Abstract: On the basis of a scaling hypothesis for the criticality of fluids, or fluid mixtures, held between parallel plates, we analyze the combined effects of finite wall separation and ’’symmetry breaking’’ boundary conditions at the walls (i.e., adsorption, or preferential adsorption of one component, at the surfaces). The resulting bulk (but ’’two‐dimensional’’) critical point is shifted both in temperature and in density or composition (and in the conjugate field variables) by amounts described by scaling. The scaled form of the phase boundary is consistent with results from a low temperature Ising lattice gas model expansion.

Transformation of boundary problems

Abstract: II. PSEUDODIFFEI%ENTIAL OPERATORS i. Symbol spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 2. Operators on open sets . . . . . . . . . . . . . . . . . . . . . . . . . . 170 3. Definition on Z . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 4. Kernels and adjoints . . . . . . . . . . . . . . ~ . . . . . . . . . . . . 176 5. Boundary values . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 6. Symbols and residual operators . . . . . . . . . . . . . . . . . . . . . . 187 7. Composition and el]iptieity . . . . . . . . . . . . . . . . . . . . . . . . 194 8. Wavefront set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 9. Normal regularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203 i0. L ~ estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206