A Numerical Method for Singular Two Point Boundary Value Problems

The numerical solution of boundary value problems for linear systems of first order equations with a regular singular point at one endpoint is considered. The standard procedure of expanding about the singularity to get a nonsingular problem over a reduced interval is justified in some detail. Quite general boundary conditions are included which permit unbounded solutions. Error estimates are given and some numerical calculations are presented to check the theory

Arc-Length Continuation and Multigrid Techniques for Nonlinear Elliptic Eigenvalue Problems

We investigate multi-grid methods for solving linear systems arising from arc-length continuation techniques applied to nonlinear elliptic eigenvalue problems. We find that the usual multi-grid methods diverge in the neighborhood of singular points of the solution branches. As a result, the continuation method is unable to continue past a limit point in the Bratu problem. This divergence is analyzed and a modified multi-grid algorithm has been devised based on this analysis. In principle, this new multi-grid algorithm converges for elliptic systems, arbitrarily close to singularity and has been used successfully in conjunction with arc-length continuation procedures on the model problem. In the worst situation, both the storage and the computational work are only about a factor of two more than the unmodified multi-grid methods

Homotopy Method for the Large, Sparse, Real Nonsymmetric Eigenvalue Problem

A homotopy method to compute the eigenpairs, i.e., the eigenvectors and eigenvalues, of a given real matrix A1 is presented. From the eigenpairs of some real matrix A0, the eigenpairs of A(t) ≡ (1 − t)A0 + tA1 are followed at successive "times" from t = 0 to t = 1 using continuation. At t = 1, the eigenpairs of the desired matrix A1 are found. The following phenomena are present when following the eigenpairs of a general nonsymmetric matrix: • bifurcation, • ill conditioning due to nonorthogonal eigenvectors, • jumping of eigenpaths. These can present considerable computational difficulties. Since each eigenpair can be followed independently, this algorithm is ideal for concurrent computers. The homotopy method has the potential to compete with other algorithms for computing a few eigenvalues of large, sparse matrices. It may be a useful tool for determining the stability of a solution of a PDE. Some numerical results will be presented

Convergence Rates for Newton’s Method at Singular Points

If Newton’s method is employed to find a root of a map from a Banach space into itself and the derivative is singular at that root, the convergence of the Newton iterates to the root is linear rather than quadratic. In this paper we give a detailed analysis of the linear convergence rates for several types of singular problems. For some of these problems we describe modifications of Newton’s method which will restore quadratic convergence

Design guide for helicopter transmission seals

A detailed approach for the selection and design of seals for helicopter transmissions is presented. There are two major types of seals presently being used and they are lip type seals and mechanical type seals. Lip type seals can be divided in conventional lip seals and hydrodynamic lip seals. Conventional lip seals can be used for slow-speed, low-pressure, low-temperature sealing. Hydrodynamic lip seals although they are as pressure and temperature limited as conventional lip seals, can operate at a higher speed. Mechanical types seals are comprised of face seals and circumferential seals. Face seals are capable of high speed, high pressure, and high temperature. Circumferential seals can be used in high-speed and high-temperature applications, but will leak excessively at moderate pressures. The performance goals of transmission seals are a life that is at least equal to the scheduled overhaul interval of the gearbox component and a leakage rate of near zero

The Stability of One-Step Schemes for First-Order Two-Point Boundary Value Problems

The stability of a finite difference scheme is related explicitly to the stability of the continuous problem being solved. At times, this gives materially better estimates for the stability constant than those obtained by the standard process of appealing to the stability of the numerical scheme for the associated initial value problem

Constraining the Circumbinary Envelope of Z CMa via imaging polarimetry

Z CMa is a complex binary system, composed of a Herbig Be and an FU Ori star. The Herbig star is surrounded by a dust cocoon of variable geometry, and the whole system is surrounded by an infalling envelope. Previous spectropolarimetric observations have reported a preferred orientation of the polarization angle, perpendicular to the direction of a large, parsec-sized jet associated with the Herbig star. The variability in the amount of polarized light has been associated to changes in the geometry of the dust cocoon that surrounds the Herbig star. We aim to constrain the properties of Z CMa by means of imaging polarimetry at optical wavelengths. Using ExPo, a dual-beam imaging polarimeter which operates at optical wavelengths, we have obtained imaging (linear) polarimetric data of Z CMa. Our observations were secured during the return to quiescence after the 2008 outburst. We detect three polarized features over Z CMa. Two of these features are related to the two jets reported in this system: the large jet associated to the Herbig star, and the micro-jet associated to the FU Ori star. Our results suggest that the micro-jet extends to a distance ten times larger than reported in previous studies. The third feature suggests the presence of a hole in the dust cocoon that surrounds the Herbig star of this system. According to our simulations, this hole can produce a pencil beam of light that we see scattered off the low-density envelope surrounding the system.Comment: Accepted for publication in A\&

A dynamical approximation for stochastic partial differential equations

Random invariant manifolds often provide geometric structures for understanding stochastic dynamics. In this paper, a dynamical approximation estimate is derived for a class of stochastic partial differential equations, by showing that the random invariant manifold is almost surely asymptotically complete. The asymptotic dynamical behavior is thus described by a stochastic ordinary differential system on the random invariant manifold, under suitable conditions. As an application, stationary states (invariant measures) is considered for one example of stochastic partial differential equations.Comment: 28 pages, no figure

Finite-size and pressure effects in YBa_2Cu_4O_8 probed by magnetic field penetration depth measurements

We explore the combined pressure and finite-size effects on the in-plane penetration depth \lambda_{ab} in YBa_2Cu_4O_8. Even though this cuprate is stoichiometric the finite-size scaling analysis of \lambda_{ab}^{-2}(T) uncovers the granular nature and reveals domains with nanoscale size L_{c} along the c-axis. L_{c} ranges from 33.2 Angstrom to 28.9 Angstrom at pressures from 0.5 to 11.5 kbar. These observations raise serious doubts on the existence of a phase coherent macroscopic superconducting state in cuprate superconductors.Comment: 7 pages, 6 figure

Loss of the Surface Layers of Comet Nuclei

The Deep Impact observations of low thermal inertia for comet 9P/Tempel 1 are of profound importance for the observations to be made by the Rosetta spacecraft at comet 67P/Churyumov-Gerasimenko. While sub-surface sublimation is necessary to explain the observations, the depth at which this occurs is no more than 2-3cm and possibly less. The low thermal conductivity when combined with local surface roughness (also observed with Deep Impact) implies that local variations in outgassing rates can be substantial. These variations are likely to be on scales smaller than the resolution limits of all experiments on the Rosetta orbiter. The observed physico-chemical inhomogeneity further suggests that the Rosetta lander will only provide a local snapshot of conditions in the nucleus laye