Linear Fractionally Damped Oscillator

The linearly damped oscillator equation is considered with the damping term generalized to a Caputo fractional derivative. The order of the derivative being considered is 0≤≤1. At the lower end (=0) the equation represents an undamped oscillator and at the upper end (=1) the ordinary linearly damped oscillator equation is recovered. A solution is found analytically, and a comparison with the ordinary linearly damped oscillator is made. It is found that there are nine distinct cases as opposed to the usual three for the ordinary equation (damped, over-damped, and critically damped). For three of these cases it is shown that the frequency of oscillation actually increases with increasing damping order before eventually falling to the limiting value given by the ordinary damped oscillator equation. For the other six cases the behavior is as expected, the frequency of oscillation decreases with increasing order of the derivative (damping term)

Generalized Bagley-Torvik Equation and Fractional Oscillators

In this paper the Bagley-Torvik Equation is considered with the order of the damping term allowed to range between one and two. The solution is found to be representable as a convolution of trigonometric and exponential functions with the driving force. The properties of the effective decay rate and the oscillation frequency with respect to the order of the fractional damping are also studied. It is found that the effective decay rate and oscillation frequency have a complex dependency on the order of the derivative of the damping term and exhibit properties one might expect of a thermodynamic Equation of state: critical point, phase change, and lambda transition

Analysis of experimental variables for the Kolbe electrolysis of organic acids to hydrocarbons

Thesis (B.S.)--Massachusetts Institute of Technology, Dept. of Chemical Engineering, 1980.Includes bibliographical references (leaves 61-62).by Mark Ramond Naber.B.S

Asymptotic flatness and peeling.

The properties of Asymptotically flat space-times are considered with emphasis on Peeling. Penrose\u27s method of conformal mapping is used to define Asymptotes, Asymptotic Simplicity and Asymptotically flat space-times. The boundary manifold is examined by the behavior of its geometrical properties and by group theoretic means. The behavior of massless fields on flat and curved space-times is considered with the final results being the Peeling theorems. The Peeling theorems are used to examine the asymptotic behavior of physical quantities associated with massless fields. Asymptotically flat space-times which do not peel are also briefly considered.Dept. of Physics. Paper copy at Leddy Library: Theses & Major Papers - Basement, West Bldg. / Call Number: Thesispaper 1990 .N334. Source: Masters Abstracts International, Volume: 30-03, page: 0763. Thesis (M.Sc.)--University of Windsor (Canada), 1990

Taub numbers.

Taub numbers are studied as a set of tensorial conservation laws derivable from curves of solutions to the vacuum Einstein equations. A formulation for Taub numbers of all orders is provided as well as a derivation of the Xanthopoulos theorem. Taub numbers are computed for the Schwarzschild and Kerr solutions viewed as perturbations of Minkowski spacetime and the Schwarzschild solution. They are found to give a measure of the mass and angular momentum and are free of the factor of 2 anomaly associated with the Komar numbers. Taub numbers are also computed for the stationary perturbations of the Schwarzschild solution.Dept. of Physics. Paper copy at Leddy Library: Theses & Major Papers - Basement, West Bldg. / Call Number: Thesis1993 .N334. Source: Dissertation Abstracts International, Volume: 54-09, Section: B, page: 4726. Adviser: E. N. Glass. Thesis (Ph.D.)--University of Windsor (Canada), 1993

Time fractional Schrodinger equation

The Schrodinger equation is considered with the first order time derivative changed to a Caputo fractional derivative, the time fractional Schrodinger equation. The resulting Hamiltonian is found to be non-Hermitian and non-local in time. The resulting wave functions are thus not invariant under time reversal. The time fractional Schrodinger equation is solved for a free particle and for a potential well. Probability and the resulting energy levels are found to increase over time to a limiting value depending on the order of the time derivative. New identities for the Mittag-Leffler function are also found and presented in an appendix.Comment: 23 page

Fractional Differential Forms II

This work further develops the properties of fractional differential forms. In particular, finite dimensional subspaces of fractional form spaces are considered. An inner product, Hodge dual, and covariant derivative are defined. Coordinate transformation rules for integral order forms are also computed. Matrix order fractional calculus is used to define matrix order forms. This is achieved by combining matrix order derivatives with exterior derivatives. Coordinate transformation rules and covariant derivative for matrix order forms are also produced. The Poincare' lemma is shown to be true for exterior fractional differintegrals of all orders excluding those whose orders are non-diagonalizable matrices.Comment: 40 page

IMECE2003-41572 Design and Fabrication of Microtacks for Retinal Implant Applications

ABSTRACT To adhere an artificial retinal implant onto the epiretinal surface of the eye, our group has designed retinal microtacks. The microtacks were fabricated using two different micromachining techniques: 1) deep reactive ion etching (DRIE) and 2) ultrahigh precision micromilling. The DRIE process consisted of machining a double-sided polished three-inch silicon wafer using ICP with the Bosch process. For the ultra-high precision micromilling technique, titanium foil was bonded to a silicon wafer and precision machined with a 150-µm end-mill using PMAC code interfaced to a machine motion controller. Due to fabrication limitations, the tip of the DRIE fabricated Si tack was chisel-shaped, whereas versatility of the micromilling technique allowed a partially conical, tapered tip to be added to the Ti tack, which created a sharper point. For the Si tacks, the average overall length and width were measured to be within 7% and 2%, respectively, of the design while the Ti tacks were found to be within 1% and 6%, respectively. Additionally, the grip width, stop thickness, and the tip taper angle of the Ti tacks were within 3%, 9%, and 4%, respectively, of the design