Advanced methods for the solution of differential equations

This book is based on a course presented at the Lewis Research Center for engineers and scientists who were interested in increasing their knowledge of differential equations. Those results which can actually be used to solve equations are therefore emphasized; and detailed proofs of theorems are, for the most part, omitted. However, the conclusions of the theorems are stated in a precise manner, and enough references are given so that the interested reader can find the steps of the proofs

The dynamics of bottom boundary currents in the ocean

Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy at the Massachusetts Institute of Technology and the Woods Hole Oceanographic Institution September, 1973This thesis presents an investigation of the dynamics of bottom boundary currents in the ocean. The major emphasis is to develop simple mathematical models in which various dynamical features of these complex geophysical flows may be isolated and explored. Two separate models are formulated and the theoretical results are compared to observational data and/or laboratory experiments. A steady flow over a constant sloping bottom is treated in each model. A streamtube model which describes the variatlon in average cross-sectional properties of the flow is derived to examine the interaction between turbulent entrainment and bottom friction in a rotating stratified fluid. Empirical laws are used to parameterize these processes and the associated entrainment and friction coefficients (Eo,K) are evaluated from data for two bottom currents: the Norwegian Overflow and the Mediterranean Outflow. The ability to fit adequately all observations with the solutions for a single parameter pair demonstrates the dynamical consistency of the streamtube model. The solutions indicate that bottom stresses dominate the frictional drag on the dense fluid layer in the vicinity of the source whereas relatively weak entrainment slowly modulates the flow properties in the downstream region. The combined influence of entrainment and ambient stratification help limit the descent of the Mediterranean Outflow to a depth of approximately 1200 m. while strong friction acting over a long downstream scale allows the flow of Norwegian Sea water to reach the ocean floor. A turbulent Ekman layer model with a constant eddy viscosity is also formulated. The properties of the flow are defined in terms of the layer thickness variable d(x,y), whose governing equation is judged intractable for the general case. However, limiting forms of this equation may be solved when the layer thickness is much less than (weak rotation) or greater than (strong rotation) the Ekman layer length scale. In the weak rotation limit, a similarity soltition is derived which describes the flow field in an intermediate downstream range. Critical measurements in a laboratory experiment are used to establish distinctive properties of rotational perturbations to the viscous flow, such as the antisymmetric corrections to the layer thickness profile and the surface velocity distribution, which depend on downstream distance like y2/7. The constraint of weak rotational effects precludes a meaningful comparison with oceanic bottom currents. The analysis of the strong rotation limit leads to the prediction of an Ekman flux mechanism by which dense fluid is drained from the lower boundary of the thick core of the current and the geostrophic flow is extinguished. The form of a similarity solution for the downstream flow is derived subject to the specification of a single constant by the upstream boundary condition. The results of some exploratory experiments are sufficient to confirm some qualitative aspects of this solution, but transience of the laboratory flow limits a detailed comparison to theory. Some features of the Ekman flux mechanism are noted in the observational data for the Norwegian Overflow

International Conference on Mathematical Analysis and Applications in Science and Engineering – Book of Extended Abstracts

The present volume on Mathematical Analysis and Applications in Science and Engineering - Book of Extended Abstracts of the ICMASC’2022 collects the extended abstracts of the talks presented at the International Conference on Mathematical Analysis and Applications in Science and Engineering – ICMA2SC'22 that took place at the beautiful city of Porto, Portugal, in June 27th-June 29th 2022 (3 days). Its aim was to bring together researchers in every discipline of applied mathematics, science, engineering, industry, and technology, to discuss the development of new mathematical models, theories, and applications that contribute to the advancement of scientific knowledge and practice. Authors proposed research in topics including partial and ordinary differential equations, integer and fractional order equations, linear algebra, numerical analysis, operations research, discrete mathematics, optimization, control, probability, computational mathematics, amongst others. The conference was designed to maximize the involvement of all participants and will present the state-of- the-art research and the latest achievements.info:eu-repo/semantics/publishedVersio

Attenuation of leaky modes and filtering techniques in graded index multimode fibres

This work is an investigation of leaky ray (mode) attenuation in graded multimode fibres. In contrast to slab or rectangular dielectric waveguides, cylindrical fibres possess leaky modes which greatly modify their propagation properties. The power carried by such modes eventually diminishes to zero, thus adding no useful contribution to the signal transmission over fibre channels. Suitable filtering techniques are therefore proposed. [Continues.

Studies of coated and polycrystalline superconductors using the time dependant Ginzburg-Landau equations

Time-dependent Ginzburg-Land au equations are used to model 2D and 3D systems containing both superconductors and normal metals, in which both T(_c) and normal-state resistivity are spatially dependent. The equations are solved numerically using an efficient semi-implicit Crank-Nicolson algorithm. The algorithm, is used to model flux entry and exit in homogenous superconductors with metallic coatings of different resistivities. For an abrupt boundary there is a minimum field of initial vortex entry occurring at a kappa-dependent finite ratio of the normal-state resistivities of the superconductor and the normal metal. Highly reversible magnetization characteristics are achieved using a diffusive layer several coherence lengths wide between the superconductor and the normal metal. This work provides the first TD GL simulation in both 2D and 3D of current flow in polycrystalline superconductors, and provides some important new results both qualitative and quantitative. Using a magnetization method we obtain Jc for both 2D and 3D systems, and obtain the correct field and kappa dependences in 3D, given by F = 3.6 x 10-4 B}l (T) (1- b)2. The pre-factor is different (about 3 to 5 times smaller) from that observed in technological superconductors, but evidence is provided showing that this prefactor depends on the details of 1կ effects at the edges of superconducting grains. In 2D, the analytic flux shear calculation developed by Pruymboom in his thin-film work gives good agreement with our computational results.Visualization of Iぜ and dissipation (including movies in the 2D case) shows that in both 2D and 3D, Jc is determined by flux shear along grain boundaries. In 3D the moving fluxons are confined to the grain boundaries, and cut through stationary fluxons which pass through the grains and are almost completely straight

The World in Eleven Dimensions

A unified theory embracing all physical phenomena is a major goal of theoretical physics. In the early 1980s, many physicists looked to eleven-dimensional supergravity in the hope that it might provide that elusive superunified theory. In 1984 supergravity was knocked off its pedestal by ten-dimensional superstrings, one-dimensional objects whose