Regularity Theory and Superalgebraic Solvers for Wire Antenna Problems

We consider the problem of evaluating the current distribution $J(z)$ that is induced on a straight wire antenna by a time-harmonic incident electromagnetic field. The scope of this paper is twofold. One of its main contributions is a regularity proof for a straight wire occupying the interval $[-1,1]$. In particular, for a smooth time-harmonic incident field this theorem implies that $J(z) = I(z)/\sqrt{1-z^2}$, where $I(z)$ is an infinitely differentiable function—the previous state of the art in this regard placed $I$ in the Sobolev space $W^{1,p}$, $p>1$. The second focus of this work is on numerics: we present three superalgebraically convergent algorithms for the solution of wire problems, two based on Hallén's integral equation and one based on the Pocklington integrodifferential equation. Both our proof and our algorithms are based on two main elements: (1) a new decomposition of the kernel of the form $G(z) = F_1(z) \ln\! |z| + F_2(z)$, where $F_1(z)$ and $F_2(z)$ are analytic functions on the real line; and (2) removal of the end-point square root singularities by means of a coordinate transformation. The Hallén- and Pocklington-based algorithms we propose converge superalgebraically: faster than $\mathcal{O}(N^{-m})$ and $\mathcal{O}(M^{-m})$ for any positive integer $m$, where $N$ and $M$ are the numbers of unknowns and the number of integration points required for construction of the discretized operator, respectively. In previous studies, at most the leading-order contribution to the logarithmic singular term was extracted from the kernel and treated analytically, the higher-order singular derivatives were left untreated, and the resulting integration methods for the kernel exhibit $\mathcal{O}(M^{-3})$ convergence at best. A rather comprehensive set of tests we consider shows that, in many cases, to achieve a given accuracy, the numbers $N$ of unknowns required by our codes are up to a factor of five times smaller than those required by the best solvers previously available; the required number $M$ of integration points, in turn, can be several orders of magnitude smaller than those required in previous methods. In particular, four-digit solutions were found in computational times of the order of four seconds and, in most cases, of the order of a fraction of a second on a contemporary personal computer; much higher accuracies result in very small additional computing times

The Future Asymptotic Behaviour of a Non-Tilted Bianchi Type IV Viscous Model

The future asymptotic behaviour of a non-titled Bianchi Type IV viscous fluid model is analyzed. In particular, we consider the case of a viscous fluid without heat conduction, and constant expansion-normalized bulk and shear viscosity coefficients. We show using dynamical systems theory that the only future attracting equilibrium points are the flat Friedmann-LeMaitre (FL) solution, the open FL solution and the isotropic Milne universe solution. We also show the bifurcations exist with respect to an increasing expansion-normalized bulk viscosity coefficient. It is finally shown through an extensive numerical analysis, that the dynamical system isotropizes at late times

On The Dynamics of a Closed Viscous Universe

We use a dynamical systems approach based on the method of orthonormal frames to study the dynamics of a non-tilted Bianchi Type IX cosmological model with a bulk and shear viscous fluid source. We begin by completing a detailed fix-point analysis which give the local sinks, sources and saddles of the dynamical system. We then analyze the global dynamics by finding the $\alpha$-and $\omega$-limit sets which give an idea of the past and future asymptotic behavior of the system. The fixed points were found to be a flat Friedmann-LeMa\^{i}tre-Robertson-Walker (FLRW) solution, Bianchi Type $II$ solution, Kasner circle, Jacobs disc, Bianchi Type $VII_{0}$ solutions, and several closed FLRW solutions in addition to the Einstein static universe solution. Each equilibrium point was described in both its expanding and contracting epochs. We conclude the paper with some numerical experiments that shed light on the global dynamics of the system along with its heteroclinic orbits. With respect to past asymptotic states, we were able to conclude that the Jacobs disc in the expanding epoch was a source of the system along with the flat FLRW solution in a contracting epoch. With respect to future asymptotic states, we were able to show that the flat FLRW solution in an expanding epoch along with the Jacobs disc in the contracting epoch were sinks of the system. We were also able to demonstrate a new result with respect to the Einstein static universe. Namely, we gave certain conditions on the parameter space such that the Einstein static universe has an associated stable subspace. We were however, not able to conclusively say anything about whether a closed FLRW model could be a past or future asymptotic state of the model