Fast Computation of Highly Oscillatory ODE Problems: Applications in High Frequency Communication Circuits

Two types of algorithms are presented to approximate highly oscillatory and non-oscillatory first order ordinary differential equations. In the first approach, radial basis function interpolation is used to approximate the function f(t,x), then quadrature method is used to evaluate the integral part of the equation. The method is implementable to non-oscillatory first order initial value problems. The second approach is more generic and can approximate highly oscillatory and non-oscillatory initial value problems. Accordingly, the first order initial value problem with oscillatory forcing term is transformed into an integral with oscillatory Fourier kernel. The transformed oscillatory integral is then evaluated numerically by the Levin collocation method. Finally, non-linear form of the initial value problems with oscillatory forcing term is converted into a linear form using Bernoulli’s transformation. The resulting linear oscillatory problem is then computed by the new approaches. To justify accuracy of the algorithms, few numerical examples are added from the literature.publishedVersio

Further studies on relic neutrino asymmetry generation I: the adiabatic Boltzmann limit, non-adiabatic evolution, and the classical harmonic oscillator analogue of the quantum kinetic equations

We demonstrate that the relic neutrino asymmetry evolution equation derived from the quantum kinetic equations (QKEs) reduces to the Boltzmann limit that is dependent only on the instantaneous neutrino number densities, in the adiabatic limit in conjunction with sufficient damping. An original physical and/or geometrical interpretation of the adiabatic approximation is given, which serves as a convenient visual aid to understanding the sharply contrasting resonance behaviours exhibited by the neutrino ensemble in opposing collision regimes. We also present a classical analogue for the evolution of the difference in $\nu_{\alpha}$ and $\nu_s$ number densities which, in the Boltzmann limit, is akin to the behaviour of the generic reaction $A \rightleftharpoons B$ with equal forward and reverse reaction rate constants. A new characteristic quantity, the matter and collision-affected mixing angle of the neutrino ensemble, is identified here for the first time. The role of collisions is revealed to be twofold: (i) to wipe out the inherent oscillations, and (ii) to equilibrate the $\nu_{\alpha}$ and $\nu_s$ number densities in the long run. Studies on non-adiabatic evolution and its possible relation to rapid oscillations in lepton number generation also feature, with the introduction of an adiabaticity parameter for collision-affected oscillations.Comment: RevTeX, 38 pages including 8 embedded figure

Steady and Stable: Numerical Investigations of Nonlinear Partial Differential Equations

Excerpt: Mathematics is a language which can describe patterns in everyday life as well as abstract concepts existing only in our minds. Patterns exist in data, functions, and sets constructed around a common theme, but the most tangible patterns are visual. Visual demonstrations can help undergraduate students connect to abstract concepts in advanced mathematical courses. The study of partial differential equations, in particular, benefits from numerical analysis and simulation