Analyzing stability of a delay differential equation involving two delays

Analysis of the systems involving delay is a popular topic among applied scientists. In the present work, we analyze the generalized equation $D^{\alpha} x(t) = g\left(x(t-\tau_1), x(t-\tau_2)\right)$ involving two delays viz. $\tau_1\geq 0$ and $\tau_2\geq 0$. We use the the stability conditions to propose the critical values of delays. Using examples, we show that the chaotic oscillations are observed in the unstable region only. We also propose a numerical scheme to solve such equations.Comment: 10 pages, 7 figure

An alternating direction implicit spectral method for solving two dimensional multi-term time fractional mixed diffusion and diffusion-wave equations

In this paper, we consider the initial boundary value problem of the two dimensional multi-term time fractional mixed diffusion and diffusion-wave equations. An alternating direction implicit (ADI) spectral method is developed based on Legendre spectral approximation in space and finite difference discretization in time. Numerical stability and convergence of the schemes are proved, the optimal error is $O(N^{-r}+\tau^2)$, where $N, \tau, r$ are the polynomial degree, time step size and the regularity of the exact solution, respectively. We also consider the non-smooth solution case by adding some correction terms. Numerical experiments are presented to confirm our theoretical analysis. These techniques can be used to model diffusion and transport of viscoelastic non-Newtonian fluids

A Low-Noise High-Density Alkali Metal Scalar Magnetometer

We present an experimental and theoretical study of a scalar atomic magnetometer using an oscillating field-driven Zeeman resonance in a high-density optically-pumped potassium vapor. We describe an experimental implementation of an atomic gradiometer with a noise level below 10 fT/Hz^{1/2}, fractional field sensitivity below 10^{-9}/Hz^{1/2}, and an active measurement volume of about 1.5 cm^3. We show that the fundamental field sensitivity of a scalar magnetometer is determined by the rate of alkali-metal spin-exchange collisions even though the resonance linewidth can be made much smaller than the spin-exchange rate by pumping most atoms into a stretched spin state.Comment: 10 pages, 7 figures. Version 2 is longer, with more complete description of theoretical analysis and comparison between analytical and experimental result

A survey on fuzzy fractional differential and optimal control nonlocal evolution equations

We survey some representative results on fuzzy fractional differential equations, controllability, approximate controllability, optimal control, and optimal feedback control for several different kinds of fractional evolution equations. Optimality and relaxation of multiple control problems, described by nonlinear fractional differential equations with nonlocal control conditions in Banach spaces, are considered.Comment: This is a preprint of a paper whose final and definite form is with 'Journal of Computational and Applied Mathematics', ISSN: 0377-0427. Submitted 17-July-2017; Revised 18-Sept-2017; Accepted for publication 20-Sept-2017. arXiv admin note: text overlap with arXiv:1504.0515

Radiating dipoles in photonic crystals

The radiation dynamics of a dipole antenna embedded in a Photonic Crystal are modeled by an initially excited harmonic oscillator coupled to a non--Markovian bath of harmonic oscillators representing the colored electromagnetic vacuum within the crystal. Realistic coupling constants based on the natural modes of the Photonic Crystal, i.e., Bloch waves and their associated dispersion relation, are derived. For simple model systems, well-known results such as decay times and emission spectra are reproduced. This approach enables direct incorporation of realistic band structure computations into studies of radiative emission from atoms and molecules within photonic crystals. We therefore provide a predictive and interpretative tool for experiments in both the microwave and optical regimes.Comment: Phys. Rev. E, accepte