Simulation of MEMRISTORS in the presence of a high-frequency forcing function

This reported work is concerned with the simulation of MEMRISTORS when they are subject to high-frequency forcing functions. A novel asymptotic-numeric simulation method is applied. For systems involving high-frequency signals or forcing functions, the superiority of the proposed method in terms of accuracy and efficiency when compared to standard simulation techniques shall be illustrated. Relevant dynamical properties in relation to the method shall also be considered

On systems of differential equations with extrinsic oscillation

We present a numerical scheme for an efficient discretization of nonlinear systems of differential equations subjected to highly oscillatory perturbations. This method is superior to standard ODE numerical solvers in the presence of high frequency forcing terms,and is based on asymptotic expansions of the solution in inverse powers of the oscillatory parameter w, featuring modulated Fourier series in the expansion coefficients. Analysis of numerical stability and numerical examples are included

Simulation of nonlinear systems subject to modulated chirp signals

Purpose The purpose of the paper is to apply a novel technique for the simulation of nonlinear systems subject to modulated chirp signals. Design/methodology/approach The simulation technique is first described and its salient features are presented. Two examples are given to confirm the merits of the method. Findings The results indicate that the method is appropriate for simulating nonlinear systems subject to modulated chirp signals. In particular, the efficiency and accuracy of the method is seen to improve as the chirp frequency increases. In addition, error bounds are given for the method. Originality/value Chirp signals are employed in several important applications such as representing biological signals and in spread spectrum communications. Analysis of systems involving such signals requires accurate, appropriate and effective simulation techniques

Explicit representations of biorthogonal polynomials

Given a parametrised weight function $\omega(x,\mu)$ such that the quotients of its consecutive moments are M\"obius maps, it is possible to express the underlying biorthogonal polynomials in a closed form \cite{IN2}. In the present paper we address ourselves to two related issues. Firstly, we demonstrate that, subject to additional assumptions, every such $\omega$ obeys (in $x$) a linear differential equation whose solution is a generalized hypergeometric function. Secondly, using a generalization of standard divided differences, we present a new explicit representation of the underlying orthogonal polynomials

Orthogonal Systems with a Skew-Symmetric Differentiation Matrix

Funder: University of ManchesterAbstract In this paper, we explore orthogonal systems in $$\mathrm {L}_2({\mathbb R})$$L2(R) which give rise to a real skew-symmetric, tridiagonal, irreducible differentiation matrix. Such systems are important since they are stable by design and, if necessary, preserve Euclidean energy for a variety of time-dependent partial differential equations. We prove that there is a one-to-one correspondence between such an orthonormal system $$\{\varphi _n\}_{n\in {\mathbb Z}_+}$${φn}n∈Z+ and a sequence of polynomials $$\{p_n\}_{n\in {\mathbb Z}_+}$${pn}n∈Z+ orthonormal with respect to a symmetric probability measure $$\mathrm{d}\mu (\xi ) = w(\xi ){\mathrm {d}}\xi$$dμ(ξ)=w(ξ)dξ. If $$\mathrm{d}\mu$$dμ is supported by the real line, this system is dense in $$\mathrm {L}_2({\mathbb R})$$L2(R); otherwise, it is dense in a Paley–Wiener space of band-limited functions. The path leading from $$\mathrm{d}\mu$$dμ to $$\{\varphi _n\}_{n\in {\mathbb Z}_+}$${φn}n∈Z+ is constructive, and we provide detailed algorithms to this end. We also prove that the only such orthogonal system consisting of a polynomial sequence multiplied by a weight function is the Hermite functions. The paper is accompanied by a number of examples illustrating our argument.</jats:p

A recurrence relation for generalised connection coefficients

We formulate and prove a general recurrence relation that applies to integrals involving orthogonal polynomials and similar functions. A special case are connection coefficients between two sets of orthonormal polynomials, another example is integrals of products of Legendre functions

Efficient computation of delay differential equations with highly oscillatory terms.

This paper is concerned with the asymptotic expansion and numerical solution of systems of linear delay differential equations with highly oscillatory forcing terms. The computation of such problems using standard numerical methods is exceedingly slow and inefficient, indeed standard software is practically useless for this purpose. We propose an alternative, consisting of an asymptotic expansion of the solution, where each term can be derived either by recursion or by solving a non-oscillatory problem. This leads to methods which, counter-intuitively to those developed according to standard numerical reasoning, exhibit improved performance with growing frequency of oscillation