N coupled non-local harmonic oscillators leading to 2N-th order initial value problem

TL;DR

This paper studies a system of N non-local harmonic oscillators, transforming their dynamics into a 2N-th order initial value problem and proposing a precise numerical solution method tested on various examples.

Contribution

It introduces a novel formulation of coupled non-local oscillators as a high-order initial value problem and develops an accurate non-polynomial spline numerical method for solving it.

Findings

The 2N-th order formulation accurately models the oscillator system.

The non-polynomial spline method effectively solves the high-order initial value problem.

Numerical tests demonstrate high precision in solutions for different driving forces.

Abstract

We consider a set of interwoven harmonic oscillators where the acceleration of a given oscillator is determined by the position of its nearest neighbor. We show that this problem of N non-local oscillators with periodic boundary conditions leads to a 2N-th order initial value problem. We discuss the numerical solution of this using a non-polynomial spline method. A very precise numerical method that minimizes the error can be developed, which we test for a few examples of driving forces.