The vibrating inhomogeneous string: a topic for a course in Computational Physics

TL;DR

This paper compares spectral Chebyshev polynomial methods and sine-wave expansions for solving the oscillating inhomogeneous string problem, demonstrating high accuracy and innovative spectral techniques in computational physics.

Contribution

It introduces a spectral expansion approach using Chebyshev polynomials for solving the integral equation of an inhomogeneous string, enhancing computational accuracy.

Findings

Chebyshev spectral method achieves high precision in oscillating string problems

Comparison shows spectral method's accuracy surpasses traditional sine-wave expansion

Iterative method based on Hartree's approach improves solution precision

Abstract

This paper solves the integral equation which describes the oscillating inhomogeneous string, by using a spectral expansion method in terms of Chebyshev polynomials. The result is compared with the solution of the corresponding differential equation, obtained by expansion into a set of sine-wave functions, with emphasis on the accuracies of the two methods. These accuracies are determined by comparison with an iterative method which allows a precision of 1:10^11. The iterative method is based on a old method by Hartree, but contains innovative spectral expansion procedures.