Wave analysis in the complex Fourier transform domain: A new method to obtain the Green's functions of dispersive linear partial differential equations

TL;DR

This paper introduces a novel analytical method using complex Fourier transform analysis to derive Green's functions for dispersive linear PDEs, improving solution accuracy and simplifying calculations.

Contribution

A new approach to obtain Green's functions in the complex Fourier domain, including finite-domain solutions and wave-mode duality, with improved convergence and computational simplicity.

Findings

Derived Green's function for Euler-Bernoulli beam

Improved short-time response accuracy

Simplified modal expansion without inner product calculations

Abstract

This paper provides a new analytical method to obtain Green's functions of linear dispersive partial differential equations. The Euler-Bernoulli beam equation and the one-dimensional heat conduction equation (dissipation equation) under impulses in space and time are solved as examples. The complex infinite-domain Green's function of the Euler-Bernoulli beam is derived. A new approach is proposed to obtain the finite-domain Green's function from the infinite-domain Green's function by the reflection and transmission analysis in the complex Fourier transform domain. It is found that the solution obtained by this approach converges much better at short response times compared with that obtained by the traditional modal analysis. Besides, by applying the geometric summation formula for matrix series, a new modal expansion solution requiring no calculation of each mode's inner product is derived, which analytically proves the wave-mode duality and simplifies the calculation. The semi-infinite-domain cases and the coupled-domain cases are also derived by the newly developed method to show its validity and simplicity. It is found that the non-propagating waves also possess wave speed, and heat conduction can also be treated as propagating waves