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

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 (dissipation equation) under impulses in space time are solved as examples. complex infinite-domain function is derived. A approach proposed finite-domain from by reflection transmission analysis Fourier transform domain. It found that solution obtained this converges much better at short response times compared with traditional modal analysis. Besides, applying geometric summation formula for matrix series, expansion requiring no calculation each mode's inner product derived, which analytically proves wave-mode duality simplifies calculation. semi-infinite-domain cases coupled-domain also derived newly developed show its validity simplicity. non-propagating waves possess wave speed, can be treated propagating