A path integral method for solution of the wave equation with continuouslyvarying coefficients
Abstract
A new method of solution is proposed for solution of the wave equation in one space dimension with continuouslyvarying coefficients. By considering all paths along which information arrives at a given point, the solution is expressed as an infinite series of integrals, where the integrand involves only the initial data and the PDE coefficients. Each term in the series represents the influence of paths with a fixed number of turning points. We prove that the series converges and provide bounds for the truncation error. The effectiveness of the approximation is illustrated with examples. We illustrate an interesting combinatorial connection between the traditional reflection and transmission coefficients for a sharp interface, and Green's coefficient for transmission through a smoothlyvarying region.
 Publication:

arXiv eprints
 Pub Date:
 January 2019
 arXiv:
 arXiv:1901.04158
 Bibcode:
 2019arXiv190104158G
 Keywords:

 Mathematics  Analysis of PDEs