Abstract
A new method of solution is proposed for the solution of the wave equation in one space dimension with continuously varying 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 smoothly varying region.
| Original language | English (US) |
|---|---|
| Pages (from-to) | 2615-2638 |
| Number of pages | 24 |
| Journal | SIAM Journal on Applied Mathematics |
| Volume | 79 |
| Issue number | 6 |
| DOIs | |
| State | Published - Dec 17 2019 |
Bibliographical note
KAUST Repository Item: Exported on 2020-10-01Acknowledgements: We are grateful to Ernst Hairer for a comment that led us to the connection with zigzag numbers, and to Lajos L'oczi for reviewing an early
draft of this work. We also thank an anonymous referee for very helpful comments and suggestions.