A Path-Integral Method for Solution of the Wave Equation with Continuously Varying Coefficients

Jithin D. George; David I. Ketcheson; Randall J. LeVeque

doi:10.1137/19m1238630

A Path-Integral Method for Solution of the Wave Equation with Continuously Varying Coefficients

Jithin D. George, David I. Ketcheson, Randall J. LeVeque

Research output: Contribution to journal › Article › peer-review

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	https://doi.org/10.1137/19m1238630
State	Published - Dec 17 2019

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10.1137/19m1238630

https://repository.kaust.edu.sa/bitstream/10754/659992/1/Preprint%20file%201.pdf

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title = "A Path-Integral Method for Solution of the Wave Equation with Continuously Varying Coefficients",

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.",

author = "George, {Jithin D.} and Ketcheson, {David I.} and LeVeque, {Randall J.}",

note = "KAUST Repository Item: Exported on 2020-10-01 Acknowledgements: 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.",

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doi = "10.1137/19m1238630",

language = "English (US)",

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AU - George, Jithin D.

AU - Ketcheson, David I.

AU - LeVeque, Randall J.

N1 - KAUST Repository Item: Exported on 2020-10-01 Acknowledgements: 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.

PY - 2019/12/17

Y1 - 2019/12/17

N2 - 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.

AB - 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.

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JF - SIAM Journal on Applied Mathematics

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ER -

A Path-Integral Method for Solution of the Wave Equation with Continuously Varying Coefficients

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