A path-integral method for solution of the wave equation with continuously varying coefficients. (English) Zbl 1437.35440
Summary: 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.
MSC:
| 35L05 | Wave equation |
| 35L15 | Initial value problems for second-order hyperbolic equations |
| 35L45 | Initial value problems for first-order hyperbolic systems |
| 35C15 | Integral representations of solutions to PDEs |
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