Improved estimates for nonoscillatory phase functions

James  Bremer; Vladimir  Rokhlin

doi:10.3934/dcds.2016.36.4101

Article Contents

2016, Volume 36, Issue 8: 4101-4131. Doi: 10.3934/dcds.2016.36.4101

This issue Previous Article Efficient representation and accurate evaluation of oscillatory integrals and functions Next Article Low-dimensional Galerkin approximations of nonlinear delay differential equations

Improved estimates for nonoscillatory phase functions

James Bremer^1, and
Vladimir Rokhlin^2,

1.
Department of Mathematics, University of California, Davis, Davis, CA 95616
2.
Department of Computer Science, Yale University, New Haven, CT 06511

Received: May 2015

Published: May 2017

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Abstract

Recently, it was observed that solutions of a large class of highly oscillatory second order linear ordinary differential equations can be approximated using nonoscillatory phase functions. In particular, under mild assumptions on the coefficients and wavenumber $\lambda$ of the equation, there exists a function whose Fourier transform decays as $\exp(-\mu |\xi|)$ and which represents solutions of the differential equation with accuracy on the order of $\lambda^{-1} \exp(-\mu \lambda)$. In this article, we establish an improved existence theorem for nonoscillatory phase functions. Among other things, we show that solutions of second order linear ordinary differential equations can be represented with accuracy on the order of $\lambda^{-1} \exp(-\mu \lambda)$ using functions in the space of rapidly decaying Schwartz functions whose Fourier transforms are both exponentially decaying and compactly supported. These new observations are used in the analysis of a method for the numerical solution of second order ordinary differential equations whose running time is independent of the parameter $\lambda$. This algorithm will be reported at a later date.

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Mathematics Subject Classification: 34E99, 33E02.

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