Asymptotic expansions for the solution of a linear PDE with a multifrequency highly oscillatory potential
Abstract
Highly oscillatory differential equations present significant challenges in numerical treatments. The Modulated Fourier Expansion (MFE), used as an ansatz, is a commonly employed tool as a numerical approximation method. In this article, the Modulated Fourier Expansion is analytically derived for a linear partial differential equation with a multifrequency highly oscillatory potential. The solution of the equation is expressed as a convergent Neumann series within the appropriate Sobolev space. The proposed approach enables, firstly, to derive a general formula for the error associated with the approximation of the solution by MFE, and secondly, to determine the coefficients for this expansion -- without the need to solve numerically the system of differential equations to find the coefficients of MFE. Numerical experiments illustrate the theoretical investigations.
Keywords
Cite
@article{arxiv.2310.14650,
title = {Asymptotic expansions for the solution of a linear PDE with a multifrequency highly oscillatory potential},
author = {Rafał Perczyński and Antoni Augustynowicz},
journal= {arXiv preprint arXiv:2310.14650},
year = {2024}
}