Asymptotic expansions for the solution of a linear PDE with a multifrequency highly oscillatory potential

TL;DR

This paper analytically derives asymptotic expansions for solutions of linear PDEs with highly oscillatory potentials using Modulated Fourier Expansion, providing error estimates and coefficient formulas without numerical solving.

Contribution

It introduces an analytical derivation of MFE for such PDEs, enabling error analysis and coefficient determination without numerical differential equation solving.

Findings

Derived a general error formula for MFE approximation.

Established a method to compute expansion coefficients analytically.

Validated theoretical results with numerical experiments.

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.