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.

翻译：高度振荡微分方程在数值处理中面临重大挑战。调制傅里叶展开（MFE）作为一种拟设，是常用的数值近似方法。本文针对含多频高度振荡势的线性偏微分方程，解析推导了调制傅里叶展开。该方程的解在适当Sobolev空间中表示为收敛的Neumann级数。所提方法首先能够推导出用MFE近似解时误差的通用公式，其次可确定该展开的系数——无需数值求解微分方程组来获取MFE系数。数值实验验证了理论分析结果。