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：高振荡微分方程在数值处理中提出了重大挑战。调制傅里叶展开作为一种试探解形式，是常用的数值近似方法。本文针对具有多频高振荡势的线性偏微分方程，解析地推导了其调制傅里叶展开。方程的解在适当的Sobolev空间内被表达为一个收敛的Neumann级数。所提出的方法首先能够推导出用MFE近似解所关联误差的一般公式，其次能够确定该展开的系数——而无需通过数值求解微分方程组来寻找MFE的系数。数值实验验证了理论分析。