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

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 in the appropriate Sobolev space. Our 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 numerically solve the system of differential equations to find the coefficients of MFE. Numerical experiments illustrate the theoretical investigations.