Partial differential equations with highly oscillatory input terms are hardly ever solvable analytically and their numerical treatment is difficult. Modulated Fourier expansion used as an {\it ansatz} is a well known and extensively investigated tool in asymptotic numerical approach for this kind of problems. Although the efficiency of this approach has been recognised, its error analysis has not been investigated rigorously for general forms of linear PDEs. In this paper, we start such kind of investigations for a general form of linear PDEs with an input term characterised by a single high frequency. More precisely we derive an analytical form of such an expansion and provide a formula for the error of its truncation. Theoretical investigations are illustrated by computational simulations.

翻译：具有高度振荡输入项的偏微分方程几乎无法解析求解，且其数值处理也较为困难。调制傅里叶展开作为一种假设方法，是此类问题渐近数值方法中一种广为人知且经过深入研究的工具。尽管该方法的高效性已获认可，但其对于一般形式的线性偏微分方程的误差分析尚未得到严格研究。本文针对具有单高频特征输入项的一般线性偏微分方程开展此类研究：具体而言，我们推导了该类展开的解析形式，并给出了其截断误差的表达式。理论分析通过计算模拟进行了验证。