Cross-Correlation and Averaging: An Equivalence Based on the Classical Probability Density

The averaging method is a widely used technique in the field of nonlinear differential equations for effectively reducing systems with "fast" oscillations overlaying "slow" drift. The method involves calculating an integral, which can be straightforward in some cases, but can also require simplifications such as series expansions. We propose an alternative approach that relies on the classical probability density (CPD) of the "fast" variable. Further, we demonstrate the equivalence between the averaging integral and the cross-correlation product of the CPD and the target function. This equivalence simplifies handling many problems, particularly those involving piecewise-defined target functions. We propose an effective numerical method to calculate the averaged function, exploiting the well-known mathematical properties of cross-correlation products.