TY - JOUR
T1 - An alternative approach to averaging in nonlinear systems using classical probability density
AU - Genda, Attila
AU - Fidlin, Alexander
AU - Gendelman, Oleg
N1 - Publisher Copyright:
© 2024 The Authors. ZAMM - Journal of Applied Mathematics and Mechanics published by Wiley-VCH GmbH.
PY - 2024
Y1 - 2024
N2 - 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. Furthermore, 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, taking advantage of the well-known mathematical properties of cross-correlation products.
AB - 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. Furthermore, 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, taking advantage of the well-known mathematical properties of cross-correlation products.
UR - http://www.scopus.com/inward/record.url?scp=85189071828&partnerID=8YFLogxK
U2 - 10.1002/zamm.202300432
DO - 10.1002/zamm.202300432
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AN - SCOPUS:85189071828
SN - 0044-2267
JO - ZAMM Zeitschrift fur Angewandte Mathematik und Mechanik
JF - ZAMM Zeitschrift fur Angewandte Mathematik und Mechanik
ER -