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Dual approach to averaged values of functions: Advanced formulas

Averaged values play major roles in the study of dynamic processes. The use of those values allows transforming varying processes to some constant characteristics that are much easier to be investigated. In order to extend the use of averaged values one may apply the dual approach which suggests a consideration of two different aspects of a problem in question. This short communication proposes new advanced formulas for averaged values of functions based on the dual conception. | Vietnam Journal of Mechanics, VAST, Vol. 34, No. 4 (2012), pp. 321 – 325 Short Communication DUAL APPROACH TO AVERAGED VALUES OF FUNCTIONS: ADVANCED FORMULAS N. D. Anh Institute of Mechanics, VAST, Vietnam Abstract. Averaged values play major roles in the study of dynamic processes. The use of those values allows transforming varying processes to some constant characteristics that are much easier to be investigated. In order to extend the use of averaged values one may apply the dual approach which suggests a consideration of two different aspects of a problem in question. This short communication proposes new advanced formulas for averaged values of functions based on the dual conception. Key words: advanced formula, averaged value, global-local averaged, dual approach, Duffing oscillator. 1. INTRODUCTION Averaged values play major roles in the study of dynamic processes. The use of those values allows transforming varying processes to some constant characteristics that are much easier to be investigated. In order to extend the use of averaged values one may apply the dual approach recently proposed and developed in [1]-[2]. One of significant advantages of the dual conception is its consideration of two different aspects of a problem in question which allows investigation to be more appropriate. This advantage has been used in [3] to propose some new global and local averaged values of functions. The main objective of the short communication is to present a further extension of global-local averaged values of function. New advanced formulas for averaged values of functions are proposed based on the dual conception. 2. ADVANCED FORMULAS FOR AVERAGED VALUES OF DETERMINISTIC FUNCTIONS Let y(x) be an integrable deterministic function of x ∈ [a, b]. The following advanced formulas for averaged value of y(x) can be considered. 322 N. D. Anh Classical formula. Define I 0 (y(x), a, b) = y(x) (1) Classical formula for averaged value of y(x) is well-known as .