A modified and compact form of Krylov–Bogoliubov–Mitropolskii unified method for solving an nth order non-linear differential equation

Abstract

A modified and compact form of Krylov–Bogoliubov–Mitropolskii (KBM) (Introduction to Nonlinear Mechanics, Princeton University Press, Princeton, NJ, 1947; Asymptotic Methods in the Theory of Nonlinear Oscillations, Gordan and Breach, New York, 1961) unified method (J. Franklin Inst. 339 (2002) 239) is determined for obtaining the transient response of an nth order ( n⩾2) differential equation with small non-linearities. The formula presented in (J. Franklin Inst. 339 (2002) 239) is a changed form of KBM method. For n=2,3,4, some previous formulas were found separately by several authors in terms of amplitude and phase variables; but the formula of Shamsul Alam, J. Franklin Inst. 339 (2002) 239) is derived in terms of some unusual variables instead of amplitudes and phases. The formula of Shamsul Alam, J. Franklin Inst. 339 (2002) 239) is a general form and used arbitrarily to obtain asymptotic solution for n=2,3,4,…. However, a solution obtained by formula Shamsul Alam, J. Franklin Inst. 339 (2002) 239) is transformed to a formal form replacing the unusual variables by amplitude and phase variables. In the present paper, the formula of Shamsul Alam, J. Franklin Inst. 339 (2002) 239) is itself transformed to a usual form (i.e. in terms of amplitude and phase variables). The later form of the formula is similar to most of the previous formulas found by several authors when n=2,3,4. This form of the formula is also generalized and it is easier than those obtained in all previous papers (extension) and identical to that initiated by original contributors (Introduction to Nonlinear Mechanics, Princeton University Press, Princeton, NJ, 1947; Asymptotic Methods in the Theory of Nonlinear Oscillations, Gordan and Breach, New York, 1961).