Non-linear oscillations of a third-order differential equation

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

“…order) corresponding equations to Eq. (5) are constructed separately in [2][3][4][5][6][7][8][9][10][11]; but all these equations have certainly a general form, which is the main theme of this paper. Such type of general form can be found easily form Eq.…”

“…order non-linear systems, formula [1] has been presented. The method [1] covers all the previous works of [2][3][4][5][6][7][8][9][10][11]. However, formula [1] is not a classical form of KBM method and is presented in terms of some unusual variables.…”

“…Mulholland [7], Osiniskii [8], Bojadziev [9] and Sattar [10] investigated third-order non-linear differential systems. Shamsul and Sattar [11] extended Murty's [4] uniÿed method to solve a third-order non-linear di erential equation.…”

A Volterra integral formulation for determining the periodic solutions of some autonomous, nonlinear, third-order ordinary differential equations

“…., which can be substituted into Eqs. (10) and (11) to determine the periodic solution and the frequency of oscillation of Eq. (5), respectively.…”

“…This behavior has been analyzed in the past by means of harmonic balance methods [6,7], linearized harmonic balance procedures [8], asymptotic perturbation techniques which combine the harmonic balance procedure and the method of multiple time scales [9], the Linstedt-Poincaré procedure [10], the method of averaging of Krylov-Bogoliubov-Mitropolskii [10][11][12], parameter-perturbation Linstedt-Poincaré techniques [13,14] which employ an artificial or book-keeping parameter and expand both the solution and some constants that appear (or are introduced) in the differential equation in terms of this parameter, artificial parameter-Linstedt-Poincaré techniques [15,16] based on the introduction of a linear term proportional to the unknown frequency of oscillation and a new independent variable and the use of either the third-order equation or a system of a first-order and a second-order ordinary differential equations, etc. Parameter-perturbation methods are extensions of the homotopy perturbation technique which introduces an artificial parameter and expands both the solution and the unknown frequency of oscillation in terms of this parameter and has been applied to a variety of problems [17].…”