Abstract
Oscillatory systems play an important role in the nature. Many engineering problems and physical systems of fifth degrees of freedom are oscillatory and their governing equations are fifth order nonlinear differential equations. To investigate the solution of fifth order weakly nonlinear oscillatory systems, in this article the Krylov–Bogoliubov–Mitropolskii (KBM) method has been extended and desired solution is found. An example is solved to illustrate the method. The results obtain by the extended KBM method show good agreement with those obtained by numerical method.
Highlights
In oscillatory problems, the method of Krylov–Bogoliubov–Mitropolskii (KBM) [1, 2] is convenient, and is the widely used technique to obtain analytical approximate solution of nonlinear systems with a small non-linearity
In order to test the accuracy of an analytical approximate solution obtained by a certain perturbation method, we compare the approximate solution to the numerical solution
For different sets of initial conditions as well as different sets of eigenvalues we have compared the results obtained by perturbation solution (25) to those obtained by the fourth order Runge–Kutta method
Summary
The method of Krylov–Bogoliubov–Mitropolskii (KBM) [1, 2] is convenient, and is the widely used technique to obtain analytical approximate solution of nonlinear systems with a small non-linearity. Murty [5] has developed a unified KBM method for solving second-order nonlinear systems. Shamsul and Sattar [8] developed a method for thirdorder critically-damped nonlinear equations. Islam and Akbar [9] investigated a new solution of third order more critically damped nonlinear systems. Shamsul and Sattar [10] presented a unified KBM method for solving third-order nonlinear systems. Murty et al [11] extended the method to the fourth-order over-damped nonlinear systems in a way which we think too much laborious and cumbersome. Akbar et al [12] has presented a method for solving the fourth-order over-damped nonlinear systems which is easier than that of Murty et al [11]. Islam et al [14] investigated the solutions of c Vilnius University, 2011
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