Perturbation solutions of ﬁfth order oscillatory nonlinear systems

. 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.


Introduction
In oscillatory problems, the method of Krylov-Bogoliubov-Mitropolskii (KBM) [1,2] is particularly convenient, and is the widely used technique to obtain analytical approximate solution of nonlinear systems with a small non-linearity. In fact, the method developed by Krylov and Bogoliubov [2] for obtaining periodic solutions was amplified and justified by Bogoliubov and Mitropolskii [1], and later extended by Popov [3] and Meldelson [4], for damped nonlinear oscillations. Murty [5] has developed a unified KBM method for solving second-order nonlinear systems. Sattar [6] has studied a third-order over-damped nonlinear system. Bojadziev [7] studied the damped oscillations modeled by a threedimensional nonlinear system. 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]. Later, Akbar et al. [13] extended the method presented in [12] to the damped oscillatory systems. Islam et al. [14] investigated the solutions of fourth order more critically damped nonlinear systems. Akbar [15] examined a different type solution of fourth order more critically damped differential systems. Rahman et al. [16] studied fourth order nonlinear oscillatory systems when two of the eigenvalues of the corresponding linear systems are real and negative and the other two are complex numbers.
The aim of this article is to obtain the analytical approximate solutions of fifth-order weakly nonlinear oscillatory systems by extending the KBM method. The results obtained by the perturbation solution are compared with those obtained by the fourth-order Runge-Kutta method.

The method
Consider a fifth-order weakly-nonlinear oscillatory system, governed by the differential equation: where ε is a small parameter, f (x) is the nonlinear function, k 1 , k 2 , k 3 , k 4 , k 5 , are the characteristic parameters of the system defined by and where −λ 1 , −λ 2 , −λ 3 , −λ 4 and −λ 5 are the eigenvalues of the unperturbed equation of (1). Suppose four of the eigenvalues are imaginary and the other is real and negative (as the system is oscillatory). The oscillatory system is represented by these imaginary eigenvalues. Therefore, the unperturbed solution is: where a j,0 , j = 1, 2, 3, 4, 5, are arbitrary constants. When ε = 0, following Shamsul [17], we seek the solution of the nonlinear differential equation (1) of the form: where each a j (t), j = 1, 2, 3, 4, 5, satisfies the differential equatioṅ a j (t) = εA j (a 1 , a 2 , a 3 , a 4 , a 5 , t) + · · · .
Confining only to the first few terms 1, 2, 3, . . . , m in the series expansions of equation (3) and (4), we evaluate the functions u 1 and A j , j = 1, 2, 3, 4, 5, such that a j (t), appearing in equation (3) and (4), satisfy the given differential equation (1) with an accuracy of ε m+1 . Theoretically, the solution can be obtained up to the accuracy of any order of approximation. However, owing to the rapidly-growing algebraic complexity for the derivation of the formulae, the solution is, in general, confined to a lower order, usually the first [5]. In order to determine these functions, it is assumed that the functions u 1 do not contain the fundamental terms which are included in the series expansion (3) at order ε 0 . Differentiating equation (3) five times with respect to t, substituting x and the deriva- (1), utilizing the relations in equation (4) and equating the coefficients of ε, we obtain: where In general, the functional f (0) can be expanded in a Taylor series (see Murty and Deekshatulu [18] for details) as: According to KBM [1,2], Sattar [19] and Shamsul [17,20], u 1 does not contain the fundamental terms. Therefore equation (5) can be separated into six equations for unknown functions u 1 and A 1 , A 2 , A 3 , A 4 , A 5 (see [17] for details). Substituting the functional values of and equating the coefficients of e −λj t , j = 1, 2, 3, 4, 5, we obtain: and where excludes those terms for m 1 = m 2 ± 1, m 3 = m 4 ± 1, m 1 = m 2 , m 3 = m 4 . The particular solutions of the equations (6)- (11) give the functions A 1 , A 2 , A 3 , A 4 , A 5 and u 1 . Thus, the determination of the first approximate solution is completed.

Example
As an example of the method, we consider the Duffing equation type of fifth order differential equation: Here f (x) = x 3 .
Equations in (22) are nonlinear and they have no exact solutions. But since ε is a small quantity,ȧ,ḃ,ċ,φ 1 andφ 2 are slowly varying functions of time t. Therefore, we can solve (22) by assuming that a, b, c, ϕ 1 and ϕ 2 are constants in the right-hand sides of (22). This assumption was first made by Murty et al. [11,18] to solve similar type of nonlinear equations. Thus the solutions of the equations of (22) are Therefore, we obtain the solution of equation (12) in the following form x = a cos(ω 1 t + ϕ 1 ) + b cos(ω 2 t + ϕ 2 ) + ce −ξt + εu 1 .
Here equation (25) is the first order approximate solution of equation (12), where a, b, c, ϕ 1 and ϕ 2 are given by the equations of (24).

Results and discussion
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 (considered to be exact). With regard to such a comparison concerning the presented technique of this article, we refer the work of Murty et al. [11]. In the present article, 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. Beside this, we have computed the Pearson correlation between the perturbation results and the corresponding numerical results. From figures, we observed that the perturbation results from equation (25) show good coincidence with the numerical results. First of all, for ω 1 = 2.00, ω 2 = 1.00, ξ = 0.003 and ε = 0.1, x(t, ε) has been computed (25), in which a, b, c, ϕ 1 , ϕ 2 by the equations (24) with initial conditions a 0 = 0.005, b 0 = 0.005, c 0 = 0.0075, ϕ 1,0 = 2π/3 and ϕ 2,0 = π/2, i.e., In these cases, the perturbation results obtained by the solution (25) and the corresponding numerical results computed by a fourth order Runge-Kutta method with a small time increment ∆t = 0.05, are plotted in the Fig. 1. The correlation between these two results has also been calculated: which is 0.999999956. In these cases, the perturbation results obtained by the solution (25), and the corresponding numerical results computed by a fourth-order Runge-Kutta method with a small time increment∆t = 0.05, are plotted in the Fig. 2. The correlation between these two results have also been calculated: which is 0.99992479. In these cases, the perturbation results obtained by the solution (25), and the corresponding numerical results computed by a fourth-order Runge-Kutta method with a small time increment ∆t = 0.05 are plotted in Fig. 3. The correlation between these two results have also been calculated: which is 0.999879695.

Conclusion
A formula is obtained in this article for obtaining the analytical approximate solution of fifth order nonlinear differential systems, characterized by an oscillatory process, which is based on the KBM [1,2] method. The correlation between the results of the perturbation solution and the corresponding numerical solution obtain by a fourth-order Runge-Kutta method have been calculated, which shows that these two results are strongly-correlated. The results obtained for different sets of initial conditions, as well as different sets of eigenvalues, show a good coincidence with corresponding numerical results. www.mii.lt/NA