1. Introduction
In order to obtain solutions of nonlinear systems, the asymptotic method of Krylov–Bogoliubov–Mitropolskii (KBM) [1, 2] is regarded as convenient and one of the widely-used tools. For the systems with periodic solutions with small nonlinearities, the method was first extended by Krylov and Bogoliubov [2]. Later, it was amplified and justified by Bogoliubov and Mitropolskii [1]. For nonlinear systems affected by strong linear damping forces, Popov [3] extended the method. However, due to its physical significance, Popov’s method was rediscovered by Mendelson [4]. Then, this method was extended by Murty and Deekshatulu [5] for over–damped nonlinear systems. Sattar [6] studied the second order critically-damped nonlinear systems by using of the KBM method. Murty [7] proposed a unified KBM method for second order nonlinear systems which covers the undamped, over-damped and damped oscillatory cases. After that, Osiniskii [8] first developed the KBM method to solve third-order nonlinear differential systems imposing some restrictions, which makes the solution over-simplified. Mulholland [9] removed these restrictions and found desired solutions of third order nonlinear systems. Sattar [10] examined solutions of three-dimensional over-damped nonlinear systems. Shamsul [11] propounded an asymptotic method for second order over-damped and critically damped nonlinear systems. Later, Shamsul [12] extended the method presented in [11] to third-order over-damped nonlinear systems under some special conditions. Akbar et al. [13] generalized the method and showed that their method is easier than the method of Murty et al. [14]. Recently Rahaman and Rahman [15] have suggested analytical approximate solutions of fifth order more critically damped systems in the case of smaller triply repeated roots. Moreover, Rahaman and Kawser [16] have also proposed asymptotic solutions of fifth order critically damped nonlinear systems with pair wise equal eigenvalues and another is distinct. Further, Rahaman et al. [17] suggested an asymptotic method of Krylov-Bogoliubov-Mitropolskii for fifth order critically damped nonlinear systems. Again, Rahaman and Kawser [18] expounded analytical approximate solutions of fifth order more critically damped nonlinear systems
In this paper, we seeks to investigate an asymptotic solution of fifth order more critically damped nonlinear system, based upon the KBM method. In this study, we suggest that the perturbation results obtained by the presented technique reveal good coincidence with numerical results obtained by Mathematica 9.0.
2. The Method
Consider a fifth order non-linear differential system of the form
 | (1) |
where 
and 
stand for the fifth and fourth derivatives respectively, and over dots are used for the first, second and third derivatives of x with respect to t; 
are constants, 
is a sufficiently small parameter and 
is the given nonlinear function. As the unperturbed equation (1) is of fifth order, so it has five real negative eigenvalues, where four eigenvalues are equal and the other one is distinct. Suppose the eigenvalues are 
and 
When 
the equation (1) becomes linear and the solution of the corresponding linear equation is
 | (2) |
where 
and
are constants of integration.
When 
Murty [7] and Shamsul [19], we look for a solution of equation (1) in an asymptotic expansion of the form
 | (3) |
where 
and 
are the functions of t and they satisfy the first order differential equations
 | (4) |
Now differentiating (3) five times with respect to t, substituting the value of x and the derivatives 
in the original equation (1) utilizing the relations presented in (4) and, finally, extracting the coefficients of ε, we obtain
 | (5) |
where 
and 
.
We have expanded the function 
in the Taylor’s series (Sattar [20], Shamsul [19]) about the origin in power of t. Therefore, we obtain
 | (6) |
Thus, using (6), the equation (5) becomes
 | (7) |
Following the KBM method, Murty and Deekshatulu [21], Sattar [20], Shamsul [22], Shamsul and Sattar [23] imposed the condition that 
does not contain the fundamental terms of 
Therefore, equation (7) can be separated for unknown functions 
and 
in the following way:
 | (8) |
 | (9) |
Now equating the coefficients of the various power of 
from equation (8), we obtain
 | (10) |
 | (11) |
 | (12) |
 | (13) |
Here, we have four equations (10), (11), (12) and (13) for determining the unknown functions A1, B1, C1, D1 and H1. Thus, to obtain the unknown functions A1, B1, C1, D1 and H1, we need to impose some conditions (Shamsul [26, 22, 24, 25]) between the eigenvalues. Different authors have imposed different conditions according to the behavior of the systems, such as Shamsul [25] imposed the condition
 | |
In this study, we have investigated solutions for both the cases 
and 
. Therefore, we obtain the value of D1 from equation (13), and substituting the value of D1 in equation (12), we get the value of C1, and using these values of C1 and D1 in equation (11), we find the value of B1. Now we will be able to separate the equation (10) for unknown functions A1 and H1 for both the conditions 
and 
; and solving them for A1 and H1. Since 
and 
are proportional to the small parameter, so they are slowly varying functions of time t, and for first approximate solution, we may consider them as constants which are presented in the right side. This assumption was first made by Murty and Deekshatulu [21]. Thus, the solutions of the equation (4) become
 | (14) |
Equation (9) is a non-homogeneous linear ordinary differential equation; therefore, it can be solved by the well-known operator method. Substituting the values of 
and 
in the equations (3), we get the complete solution of (1). Therefore, the determination of the first approximate solution is complete.
3. Example
As an example of the above technique, we have considered the Duffing type equation of fifth order nonlinear differential system:
 | (15) |
Comparing equation (13) and equation (1), we obtain
Therefore,
 | (16) |
For equation (15), the equation (9) to equation (13) respectively become
 | (17) |
 | (18) |
 | (19) |
 | (20) |
 | (21) |
The solution of the equation (21), therefore, is
 | (22) |
Putting the value of 
from equation (22) into equation (20), we obtain
 | (23) |
Therefore, the solution of the equation (23) is
 | (24) |
Putting the value of 
and 
from equation (24) and (22) into equation (19), we obtain
 | (25) |
Thus, the solution of the equation (25) is
 | (26) |
Now applying the conditions 
in equation (18), we obtain the following equations for unknown functions 
and 
:
 | (27) |
And
 | (28) |
Solution of the equations (27) and (28) are
 | (29) |
 | (30) |
Now applying the condition 
in equation (18), we obtain the following equations for unknown functions 
and 
:
 | (31) |
 | (32) |
Thus, the solution of the equations (31) and (32) are
 | (33) |
 | (34) |
Finally, the solution of the equation (17) for 
is
 | (35) |
where
Substituting the values of 
and 
from equations (29), (33), (26), (24), (22), (32) and (34) into equation (4), we obtain, when 
then 
becomes
And when 
then 
becomes
 | (36) |
Again, when 
then 
becomes
Further, when 
then 
becomes
Here, all of the equations (36) have no exact solutions, but since 
and 
are proportional to the small parameter
, they are slowly varying functions of time t. Therefore, it is possible to replace 
and 
by their respective values obtained in linear case (i.e., the values of 
and 
obtained when
) in the right hand side of equations (36). This type of replacement was first introduced by Murty and Deekshatulu [5] and Mutry et.al. [14] in order to solve similar types of nonlinear equations.
Therefore, the solutions to the equations (36) are, when 
then 
becomes
Again, when 
then 
becomes
 | (37) |
Moreover, when 
then 
becomes
when 
then 
becomes
Hence, we obtain the first approximate solution of the equation (13) as:
 | (38) |
where 
and 
are given by the equations (37) and 
is given by (35).
4. Results and Discussion
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In order to bring more efficiency to our results, the numerical results obtained by Mathematica 9.0 are compared with the perturbation results obtained by the same program for the different sets of initial conditions. Here, we have computed 
from (38) by considering different values of 
and 
in which 
and 
are obtained from (37) and 
is calculated from equation (35) together with four sets of initial conditions. The corresponding numerical solutions have been computed by the Mathematica 9.0 program for various values of t and all the perturbation solutions have been developed by a code in Mathematica 9.0 program. All the results presented by the Figure 1 and Figure 2 for the case 
and Figure 3 and Figure 4 for the case 
show the perturbation results, which are plotted by a blue line and the corresponding numerical results, which are plotted by the red line respectively.
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5. Conclusion
In conclusion, it can be said that, in this article, we have successfully modified the KBM method and applied it to the fifth order more critically damped nonlinear systems. In relation to the fifth order more critically damped nonlinear systems, the solutions are obtained in such circumstances where the four eigenvalues are equal. Ordinarily, it is seen that, in the KBM method, much error occurs in the case of rapid changes of x with respect to time t. However, it has been observed in this study that, with respect to the different sets of initial conditions of the modified KBM method, the results obtained for both the cases (when 
and 
) correspond accurately to the numerical solutions obtained by Mathematica 9.0. We, therefore, come to the conclusion that the modified KBM method gives highly accurate results, which can be applied for different kinds of nonlinear differential systems.
Acknowledgement
The authors are grateful to Mr. Md. Mizanur Rahman, Associate Professor, Department of Mathematics, Islamic University, Bangladesh, for his invaluable comments on the earlier draft of this article. Special thanks is due to Mr. Md. Imamunur Rahman who has assisted the authors in editing this paper.
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, Md. Shajib Ali1, Md. Nurul Islam1
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