American Journal of Applied Mathematics
Volume 3, Issue 6, December 2015, Pages: 265-270

Krylov-Bogoliubov-Mitropolskii Method for Fourth Order More Critically Damped Nonlinear Systems

Md. Mahafujur Rahaman

Department of Computer Science & Engineering, Z.H. Sikder University of Science & Technology, Shariatpur, Bangladesh

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To cite this article:

Md. Mahafujur Rahaman. Krylov-Bogoliubov-Mitropolskii Method for Fourth Order More Critically Damped Nonlinear Systems.American Journal of Applied Mathematics.Vol.3, No. 6, 2015, pp. 265-270. doi: 10.11648/j.ajam.20150306.15


Abstract: With a view to obtaining the transient response of the system where triply eigenvalues are equal and another is distinct, we have considered a fourth order more critically damped nonlinear systems, and enquired into analytical approximate solution in this paper. We have also suggested that the results obtained by the proposed method correspond to the numerical solutions obtained by the fourth order Runge-Kutta method satisfactorily.

Keywords: KBM, Eigenvalues, More Critically Damped System, Nonlinearity, Runge-Kutta Method


1. Introduction

The KBM [1,2] method is a broadly exercised technique to study nonlinear oscillatory and non-oscillatory differential systems with small nonlinearities. Initially, the method was developed by Krylov and Bogoliubov [2] for finding the periodic solutions of second order nonlinear differential systems with small nonlinearities. Later on, the method was improved and justified mathematically by Bogoliubov and Mitroposkii [1]. Popov [3] extended the method to damped oscillatory nonlinear systems. Due to physical importance of the damped oscillatory systems, Popov's results were 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. Next, Osiniskii [8] first developed the KBM method to solve third-order nonlinear differential systems imposing some restrictions, which made the solution over-simplified. Mulholland [9] removed these restrictions and found desired solutions of third order nonlinear systems. Bojadziv [10] assessed solutions of nonlinear systems by converting it to a three-dimensional differential system. Sattar [11] examined solutions of three-dimensional over-damped nonlinear systems. Shamsul [12] propounded an asymptotic method for second order over-damped and critically damped nonlinear systems. Shamsul [13] then extended the method presented in [12] to third order over-damped nonlinear systems under some special conditions. Akbar et al. [14] generalized the method and showed that their method was easier than the method of Murty et al. [15]. Later, Akbar et al. [16] extended the method presented in Akbar et al. [14] for fourth order damped oscillatory systems. Again, Akbar et al. [17] investigated a technique for obtaining over-damped solutions of n-th order nonlinear differential equations under some special conditions including the case of internal resonance. A method has been established by Akbar et al. [18] for solving the fourth order more critically damped systems. Soon after Rokibul et al. [19] expounded an analytical approximate solution of fourth order more critically damped systems when the unequal eigenvalue is integral multiple of equal eigenvalues. Afterwards Hakim [20] presented a method to enquire solutions of fourth order more critically damped nonlinear systems.

In this article, we have investigated solutions of fourth order more critically damped nonlinear systems i.e. the three eigenvalues are equal and the other one is distinct, by developing a method which is different from the method of Akbar et al. [18], Rokibul et al. [19] and Hakim [20]. Finally, in this paper, we have suggested that the acquired perturbation results show good coincidence with the numerical results for different sets of initial conditions as well as different sets of eigenvalues.

2. The Method

Let us consider a weakly nonlinear fourth order ordinary differential system

(1)

In which  indicates the fourth derivative of x, over dots indicate the first, second and third derivatives with respect to t;  are characteristic parameters,  is a small parameter and is the nonlinear function.

When the equation becomes linear and the solution of the linear equation of (1) is

(2)

In which  and are constants of integration.

However, when  following Shamsul [21], the solution of the equation (1) is sought in the form

(3)

where  and  are slowly varying functions of time t and satisfy the following first order differential equations:

(4)

In this calculation, we have merely considered first few terms in the series expansion of (3) and (4) and we have calculated the functions  and  for  such that  and appearing in (3) and (4) satisfy the given differential equation (1).

With a view to ascertaining these unknown functions, the KBM method usually suggests that the correction terms,  for  should exclude terms (sometimes referred to as secular terms) that enlarge them. The solution may be, in theory, accurate for any order of approximation. But due to the rapid rise in algebraic intricacy for the derivation of the formulae, the solution is generally limited to a lower order, especially the first order (Murty [7]).

Now differentiating the equation (3) four times with respect to t, substituting the value of x and the derivatives  in the equation (1), using the relations presented in (4) and finally equating the coefficients of , we obtain

(5)

where

and  

Now we expand  in the Taylor’s series of the form

(6)

Thus we can write

(7)

We impose the condition that  cannot contain the fundamental terms of  therefore equation (7) can be separated for unknowns functions  and in the following way (see also Murty et al.[15]; Sattar [6]; Shamsul and Sattar [8]; Shamsul [13]; Shamsul [22] for details).

(8)

(9)

(10)

(11)

Solving the equation (8), we get the value

(12)

Substituting the value of  from (12) into equation (9), we obtain

(13)

Now solving equation (13), we obtain

(14)

Now using the value of  from (12) and from (14) into equation (10), we obtain

(15)

Now we have only one equation (15) for obtaining the unknown functions  and, for finding the value of  and equating the coefficient of and  from the equation (15).

Thus, the determination of the first order improved solution of the equation (1) is completed. It should be noted that the solution for higher order systems can also carried out in the same manner as has been carried out in this study.

3. Example

As an example of the above procedure, consider a fourth order weakly nonlinear system governed by the ordinary differential equation

(16)

Here

Therefore,

Thus for equation (16), the equations (8) to (11) respectively become

(17)

(18)

(19)

(20)

The solution of the equation (17) is

(21)

Consider and

Substituting the value of  from the equation (21) into the equation (18), we obtain

(22)

To separate the equation (19) for determining unknown functions  and , we equate the coefficient of and ,  we obtain  

(23)

(24)

The particular solutions of (23) and (24) respectively become

(25)

(26)

The solution of the equation (20) for  is

(27)

where  

 

 

 

Substituting the values of  and  from the equations (25), (22), (21) and (26) into equation (19) and integrating, we obtain

(28)

(29)

(30)

(31)

Therefore, we obtain the first approximate solution of the equation (16) as

(32)

where  and  are given by the equations (28) to (31) and  is given by (27).

4. Results and Discussion

To make sure the efficiency of our results, we have compared our results to the numerical results obtained by fourth order Runge-Kutta method for the different set of initial conditions.

First of all,  has been computed from (32) by considering values of  in which  and  are calculated from equations (28) to (31) with the initial conditions   and  when  Fig. 1 represents the perturbation results which are plotted by the continuous line and the corresponding numerical solution has been computed by a fourth-order Runge-Kutta method, which are plotted by a dotted line as follows:

Fig. 1. Perturbation results are plotted by continuous line and numerical results are plotted by dotted line.

Secondly,  has been computed from (32) by considering values of in which  and  are calculated from equations (28) to (31) with the initial conditions   and  when Fig. 2 represents the perturbation results which are plotted by the continuous line and the corresponding numerical solution has been computed by a fourth-order Runge-Kutta method, which are plotted by a dotted line as follows:

Fig. 2. Perturbation results are plotted by continuous line and numerical results are plotted by dotted line.

Finally,  has been computed from (32) by considering values of  in which  and  are calculated from equations (28) to (31) with the initial conditions   and  when Fig. 3 represents the perturbation results which are plotted by the continuous line and the corresponding numerical solution has been computed by a fourth-order Runge-Kutta method, which are plotted by a dotted line as follows:

Fig. 3. Perturbation results are plotted by continuous line and numerical results are plotted by dotted line.

5. Conclusion

Based upon the KBM method of fourth order more critically damped nonlinear systems, we have been able to obtain an analytical approximate solution in this study. Moreover, we have shown in this study that the results obtained by the proposed method correspond satisfactorily to the numerical results obtained by the fourth order Runge-Kutta method. It is, therefore, concluded that the modified KBM method provides 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 early draft of this paper. The authors are also thankful to Mr. Md. Imamunur Rahman for his assistance in editing this paper.


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