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
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
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  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 . Popov  extended the method to damped oscillatory nonlinear systems. Due to physical importance of the damped oscillatory systems, Popov's results were rediscovered by Mendelson . Then, this method was extended by Murty and Deekshatulu  for over–damped nonlinear systems. Sattar  studied the second order critically-damped nonlinear systems by using of the KBM method. Murty  proposed a unified KBM method for second order nonlinear systems which covers the undamped, over-damped and damped oscillatory cases. Next, Osiniskii  first developed the KBM method to solve third-order nonlinear differential systems imposing some restrictions, which made the solution over-simplified. Mulholland  removed these restrictions and found desired solutions of third order nonlinear systems. Bojadziv  assessed solutions of nonlinear systems by converting it to a three-dimensional differential system. Sattar  examined solutions of three-dimensional over-damped nonlinear systems. Shamsul  propounded an asymptotic method for second order over-damped and critically damped nonlinear systems. Shamsul  then extended the method presented in  to third order over-damped nonlinear systems under some special conditions. Akbar et al.  generalized the method and showed that their method was easier than the method of Murty et al. . Later, Akbar et al.  extended the method presented in Akbar et al.  for fourth order damped oscillatory systems. Again, Akbar et al.  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.  for solving the fourth order more critically damped systems. Soon after Rokibul et al.  expounded an analytical approximate solution of fourth order more critically damped systems when the unequal eigenvalue is integral multiple of equal eigenvalues. Afterwards Hakim  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. , Rokibul et al.  and Hakim . 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
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
In which and are constants of integration.
However, when following Shamsul , the solution of the equation (1) is sought in the form
where and are slowly varying functions of time t and satisfy the following first order differential equations:
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 ).
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
Now we expand in the Taylor’s series of the form
Thus we can write
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.; Sattar ; Shamsul and Sattar ; Shamsul ; Shamsul  for details).
Solving the equation (8), we get the value
Substituting the value of from (12) into equation (9), we obtain
Now solving equation (13), we obtain
Now using the value of from (12) and from (14) into equation (10), we obtain
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.
As an example of the above procedure, consider a fourth order weakly nonlinear system governed by the ordinary differential equation
Thus for equation (16), the equations (8) to (11) respectively become
The solution of the equation (17) is
Substituting the value of from the equation (21) into the equation (18), we obtain
To separate the equation (19) for determining unknown functions and , we equate the coefficient of and , we obtain
The particular solutions of (23) and (24) respectively become
The solution of the equation (20) for is
Substituting the values of and from the equations (25), (22), (21) and (26) into equation (19) and integrating, we obtain
Therefore, we obtain the first approximate solution of the equation (16) as
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:
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:
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:
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.
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.