I hereby declare that this thesis entitled "On Some Asymptotic Solutions of Fourth Order Critical Damped Non-linear Systems", submitted for the partial fulfillment of the degree of Master of Philosophy, was done by myself under the supervision of Dr. Fouzia Rahman, Professor, Department of Mathematics, Khulna University of Engineering and Technology (KUET), Khulna under whose guidance the work was accomplished. Ali Akbar, Assistant Professor, Department of Applied Mathematics, Rajshahi University, Rajshahi, for agreeing to accept me as a research student.
Mohammad Arif Hossain, Professor, Department of Mathematics, KUET for his help in revising thesis. I would like to express the greatest pleasure to acknowledge my respected and honorable teacher, Md. Now the method is used to obtain the solutions of oscillating, damped oscillating, overdamped, critically damped and more critically damped systems with second, third, fourth etc.
In this thesis, we have modified and extended the Krylov-Bogoliubov-Mitropolskii (KBM) method to investigate fourth-order critically damped and more critically damped nonlinear systems. To obtain the solutions of the variational equations, we have replaced the variables with their corresponding linear values.
Introduction
Among the methods, the perturbation method takes the lead, i.e. asymptotic expansions with respect to a small parameter. In the thesis, we will consider problems that can be described by dynamic systems of nonlinear autonomous differential equations of the fourth order with small nonlinearities using the KryIovBogoliubOV-MitrOP0lSki (KBM) method. The method was developed only for obtaining periodic solutions of nonlinear differential equations of the second order.
Now the method is used to obtain solutions of oscillatory, damped oscillatory, critically damped, more critically damped and non-oscillatory systems with the second, third, fourth, etc. In the KBM method, the solution begins by solving a linear equation (sometimes called generating a linear equation solution), only using the amplitude and phase of the linear differential equation solution, which are assumed to be time-dependent functions instead of constants. This method introduces an additional condition for the first derivative of the generated solution to determine the solution of the second-order equation.
These assumptions are certainly valid for second- and third-order equations. But for the fourth-order equations, the correction terms sometimes contain secular terms, although the solution is generated by the classical asymptotic method of KBM. For this reason, traditional solutions fail to explain the correct situation of the systems.
The Survey and the Proposal
The Survey
When # 0, but is small enough, then Krylov and Bogoliubov assumed that the solution of (1.2) is still given by (1.3) together with the derivative of the form. The interesting feature of this transformation lies in the fact that these first-order equations are now written in terms of amplitude a and phase p as dependent variables. The solution of this equation is based on recurrence relations and is given as a power series of the small parameter.
Shmsul [79, 85] also extended the KBM method for some nonlinear non-oscillatory systems when the eigenvalues of the unperturbed equation are real and non-positive. Using the KBM method, Bojadziev [14] investigated the solutions of nonlinear damped oscillatory systems with small time delay. Shamsul and Sattar [70] studied time-dependent third-order oscillatory systems with damping based on an extension of the asymptotic Krylov-Bogoliubov-Mitropolskii method.
Shamsul [75] has also presented a unified Krylov-Bogoliubov-Mitropolskii method, which is not the formal form of the original KBM method, for solving nonlinear systems of nth order. Shamsul [87] has also presented a modified and compact form of the uniform Krylov-Bogoliubov-Mitropolskii method for solving a nonlinear differential equation of nth order.
The Proposal
Introduction
Shamsul [77] also studied a critically damped third-order nonlinear system whose unequal eigenvalues are integrally multiple. Shamsul and Sattar [66] extended Bogoliubov's asymptotic method to a critically damped third-order nonlinear system. Shamsul and Sattar [67] have also presented a unified method for obtaining approximate solutions of damped and overdamped third-order oscillating nonlinear systems based on the KBM method.
2] again presented an asymptotic method for overdamped fourth order nonlinear systems, which is simple and easier than the method presented by Murty ci' al. but the results obtained by Ali Akbar ci al. method is the same as the results obtained by Murty el al. 4] also presented a simple technique for obtaining certain overdamped solutions of a nonlinear differential equation of nth order. In this chapter we have extended the KBM method for solving critically damped fourth-order nonlinear differential systems, which is different from the technique presented by Rokibul ci' al.
The solutions obtained by the presented method show a good agreement with the solutions obtained by the numerical method. 2.3) where x represents the fourth derivative of x with respect to t, dots are used for the first, second and third derivative with respect to t; e is a small parameter; k1 , k2 , k35 k4 are constants, and f is a given non-linear function. Now we differentiate the equation (2.5) four times with respect to t, substitute the value of x and the derivatives , 1, x, x in the original equation (2.3), use the relation presented in (2.6) and finally equate the coefficients 6 , we get.
Bogoliubov and Mitropolskii [13], Krylov and Bogoliubov [31], Sattar [63], Shamsul Shamsul and Sattar[66] imposed the condition that u1 cannot contain the fundamental terms (the solution presented in equation (2.4) is called the generating solution and the terms of it are called fundamental terms) of f(0). Now, equating the coefficients of tO and t' from both sides of equation (2.10), we get. These relations are important, since in these relations the coefficients in the solution of A1, A3 and A4 do not become large.
When the nonlinear function f is given, we can find a particular solution of the equation (2.11) for the unknown function u1 using the known operator method. Since a1, a25 a35 a4 are proportional to small parameters, they are therefore slowly varying functions of time i. The method can be implemented for higher-order nonlinear systems in the same way.
Example
Results and Discussion
Conclusion
Introduction
2] have found an asymptotic solution of fourth-order overdamped nonlinear systems, which is simple and easier than the method presented by Murty et al. 4] have also presented a simple technique to obtain certain overdamped solutions of an nth order nonlinear differential equation. In this chapter, a fourth order more critically damped nonlinear system is considered and an asymptotic solution is found by extending the KBM method.
The results obtained with the presented method are in good agreement with those obtained with the numerical method.
The method
To determine these unknown functions, it is common in the KBM method that the correction terms must exclude terms (so-called secular terms) that make them large. Theoretically, the solution can be obtained with the accuracy of any order of approximation. 1-lower, due to the rapidly increasing algebraic complexity for deriving the formulas, the solution is generally limited to a lower order, usually the first (Murty et al.
Now differentiating equation (3.3) four times with respect to t, substituting the value of x and the derivatives ±, 1, 1, x into the original equation (3.1), using the relation presented in (3.4) and finally equating coefficients of s, we get So, we need to set some restrictions to separate the equation (3.15) for the determination of the unknown functions A. For the sake of definiteness of the eigenvalues, it is possible to have the relation 23 > 24 between them.
This restriction is important, since under this restriction the coefficients of A1 , and A4 do not become large (the principle of the KBM method is that the coefficients of A1 , A2, A3 and A4 must be small) as well as the unknown functions A , and A4 can be determined very quickly. This restriction has another importance, that the solution is also useful in the case of more critically (when three eigenvalues are equal) damped systems. Equation (3.9) is a non-homogeneous linear differential equation, so it can be solved for u1 by the well-known operator method.
Example
Results and Discussion
Conclusion
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