This confirms that the thesis entitled "Analytical Solutions of Third and Fourth Order Weakly Nonlinear Ordinary Differential Systems with Damping and Slowly Varying Coefficients by the Unified KBM Method" was carried out by Md. Eabad Au, from Department of Mathematics, Khulna University of Engineering and Technology, Khulna, Bangladesh. Eabad Ali entitled Analytica1 Solutions of Third and Fourth Order Weakly Nonlinear Ordinary Differential Systems with Damping and Slowly Varying Coefficients by the Unified KBM Method" has been approved by the Examination Board for partial fulfillment of the requirements for the degree of Master of Philosophy in the Department of Mathematics, Khulna University of Engineering & 'I'echnology, Khulna.
Touhid Hossain, Professor, Department of Mathematics, Khulna University of Engineering & Technology, Khulna, for providing all kinds of cordial suggestions and authentic information. First, the method was developed only to obtain the periodic solutions of weakly nonlinear second-order differential systems. Nowadays this method is used to obtain the solutions of second, third and fourth order nonlinear differential systems for oscillating, damped oscillating, overdamped, critically damped and more critically damped cases by imposing some good constraints. This thesis provides an analytical approach. The technique is extended to find the second approximation solutions of weakly nonlinear third-order differential systems in the presence of strong linear damping and slowly varying coefficients, based on the KBM method.
The KBM method is also presented to find out the solutions of fourth-order weakly nonlinear differential systems in the presence of strong linear damping and slowly varying coefficients, including some constraints. Wali Ullah and Rehana Sultana Bipasha, Analytical approximate solution of fourth order weakly nonlinear differential systems based on unified KBM method with strong damping and slowly varying coefficients including some constraints.
Introduction
Introduction
The study of nonlinear problems is of crucial importance not only in various fields of physics, but also in engineering and applied mathematics, since most phenomena in our world are nonlinear and are described by nonlinear differential equations. IF nonlinear problems are difficult to solve and, in general, it is often more difficult to obtain an analytical approximate solution than a numerical one for a given nonlinear problem. The various methods are used to find approximate solutions to nonlinear problems, such as perturbation techniques [24, 45] and harmonic balance method [68], etc.
Arya and Bojadziev [26] studied a system of second order nonlinear hyperbolic differential equation with slowly varying coefficients. Arya and Bojadziev [27] also studied a time-dependent nonlinear oscillatory system with damping, slowly varying coefficients, and delay. Alam [45] investigated a unified KBM method for obtaining the first approximate solution of nih-order nonlinear systems with slowly varying coefficients.
Uddin and Sattar [55] have obtained an approximate solution of a weakly nonlinear fourth-order differential system with strong dampers and slowly varying coefficients by the unified KBM method. Roy and Alam [69] have studied the effect of the higher approximation of the Krylov-Bogol and ubov-Mitropolski solution and the matched asymptotic solution for a second-order nonlinear differential system with slowly varying coefficients and damping near a turning point . Alam and Sattar [70] have also presented an asymptotic method for obtaining the first approximate solution of a third order nonlinear differential system with different coefficients.
71] established a technique for solving nih-order nonlinear differential equation under certain special conditions, including the case of internal resonance. Al presented a short way to determine the KBM solution (first order) of nih, n = 2, 3, order linear differential systems. Otherwise, the solutions give incorrect results after a long time I >> 1, where the reduced frequency becomes small.
From our study, it is seen that most of the researchers have been carried out to obtain the first approximate solutions for both constants and varying coefficients. The complicated but no less important case of the second-order approximate solution of a third-order nonlinear differential systems in the presence of strong linear damping and slowly varying coefficients is almost untouched. The main aim of this chapter is to fill this gap based on the KBM method.
The method
Theoretically the solution can be obtained to any order of approximations, but due to the rapidly growing algebraic complexity for deriving the formula, the solution is generally limited to low order, usually first order. To determine these functions, it is assumed that the functions u1, u., do not contain the fundamental terms included in the series expansion equation (3.3) of order e°. Now differentiate equation (3.3) three times with respect to time i and use the relationship equation (3.4) and substitute the values of.
We have already assumed that u1 and u2 do not contain the fundamental terms and for this reason the solution will be free of secular terms, namely /cos/,I sini and /e'. Under these constraints, we are able to solve equation (3.5) and equation (3.6) by separating them into n + 1 individual equations for the unknown functions u1, u., A and B,. In general, the functions f(0), fW, u1 and u in Taylor series are expanded into the following forms.
Furthermore, it is bounded (by Krylov and Bogoliubov [3]) that the functions A1 and B1 are independent of the basic terms. But it is noted that the solution equation (3.3) is not a standard form for the KBM method. To reduce the solution of equation (3.3) to the standard form of the KBM method, we must use the following substitutions.
Example
By substituting the values of n, j and f in equation (3.5) and according to our constraints [equations, we obtain four equations for A1, A2, A3 and u1 whose solutions are respectively given by the following equations.
Results and Discussion
JPO =0
- The method
- Example
- Results and Discussion
- Series Series2
Ir The first approximate solution of a fourth-order weak nonlinear differential in the presence of strong linear damping and slowly varying coefficients based on the KBM method. The KBM method [3, 4] is suitable and one of the widely used and popular techniques to obtain approximate analytical solutions of weak nonlinear differential systems. Murty [25] has used Popov's method to obtain fuzzy solutions of weak nonlinear differential equations based on the unified theory of Murty et al.
26] have studied a system of second-order weakly nonlinear hyperbolic partial differential equations with slowly varying coefficients. 41 order weakly nonlinear differential system with strong damping and slowly varying coefficients by uniform KBM method, but they have not strictly followed their imposed restrictions. 70] have also presented an asymptotic method for the third-order weakly nonlinear system with varying coefficients.
Most of the researchers have studied second and third order weakly nonlinear differential systems for both constant and varying coefficients to obtain the first approximate solutions. Ir Let us assume a fourth-order weakly nonlinear ordinary differential equation with slowly varying coefficients of the following form. The coefficients vary slowly in the sense that their time derivatives are proportional to s [57].
Ir functions, it is assumed that the functions u1, u2,••• do not contain the fundamental terms that are included in the series expansion equation (4.3) according to order Co. In the presence of secular terms, the solutions will be non-uniform, so we must ignore these terms. Under these constraints, we are able to solve equation (4.5) by splitting it into five individual equations for the unknown functions u1 and A,.
Therefore, the solution obtained by the formula of equation (4.1) is transformed into the formal form by replacing the unusual variables with amplitudes and phases i. Let us assume the following fourth-order weakly nonlinear differential equation with slowly varying coefficients. Based on the KBM method, the first-order analytical approximate solution is obtained for fourth-order weakly nonlinear differential equations in the presence of strong linear damping and slowly varying coefficients.
It is also noticeable that the presented method is only valid for strong linear damping and deviates from the numerical solutions for small linear damping. it can be seen that the first approximate solutions agree well with the corresponding numerical solutions obtained by the fourth-order Runge-Kutta method in the presence of strong linear damping effects, and deviate from the numerical solutions in the presence of small linear damping effects (Fig.
CONCLUSIONS
Osiniskii, Z., Variation of a single degree of freedom system with nonlinear internal friction and relaxation, Proceedings of International Symposia on Nonlinear Oscillations, Vol.
