Approximate Solutions of Second and Fourth Order Ordinary Differential Systems with Strong Generalized Nonlinearity

In this thesis, approximate analytical techniques will be presented by combining the He's homotopy perturbation technique and the extended form of the KBM method for solving second- and fourth-order nonlinear ordinary differential systems with strong generalized nonlinearity. Several authors [5-73] have investigated and developed many significant results regarding the solutions of the weakly nonlinear differential systems. In this thesis, He's hornotopy perturbation method has been extended to obtain analytical approximate solutions of highly generalized second- and fourth-order nonlinear differential systems with small damping, based on the extended form of the KBM method.

Also, a coupled analytical approximation technique has been extended to obtain the solutions of certain types of highly generalized fourth-order nonlinear oscillatory differential systems with small damping, based on the He's homotopy perturbation and the extended form of the famous KBM methods in Chapter IV. Duffing [14] has investigated many significant results for the periodic solutions of the following damped nonlinear differential. Kruskal [15] extended the KB [2] method to solve the completely nonlinear differential equation of the following form.

Alam and Sattar [51] studied time-dependent third-order oscillatory systems with damping based on the extended form of the KBM method.

Table 4.1 Average percentage error between numerical and approximate 39 solutions.

Introduction

An approximate analytical technique for solving second-order strongly generalized nonlinear differential systems with small damping. Uddin and Sattar [79] developed an approximate technique to solve strongly nonlinear biological systems with small damping effects. Uddin and Sattar [80] also presented an approximate technique for Duffing equation with small damping and slowly varying coefficients.

Uddin [81] applied 1-le's homotopy perturbation method to Duffing equation with small damping and high order strong nonlinearities. 82] developed an approximate analytical technique to solve a certain type of fourth-order strongly nonlinear oscillatory differential system with small damping and cubic nonlinearity. 83] presented the frequency analysis of strongly generalized nonlinear Duffing oscillators using 1-Le's frequency-amplitude formulation and 1-le's energy balance method.

From the earlier discussion, it is seen that most authors have studied nonlinear differential systems without considering any damping effect. But most physical and engineering problems occur in nature as nonlinear differential systems with small damping and the damping term plays an important role for the systems. It is noted that from our earlier discussion, the second-order strongly generalized nonlinear differential system with small damping has remained almost intact.

The advantage of the presented coupling technique is that the first-order analytical approximation solutions show good agreement with the corresponding numerical solutions and the highly generalized nonlinear differential equation can be handled easily, while the classical perturbation methods fail to handle such generalized nonlinear differential equations. to solve linear differential systems.

The Method

Now differentiating equation (3.4) twice with respect to time t and substituting 1, along with x in equation (3.1), we get 3.5) According to the homotopy perturbation method, equation (3.5) can be rewritten as Here &j is a constant for non-defeated nonlinear oscillators and known as the angular frequency of nonlinear systems and 2 is an unknown function that can be determined by eliminating secular terms. But for damped nonlinear differential systems w is a time-dependent function and it varies slowly with time t.

To handle this situation, we will use the extended form of the KBM [2-3] method. In accordance with this method, we choose the solution of equation (3.6) in the following. where b and q vary slowly with respect to time t. In the literature, b and ço are known as the amplitude and phase variables, respectively, and they play an important role for non-linear physical systems.

Examples

In accordance with the extended form of the KBM [2-3] method, the solution of equation (3.16) is obtained in the form. Now taking the value of y from equation (3.18) into the right hand side of equation (3.16) and using and rearranging the trigonometric identity equation (3.20) we obtain. The requirement that there be no secular terms, in particular the solution of equation (3.16), implies that the coefficient of the cos term is zero.

From equation (3.24) it is clear that the frequency of the damped nonlinear differential systems depends on both amplitude b and time 1. Now, when equation (3.26) is inserted into equation (3.24), we get a biquadratic algebraic equation in CO in the following form. In accordance with the extended form of the KBM [2-3] method, the solution of equation (3.34) is considered as the following form.

The requirement of no secular terms in a particular solution of equation (3.38) implies that the coefficient of the cosq term is zero. Using equation (3.43) into equation (3.41) we obtain a fifth degree polynomial in w of the following form.

Results and Discussion

On the other hand, our proposed technique can take full advantage of the classical perturbation method. The solutions obtained by the presented method show a good agreement with those obtained by the numerical procedure with multiple damping effects. It is also noted that the presented method is also able to handle second-order weakly generalized nonlinear differential system with damping effects and high-order nonlinearities.

The comparison is made between the solutions obtained by the presented technique and those obtained by the numerical procedure in Fig. The average percentage errors between the numerical and approximate solutions in table 3.1 were also calculated. From table 3.1, it is clear to us that the average percentage errors, except for a few cases, are negligible.

3.1 (b) First approximate solution of equation (3.13) is indicated by -.- (dashed lines) by the presented analytical technique with the initial conditions. Corresponding numerical solution is indicated by - (solid line). dashed lines) by the presented analytical technique with the initial conditions. An approximate coupling analysis technique for solving a particular type of fourth-order strongly generalized nonlinear damped oscillatory differential.

Fig. 3.1 (a) First approximate solution of equation (3.13) is denoted by -.- (dashed lines) by the presented analytical technique with the initial conditions b0 = 0.5, = 0 or [x(0) = 0.5, ±(0) = —0.07194] when k = 0.15, c = 1.0, a3 = 1.

Introduction

Kam Hy's homotopy perturbation and the extended form of the KBM methods, Uddin ci al. 78], Uddin and Sattar [79-80] presented approximate techniques for solving second order strongly nonlinear damped oscillatory differential systems for both cubic and quadratic nonlinearities. 83] developed frequency analysis of strongly nonlinear generalized Duffing oscillators using He's frequency-amplitude formulation and He's energy balance method.

More difficult and no less important cases, strongly generalized fourth-order nonlinear damped oscillatory differential systems, remained almost untouched. Therefore, in this chapter we are interested in the extension of the approximate analytical technique based on He homotopy perturbation [74-76] and the extended form of KBM methods [2-3] to solve a certain type of strongly generalized fourth-order nonlinear damped oscillators. differential systems. This method transforms the difficult problem under simplification into a simple problem that is easy to solve, but the classical perturbation techniques are almost unable to deal with strongly and weakly generalized fourth-order nonlinear damped oscillatory differential systems.

The presented method has been successfully implemented to solve fourth order strongly generalized nonlinear damped oscillatory differential systems with example. The advantage of this method is that the first-order approximate analytical solutions show good agreement with the corresponding numerical solutions. Furthermore, the presented method is also able to provide the desired results for fourth-order weakly generalized nonlinear damped oscillatory differential systems.

It is also noted that the proposed method is able to handle the fourth order strongly and weakly nonlinear oscillatory differential systems with cubic nonlinearity for various damping effects.

The Method

Here w is a constant for an undamped nonlinear oscillator and is known in the literature as the angular frequency, and 2 is an unknown function that can be determined by eliminating the secular terms. But w is a time dependent function and varies slowly with time I for a nonlinear damped oscillating differential system. To address this situation, the extended form of the KBM [2-3] method of Mitropolskii [4] is applied.

According to this method, we consider the solution (for a single oscillation mode) of equation (4.6) in the following form. Now we differentiate equation (4.8) four times with respect to time I and use equation (4.9) and then sum the terms up to 0(c) and neglect 0(2) and higher terms, we get If we equate the coefficients sinq and cosq from both sides of equation (4.13), we get the following functions.

Thus, we obtain an approximate first-order analytical solution of equation (4.1) with equation (4.20) using the presented coupling technique. Therefore, the determination of the first-order analytical approximate solution of equation (4.1) is completed by the proposed method.

Example

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