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We introduce a new method, namely, the Optimal Iteration Perturbation Method (OIPM), to solve nonlinear differential equations of oscillators with cubic and harmonic restoring force. We illustrate that OIPM is very effective and convenient and does not require linearization or small perturbation. Contrary to conventional methods, in OIPM, only one iteration leads to high accuracy of the solutions. The main advantage of this approach consists in that it provides a convenient way to control the convergence of approximate solutions in a very rigorous way and allows adjustment of convergence regions where necessary. A very good agreement was found between approximate and numerical solutions, which prove that OIPM is very efficient and accurate.

Mathematical modelling of many physical systems leads to nonlinear ordinary or partial differential equations in various fields of physics, mathematics, or engineering. An effective method is required to analyze the mathematical model which provides solutions conforming to physical reality. In many cases, it is possible to replace a nonlinear differential equation by a corresponding linear differential equation that approximates closely the original one to give useful results. In general, the study of nonlinear differential equations is restricted to a variety of special classes of equations and the method of solution usually involves one or more techniques to achieve analytical approximations to the solutions. Solving the governing equations of nonlinear oscillators has been one of the most time-consuming and difficult affairs among researchers. Therefore, many researchers and scientists of both vibrations and mathematics have recently paid much attention to find and develop approximate solutions. Perturbation methods are well established tools to study diverse aspects of nonlinear problems [

It is noted that several methods have been used to obtain approximate solutions for strongly nonlinear oscillators. An interesting approach which combines the harmonic balance method and linearization of nonlinear oscillation equation was proposed in [

In this paper, coupling the iteration perturbation method [

The efficiency of the present procedure is proved while an accurate solution is explicitly analytically obtained in an iterative way after only one iteration. The proposed method does not require a small parameter into the equation and provides a convenient and rigorous way to optimally control the convergence of the solutions by means of a finite number of unknown parameters.

In this work, we consider a nonlinear oscillator in the form

For (_{i}

Integrating (

From the initial conditions (

In this way, the approximate solution of _{i},

Therefore, the solution (

In the present paper we consider a nonlinear oscillator with cubic and harmonic restoring force

If Ω is the frequency of the system described by (

The initial conditions (

We consider the initial approximation in the form
_{1}, _{2}, _{3}, and _{4} are unknown constants at this moment.

For _{0} given by (

The first term in the right-hand side of (

The last term in (

Substituting (

The last term in the right-side of (

In (

Substituting (

Substituting (

Finally, (

The frequency Ω and the constants

We will illustrate the applicability, accuracy, and effectiveness of the proposed approach by comparing the analytical approximate periodic solution with numerical integration results obtained using a fourth-order Runge-Kutta method. The comparison is made in terms of displacements and phase plane. The error of the solution has been also computed. The results of these comparisons are presented in Figures

Comparison between the approximate solution (

Comparison between the approximate solution (

The error between the numerical and approximate solution (

Comparison between the approximate solution (

Comparison between the approximate solution (

The error between the numerical and approximate analytical solution (

For

For

In this paper we have developed an analytical treatment of strongly nonlinear oscillators with cubic and harmonic restoring force using a new approximate analytical technique, namely, the Optimal Iteration Perturbation Method (OIPM). This method accelerates the convergence of the solutions since after only one iteration we achieved very accurate results. The proposed approach is an iterative procedure, and iterations are preformed in a very simple manner by identifying optimally some coefficients and therefore very good approximations are obtained in few terms. Actually, the capital strength of OIPM is its fast convergence. An excellent agreement of the approximate periodic solutions and frequencies with the exact ones has been demonstrated. Two examples are given, and the results reveal that our procedure is very effective, simple, and accurate. This paper demonstrates the general validity and the great potential of the OIPM for solving strongly nonlinear problems.