A new technique for solving a class of strongly nonlinear oscillatory equations

Highlights

• A new method based differential transformation method is presented in this study.

• The proposed method reduces the algebraic complexity.

• All the related equations are linear and easy to solve.

• Some nonlinear problems are studied.

Abstract

A new technique has been presented for solving strong nonlinear oscillatory problems. The solution is independent of trigonometric functions, and the determination of unknown coefficients related to the proposed solution is very simple. The method is illustrated by examples. For a large amplitude of oscillation of some nonlinear oscillators, the solution nicely agrees with the corresponding numerical solution.

Introduction

The classical perturbation methods [1], [2], [3], [4], [5] have been developed for solving nonlinear oscillators

$$\ddot{x}+{\omega }_0^{2}x=\epsilon f\left(x,\dot{x}\right),\left[x\left(0\right)=0,\dot{x}\left(0\right)=b\right]$$

where $\epsilon \ll {\omega }_0^2$. But modifications of some classical perturbation methods [6], [7], [8] overcome this limitation. On the other hand, some analytical techniques have been presented for handling strong nonlinear oscillators such as homotopy perturbation method [9], [10], [11], homotopy analysis method (HAM) [12], variational iteration method (VIM) [13], generalized homotopy method [14], harmonic balance method (HBM) [15], energy balance method [16], etc. However, most of the methods were formulated based on trigonometric functions. The differential transform method (DTM) [17], [18], [19] is a semi-analytical approach which is applied for solving strong nonlinear oscillators. Zhou [20] first introduced DTM and applied it to analyse linear and nonlinear electric circuit problems. Further the method is modified and applied to solve various differential equations. Shahed [21] also modified DTM based on Pade´ approximants. Combining Laplace transform and Pade ́ approximants, Momani and Erturk [22] presented a modified DTM method. Gokdogana et al. [23] proposed an adaptive multi-step differential transformation method to solve nonlinear differential equations. Besides, the DTM method is used by several researchers [24], [25], [26] to investigate nonlinear problems.

In this article, a new technique (like DTM) is presented for solving Eq. (1.1). The proposed solution is independent of trigonometric functions. The formulation as well as determination of the solution is very simple. Moreover, the solution is valid for all values of ${\omega }_0\ge 0$ and .