Study of nonlinear systems using approximation methods

Abstract

Physical systems are, in general, nonlinear in nature. Therefore, governing equations corresponding to such systems are complicated. A vibration of nonlinear system is a deviation from the exactly solvable harmonic oscillator (HO) and is termed as an anharmonic oscillator (AHO). Nonlinear systems are usually challenging to solve, either numerically or analytically. Numerical solutions of AHOs are sometimes easy, but one desires to get the analytical solutions to such problems as they carry more information and give a better insight into the system. The homotopy perturbation method (HPM) yields approximate analytical solutions for the nonlinear oscillators using simple calculations, but it compromises the accuracy for strongly nonlinear cases. Therefore, one strives to find an improved HPM. newlineIn a recent article Manimegalai et al. [2019], Aboodh transform-based homotopy perturbation method (AT) has been found to produce approximate analytical solutions to strongly nonlinear oscillator in a simple way but with better accuracy in comparison to those obtained from some of the established approximation methods Mehdipour et al. [2010], Nofal et al. [2013] for some physically relevant AHOs such as autonomous conservative oscillator (ACO). In this thesis, expansion of frequency and a convergence control parameter (h) are adopted in the framework of the HPM to improve the accuracy by retaining its simplicity. Laplace transform is used to make the calculation simpler. This improved HPM (LH) is simple but provides highly accurate (order of magnitude higher) results for the ACO compared to AT. LH is applied to differet kind of nonlinear oscillators such as forced nonlinear oscillators, fractional nonlinear oscillators and damped nonlinear oscillators and noticed that LH produces highly accurate result compared to the available results obtained from other approximation methods such as HPM, enhanced HPM(EHPM), Lindstedt Poincaré (LP) method. newline