Turgut Öziş

A Comparative Study of He's Homotopy Perturbation Method for Determining Frequency-amplitude Relation of a Nonlinear Oscillator with Discontinuities

In nonlinear analysis, perturbation methods are well established tools to study diverse aspects of nonlinear problems. Surveys of the early literature with numerous references, and useful bibliographies, have been given by Nayfeh [1], Mickens [2], Jordan and Smith [3] and Hagedorn [4], However, the use of perturbation theory in many important practical problems is invalid, or it simply breaks down for parameters beyond a certain specified range. Therefore, new analytical techniques should be developed to overcome these shortcomings. Such a new technique should work over a large range of parameters and yield accurate analytical approximate solutions beyond the coverage and ability of the classical perturbation methods. For example, homotopy perturbation method introduced by He [5-14] can readily eliminate the limitations of the traditional perturbation techniques. Moreover, this method was first applied to nonlinear oscillators with discontinuities[15]. For more detailed information please refer to the comprehensive book[16] or the review articles[ 17,18] by He. There also exists a wide range of literature dealing with the approximate determination of periodic solutions for nonlinear problems by using a mixture of methodologies [19-37], The purpose of this paper is to make the comparison of two distinct adaptations of Hes homotopy perturbation method (HPM) for determining frequency-amplitude relation of the nonlinear oscillator with discontinuities.

Traveling Wave Solution of Korteweg-de Vries Equation using He's Homotopy Perturbation Method

Hes Homotopy Perturbation Method combined with modified Lindstedt-Poincare method is applied to the search for traveling wave solutions of Korteweg-de Vries (KdV) equation. The work emphasizes the power of the method that can be used in problems of identical nonlinearity.

Analytic study on two nonlinear evolution equations by using the (G′/G)-expansion method

Abstract The validity and reliability of the so-called (G′/G)-expansion method is tested by applying it to two nonlinear evolutionary equations. Solutions in more general forms are obtained. When the parameters are taken as special values, it is observed that the previously known solutions can be recovered. New rational function solutions are also presented. Being concise and less restrictive, the method can be applied to many nonlinear partial differential equations.

Determination of the frequency-amplitude relation for a Duffing-harmonic oscillator by the energy balance method

This paper applies Hes energy balance method to determine the frequency-amplitude relation of the Duffing-harmonic oscillator, which gives a good estimate for the angular frequency.

Comparison between Adomian's method and He's homotopy perturbation method

In this paper, it is revealed that modified form of Hes homotopy perturbation method corresponds to Adomians decomposition method for certain nonlinear problems.

Application of He's semi-inverse method to the nonlinear Schrödinger equation

A variational theory is established for the nonlinear Schrodinger equation using Hes semi-inverse method. On the basis of the variational principle obtained, a solitary solution is obtained, which is the same as Debnaths result [L. Debnath, Nonlinear Partial Differential Equations for Scientists and Engineers, Birkhauser, Boston, 1997], confirming the validity of our analysis. The method is straightforward and concise.

Application of the G′/G-expansion method to Kawahara type equations using symbolic computation

Abstract In this paper, Kawahara type equations are selected to illustrate the effectiveness and simplicity of the G ′ / G -expansion method. With the aid of a symbolic computation system, three types of more general traveling wave solutions (including hyperbolic functions, trigonometric functions and rational functions) with free parameters are constructed. Solutions concerning solitary and periodic waves are also given by setting the two arbitrary parameters, involved in the traveling waves, as special values.

An analytical study for Fisher type equations by using homotopy perturbation method

In this paper, homotopy perturbation method is applied to Fisher type equations. The solutions introduced in this study are in recursive sequence forms which can be used to obtain the closed form of the solutions if they are required. The method is tested on various examples which are revealing the effectiveness and the simplicity of the method.

A smart nonstandard finite difference scheme for second order nonlinear boundary value problems

A new kind of finite difference scheme is presented for special second order nonlinear two point boundary value problems. An artificial parameter is introduced in the scheme. Symbolic computation is proposed for the construction of the scheme. Local truncation error of the method is discussed. Numerical examples are illustrated. Numerical results show that the method is very effective.

Homotopy perturbation method for the nonlinear dispersive K(m, n, 1) equations with fractional time derivatives

Purpose – This paper aims to apply Hes homotopy perturbation method (HPM) to obtain solitary solutions for the nonlinear dispersive equations with fractional time derivatives.Design/methodology/approach – The authors choose as an example the nonlinear dispersive and equations with fractional time derivatives to illustrate the validity and the advantages of the proposed method.Findings – The paper extends the application of the HPM to obtain analytic and approximate solutions to the nonlinear dispersive equations with fractional time derivatives.Originality/value – This paper extends the HPM to the equation with fractional time derivative.