Ahmet Yildirim

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.

Solution of BVPs for fourth-order integro-differential equations by using homotopy perturbation method

In this study, the homotopy perturbation method proposed by Ji-Huan He is applied to solve both linear and nonlinear boundary value problems for fourth-order integro-differential equations. The analysis is accompanied by numerical examples. The results show that the homotopy perturbation method is of high accuracy, more convenient and efficient for solving integro-differential equations.

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.

On the coupling of the homotopy perturbation method and Laplace transformation

In this paper, a Laplace homotopy perturbation method is employed for solving one-dimensional non-homogeneous partial differential equations with a variable coefficient. This method is a combination of the Laplace transform and the Homotopy Perturbation Method (LHPM). LHPM presents an accurate methodology to solve non-homogeneous partial differential equations with a variable coefficient. The aim of using the Laplace transform is to overcome the deficiency that is mainly caused by unsatisfied conditions in other semi-analytical methods such as HPM, VIM, and ADM. The approximate solutions obtained by means of LHPM in a wide range of the problems domain were compared with those results obtained from the actual solutions, the Homotopy Perturbation Method (HPM) and the finite element method. The comparison shows a precise agreement between the results, and introduces this new method as an applicable one which it needs fewer computations and is much easier and more convenient than others, so it can be widely used in engineering too.

The effects of variable viscosity and thermal conductivity on a thin film flow over a shrinking/stretching sheet

The effects of variable viscosity and thermal conductivity on the flow and heat transfer in a laminar liquid film on a horizontal shrinking/stretching sheet are analyzed. The similarity transformation reduces the time independent boundary layer equations for momentum and thermal energy into a set of coupled ordinary differential equations. The resulting five-parameter problem is solved by the homotopy perturbation method. The results are presented graphically to interpret various physical parameters appearing in the problem.

Analytical solutions to the pulsed Klein-Gordon equation using Modified Variational Iteration Method (MVIM) and Boubaker Polynomials Expansion Scheme (BPES)

In this study, we propose an analytical solution to the Klein-Gordon equation in a pulsed stationary regime. The performed protocols are based on the modified variational iteration method MVIM and Boubaker polynomials expansion scheme BPES. The results are presented, and compared with some solutions proposed later in order to confirm the good accuracy of the protocols used.

Application of He's homotopy perturbation method for solving the Cauchy reaction-diffusion problem

In this paper, the solution of Cauchy reaction-diffusion problem is presented by means of the homotopy perturbation method. Reaction-diffusion equations have special importance in engineering and sciences and constitute a good model for many systems in various fields. Application of homotopy perturbation method to this problem shows the rapid convergence of the sequence constructed by this method to the exact solution.

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.

Effects of partial slip on the peristaltic flow of a MHD Newtonian fluid in an asymmetric channel

In this study, the effects of partial slip on the peristaltic flow of a MHD Newtonian fluid in an asymmetric channel are studied analytically and numerically. The governing equations of motion and energy are simplified using a long wavelength approximation. A closed form solution of the momentum equation is obtained by the homotopy perturbation method (HPM) and an exact solution of the energy equation is presented in the presence of the viscous dissipation term. The expression for pressure rise is calculated using numerical integration. Also, we discussed the trapping phenomena. The graphical results are presented to interpret various physical parameter of interest.

He's homotopy perturbation method for solving the space-and time-fractional telegraph equations

In this study, homotopy perturbation method (HPM) is used to obtain analytic and approximate solutions of the space- and time-fractional telegraph equations. The space- and time-fractional derivatives are considered in the Caputo sense. The analytic solutions are calculated in the form of a series with easily computable terms. Some examples are given. The results reveal that HPM is very effective and convenient.