TOPOLOGICAL METHODS IN NONLINEAR ANALYSIS

TABLE OF CONTENTS

	Title and Author(s)	Page
	Recent Development of the Homotopy Perturbation Method Ji-Huan He ABSTRACT. The homotopy perturbation method is extremely accessible to non-mathematicians and engineers. The method decomposes a complex problem under study into a series of simple problems that are easy to be solved. This note gives an elementary introduction to the basic solution procedure of the homotopy perturbation method. Particular attention is paid to constructing a suitable homotopy equation.	205
	Algorithms for nonlinear fractional PDE Shaher Momani, Zaid Odibat and Ishak Hashim ABSTRACT. Fractional order partial differential equations, as generalization of classical integer order partial differential equations, are increasingly used to model problems in fluid flow, finance and other areas of applications. In this paper we present a collection of numerical algorithms for the solution of nonlinear partial differential equations with space- and time-fractional derivatives. The fractional derivatives are considered in the Caputo sense. Two numerical examples are given to demonstrate the effectiveness of the present methods. Results show that the numerical schemes are very effective and convenient for solving nonlinear partial differential equations of fractional order.	211
	Applications of VIM and HPM Zaid Odibat and Shaher Momani ABSTRACT. In this paper, variational iteration and homotopy perturbation methods that developed for integer-order differential equations are directly extended to derive explicit and numerical solutions of various evolution equations with time-fractional derivatives. The results reveal that the two methods are very effective and convenient for solving nonlinear partial differential equations of fractional order.	227
	New Application of HPM to ZK-MEW equation Jia-Cheng Lan, Jia-Min Zhu and Zheng-Yi Ma ABSTRACT. The work presents a derivation of solitary solutions of the two-dimensional Zakharov-Kuznetsov Modified Equal Width (ZK-MEW) equation using the homotopy perturbation method.	235
	Application of HPM to the Bratu-Type Equations Xinlong Feng, Yinnian He and Jixiang Meng ABSTRACT. A new algorithm is presented for solving the Bratu-type equations. The numerical scheme based on the homotopy perturbation method is deduced. Two boundary value problems and an initial value problem are given to illustrate effectiveness and convenience of the proposed scheme. Our results agree very well with the numerical solutions showing that the homotopy perturbation method is a promising method.	243
	He's HPM for the Temperature Distribution Lan Xu ABSTRACT. This paper applies J. H. He's homotopy perturbation method (HPM) to calculate the temperature distribution in convective straight fins with temperature-dependent thermal conductivity. The temperature distribution of straight fins is obtained as a function of thermo-geometric fin parameter. Comparison with the exact solution shows that the method is very effective and convenient, only one iteration leads to an accurate solution.	253
	Simulation of the Predator-Prey Problem by the HPM M. S. H. Chowdhury, Ishak Hashim and R. Roslan ABSTRACT. In this paper, the predator-prey problem is revisited. Previous solution by homotopy-perturbation method (HPM) is improved by treating the homotopy-perturbation method as an algorithm in a sequence of intervals (time steps) called the multistage homotopy-perturbation method (shortly MHPM). Numerical results show that the multistage homotopy-perturbation method and the classical fourth-order Rungge-Kutta (RK4) methods are in complete agreement.	263
	Large deflection of a Cantilever beam under point load D. D. Ganji, A. Sadighi, H. Tari, M. Gorji and N. Haghparast ABSTRACT. In this study, the homotopy-perturbation method (HPM) is used to investigate the large deflection of a cantilever beam under point load at the free end. The vertical and horizontal displacements of the cantilever beam are conveniently obtained in explicit analytical forms The main objective of this study is to propose an alternative method of solution, which does not require small parameters and avoid linearization and physically unrealistic assumptions. The results show that this method is very efficient and convenient and can be applied to a large class of practical problems.	271
	Frequency-Amplitude Relationship of the Duffing-Harmonic Oscillator Zh.-L. Tao ABSTRACT. The variational iteration method, the variational method and the parameter-expanding method are applied to obtain the frequency-amplitude relationship of the Duffing-harmonic oscillator. The obtained results reveal that all the three methods are very effective and convenient.	279
	Analytical Approach to Kawahara Equation J. Lu ABSTRACT. Variational iteration method and homotopy perturbation method are introduced to solve the Kawahara equation. Comparison of the obtained results with the numerical solution shows that both methods lead to remarkably accurate solutions. The main property of the both methods is its flexibility and ability to solve nonlinear equations accurately and conveniently.	287
	HPM for Multi-Dimensional Nonlinear Coupled System N. H. Sweilam, Mohamed Meabed Khader and R. F. Al-Bar ABSTRACT. In this paper, the homotopy perturbation method (HPM) proposed by J. H. He is adopted for solving multi-dimensional nonlinear coupled system of parabolic and hyperbolic equations. The numerical results of the present method are compared with the exact solution of an artificial multi-dimensional nonlinear coupled system of parabolic and hyperbolic model to show the efficiency of the method. Moreover, comparison is made between the results obtained by the present method and that obtained by the Adomian decomposition method (ADM). It is found that the present method works extremely well, very efficient, simple and convenient.	295
	Application of HPM to Regularization of Scalar Images Q. Ma, R.-Y. Xing and S.-L. Mei ABSTRACT. The homotopy perturbation method is implemented to solve nonlinear equations. Based on this method, a multi-step scheme is constructed for a kind of Hamilton-Jacobi formulations by assuming the homotopy parameter is a linear function of time. Using this multi-step scheme, a minimal surface regularization equation is solved, which designates a regularization process that doesn't smooth the image with the same weight in all the spatial directions. Some image denoisying examples illustrate its effectiveness and convenience.	305
	On the Solution of Stochastic Oscillatory Quadratic Nonlinear Equations Magdy A. El-Tawil and A. S. Al-Jihany ABSTRACT. In this paper, nonlinear oscillators under quadratic nonlinearity with stochastic inputs are considered. Different methods are used to obtain first order approximations, namely the WHEP technique, the perturbation method, the Pickard approximations, the Adomian decompositions and the homotopy perturbation method (HPM). Some statistical moments are computed for the different methods using Mathematica 5. Comparisons are illustrated by figures for different case-studies.	315
	HPM Solution for Peristaltic Flow of a Third Order Fluid A. M. Siddiqui, Q. A. Azim, A. Ashraf and Q. K. Ghori ABSTRACT. The peristaltic transport of a third order fluid in a planar channel as well as in an axisymmetric tube having walls that are transversely displaced by an infinite, harmonic travelling wave of large wavelength and negligibly small Reynolds number, has been analyzed using homotopy perturbation technique. Unlike perturbation method, this method does not restrict the Deborah number $\Gamma $ to be very large or small and works fairly well for any choice of $\Gamma $. The expressions for stream function and pressure rise per wavelength have been obtained up to second order of approximation.	331
	Application of the HPM Z. Z. Ganji, D. D. Ganji, H. Jafari and M. Rostamian ABSTRACT. The homotopy perturbation method (HPM) is applied to solve nonlinear partial differential equations of fractional orders. The corresponding solutions for integer orders of the fractional derivatives are found to be special cases of the fractional differential equations. It is predicted that HPM can be found widely applicable in engineering.	341
	Determination of Limit Cycles by Iterated Homotopy Perturbation Method Turgut Ozis and Ahmet Yildirim ABSTRACT. He's Homotopy Perturbation Method which reduced to an Iterative Scheme is applied to nonlinear oscillators with strong nonlinearity. With the method, the iteration scheme provides excellent approximations to the solutions even though the iteration can only be done to the first stage.	349
	Solitary wave solutions for a coupled MKdV system Yong-Qing Jiang and Jia-Min Zhu ABSTRACT. The work presents a derivation of solitary solutions of a coupled MKdV system using the homotopy perturbation method.	359
	HPM for two point boundary value problems Shun-Dong Zhu ABSTRACT. The homotopy perturbation method is applied for solving two point boundary value problems. In this method a trial function (initial solution) is chosen with some unknown parameters, which are identified using the method of weighted residuals. An example is given, the obtained result is compared with the exact solution, revealing that this method is very efficient and the obtained solution is of high accuracy.	369
	HPM for the nonlinear relativistic toda Lattice equations Jiamin Zhu ABSTRACT. The work presents a derivation of solitary wave solutions of the nonlinear relativistic Toda lattice equations using the homotopy perturbation method. The work presents a derivation of solitary wave solutions of the nonlinear relativistic Toda lattice equations using the homotopy perturbation method.	373
	Chinese mathematics for nonlinear oscillators Ling Zhao ABSTRACT. Ancient Chinese mathematicians made dramatic progress toward answering one of the oldest, most fundamental problem of how to solve approximately a real root of a nonlinear algebra equation in about 2nd century BC. The idea was further extended to nonlinear differential equations by J. H. He in 2002. In this paper, J. H. He's frequency-amplitude formation is used to find periodic solution of a pure nonlinear oscillator (without a linear term). The obtained result is of remarkable accuracy.	383
	Application of He's frequency-amplitude formulation Jie Fan ABSTRACT. The work presents a derivation of frequency-amplitude of the Duffing-harmonic oscillator from a formulation suggested by Ji-Huan He. The obtained result is valid for all amplitudes, and its maximal error is less than 2.2%.	389