Construction of solitary solution and compacton-like solution by variational iteration method

Pascual

Nonlinear Analysis: Real World Applications

et al. 2009

In this paper He's homotopy perturbation method has been adapted to calculate higherorder approximate periodic solutions of a nonlinear oscillator with discontinuity for which the elastic force term is anti-symmetric and quadratic term. We find He's homotopy perturbation method works very well for the whole range of initial amplitudes, and the excellent agreement of the approximate frequencies and periodic solutions with the exact ones has been demonstrated and discussed. Only one iteration leads to high accuracy of the solutions with a maximal relative error for the approximate period of less than 0.73%for all values of oscillation amplitude, while this relative error is as low as 0.040% when the second iteration is considered. Comparison of the result obtained using this method with those obtained by harmonic balance method reveals that the former is very effective and convenient.

Section: Introductionmentioning

confidence: 99%

Solution for an anti-symmetric quadratic nonlinear oscillator by a modified He’s homotopy perturbation method

Pascual

Nonlinear Analysis: Real World Applications

et al. 2009

“…He [1,2] developed the variational iteration and homotopy perturbation methods for solving linear, nonlinear partial integro-differential equation. It is worth mentioning that the origin of variational iteration method can be traced to Inokuti, Sekine and Mura [3], but the real potential of this technique was explored by He [4][5][6][7][8][9][10]. Moreover, He realized the physical significance of the variational iteration method, its compatibility with the physical problems and applied this promising technique to a wide class of linear and nonlinear, ordinary, partial integro-differential equation [1,2].…”

Section: Introductionmentioning

confidence: 99%

A comparison of Semi-analytical Methods for Solving Partial Integro-Differential Equations

Soliman¹,

El–Azab²,

El-Gamel³

et al. 2013

Math. Sci. Lett.

Abstract:In this paper, we apply the Variational iteration method and homotopy perturbation method for solving linear and nonlinear partial integro-differential equation (PIDE). The efficiency and accuracy of the methods is validated by its application to several distinct test problems which have exact solutions. The results of applying these methods show the simplicity and efficiency of these methods.

“…When a NLEE is analysed, one of the most important question is the construction of the exact solutions for equation [1]. In the open literature, quite a few methods for obtaining explicit travelling and solitary wave solutions to NLEEs have been suggested such as the inverse scattering method [2], the bilinear transformation method [3], the tanh-sech method [4,5], the extended tanh method [6,7], the sine-cosine method [8][9][10], the homogeneous balance method [11,12], the pseudo spectral method [13], the G ′ /G -expansion method [14][15][16], exp-function method [17], variational iteration method [18], homotopy perturbation method [19], the Jacobi elliptic function method [20], Lie group analysis method [21] and so on.…”

Section: Introductionmentioning

confidence: 99%

Nonlinear self adjointness, conservation laws and exact solutions of ill-posed Boussinesq equation

San

Yaşar

2016

Open Physics

Abstract:In this work, we consider the ill-posed Boussinesq equation which arises in shallow water waves and non-linear lattices. We prove that the ill-posed Boussinesq equation is nonlinearly self-adjoint. Using this property and Lie point symmetries, we construct conservation laws for the underlying equation. In addition, the generalized solitonary, periodic and compact-like solutions are constructed by the exp-function method.