An adaptation of homotopy analysis method for reliable treatment of strongly nonlinear problems: construction of homotopy polynomials

2014 ◽  
Vol 38 (5) ◽  
pp. 991-1000 ◽  
Author(s):  
Zaid Odibat ◽  
A. Sami Bataineh
Keyword(s):  
Homotopy Analysis Method ◽  
Nonlinear Problems ◽  
Analysis Method ◽  
Strongly Nonlinear ◽  
Homotopy Analysis ◽  
Homotopy Polynomials

The Analysis Approach of Boundary Layer Equations of Power-Law Fluids of Second Grade

2008 ◽  
Vol 63 (9) ◽  
pp. 564-570 ◽  
Author(s):  
Saeid Abbasbandy ◽  
Muhammet Yürüsoy ◽  
Mehmet Pakdemirli
Keyword(s):  
Power Law ◽  
Homotopy Analysis Method ◽  
Nonlinear Problems ◽  
Analytic Solutions ◽  
Second Grade ◽  
Analysis Method ◽  
Boundary Layer Equations ◽  
Power Law Fluids ◽  
Homotopy Analysis ◽  
Constant Coefficients

A powerful analytic technique for nonlinear problems, the homotopy analysis method (HAM), is employed to give analytic solutions of power-law fluids of second grade. For the so-called secondorder power-law fluids, the explicit analytic solutions are given by recursive formulas with constant coefficients. Also, for the real power-law index in a quite large range an analytic approach is proposed. It is demonstrated that the approximate solution agrees well with the finite difference solution. This provides further evidence that the homotopy analysis method is a powerful tool for finding excellent approximations to nonlinear equations of the power-law fluids of second grade.

scholarly journals The Application of the Homotopy Perturbation Method and the Homotopy Analysis Method to the Generalized Zakharov Equations

2012 ◽  
Vol 2012 ◽  
pp. 1-19 ◽  
Author(s):  
Hassan A. Zedan ◽  
Eman El Adrous
Keyword(s):  
Perturbation Method ◽  
Exact Solutions ◽  
Homotopy Analysis Method ◽  
Homotopy Perturbation Method ◽  
Initial Conditions ◽  
Nonlinear Problems ◽  
Analysis Method ◽  
Homotopy Perturbation ◽  
Homotopy Analysis ◽  
The Absolute

We introduce two powerful methods to solve the generalized Zakharov equations; one is the homotopy perturbation method and the other is the homotopy analysis method. The homotopy perturbation method is proposed for solving the generalized Zakharov equations. The initial approximations can be freely chosen with possible unknown constants which can be determined by imposing the boundary and initial conditions; the homotopy analysis method is applied to solve the generalized Zakharov equations. HAM is a strong and easy-to-use analytic tool for nonlinear problems. Computation of the absolute errors between the exact solutions of the GZE equations and the approximate solutions, comparison of the HPM results with those of Adomian’s decomposition method and the HAM results, and computation the absolute errors between the exact solutions of the GZE equations with the HPM solutions and HAM solutions are presented.

scholarly journals Homotopy Analysis Method for the Rayleigh Equation Governing the Radial Dynamics of a Multielectron Bubble

2012 ◽  
Vol 2012 ◽  
pp. 1-20 ◽  
Author(s):  
F. A. Godínez ◽  
M. A. Escobedo ◽  
M. Navarrete
Keyword(s):  
Dynamical Systems ◽  
Liquid Helium ◽  
Analytical Solutions ◽  
Homotopy Analysis Method ◽  
Nonlinear Dynamical Systems ◽  
Rayleigh Equation ◽  
Analysis Method ◽  
Nonlinear Dynamical ◽  
Strongly Nonlinear ◽  
Homotopy Analysis

The homotopy analysis method is used to obtain analytical solutions of the Rayleigh equation for the radial oscillations of a multielectron bubble in liquid helium. The small order approximations for amplitude and frequency fit well with those computed numerically. The results confirm that the homotopy analysis method is a powerful and manageable tool for finding analytical solutions of strongly nonlinear dynamical systems.

scholarly journals Wavelet-Based Homotopy Analysis Method for Nonlinear Matrix System and Its Application in Burgers Equation

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
Xing Ruyi
Keyword(s):  
Collocation Method ◽  
Homotopy Analysis Method ◽  
Burgers Equation ◽  
Nonlinear Problems ◽  
Nonlinear Pdes ◽  
Analysis Method ◽  
Precise Integration ◽  
Wavelet Collocation Method ◽  
Homotopy Analysis ◽  
Shannon Wavelet

To generalize the homotopy analysis method (HAM) to multidegree-of-freedom nonlinear system, the adaptive precise integration method (APIM) is introduced into the HAM, with which the almost exact value of the exponential matrix can be obtained. Combining the interval interpolation wavelet collocation method, HAM-based APIM can be employed to solve the nonlinear PDEs. As an example, Burgers equation is spatially discretized by the interval quasi-Shannon wavelet collocation method and solved by the proposed method to illustrate the effectiveness and great potential of the homotopy analysis method in nonlinear problems.

Solitary Solution of Nonlinear Equation by Homotopy Analysis Method

2011 ◽  
Vol 130-134 ◽  
pp. 3668-3671
Author(s):  
Xiu Rong Chen ◽  
Wen Shan Cui
Keyword(s):  
Exact Solution ◽  
Nonlinear Equation ◽  
Homotopy Analysis Method ◽  
Nonlinear Problems ◽  
Analysis Method ◽  
Solitary Solution ◽  
Homotopy Analysis

In this paper, we apply homotopy analysis method to solve nonlinear equation and successfully obtain the bell-shaped solitary solution to the nonlinear equation. Comparison between our solution and the exact solution shows that homotopy analysis method is effective and valid for nonlinear problems.

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