scholarly journals Approximate Analytic Solution for the KdV and Burger Equations with the Homotopy Analysis Method

2012 ◽  
Vol 2012 ◽  
pp. 1-13 ◽  
Author(s):  
Mojtaba Nazari ◽  
Faisal Salah ◽  
Zainal Abdul Aziz ◽  
Merbakhsh Nilashi
Keyword(s):  
Homotopy Analysis Method ◽  
Analytic Solution ◽  
Order Approximation ◽  
Soliton Solutions ◽  
Nonlinear Problems ◽  
Approximate Analytic Solution ◽  
Analysis Method ◽  
Convergence Region ◽  
Burgers Equations ◽  
Homotopy Analysis

The homotopy analysis method (HAM) is applied to obtain the approximate analytic solution of the Korteweg-de Vries (KdV) and Burgers equations. The homotopy analysis method (HAM) is an analytic technique which provides us with a new way to obtain series solutions of such nonlinear problems. HAM contains the auxiliary parameterħ, which provides us with a straightforward way to adjust and control the convergence region of the series solution. The resulted HAM solution at 8th-order and 14th-order approximation is then compared with that of the exact soliton solutions of KdV and Burgers equations, respectively, and shown to be in excellent agreement.

Approximate Analytical Solutions of KdV and Burgers’ Equations via HAM and nHAM

2014 ◽  
Vol 67 (1) ◽  
Author(s):  
Mojtaba Nazari ◽  
Vahid Baratie ◽  
Vincent Daniel David ◽  
Faisal Salah ◽  
Zainal Abdul Aziz
Keyword(s):  
Comparative Study ◽  
Analytical Solutions ◽  
Homotopy Analysis Method ◽  
Soliton Solutions ◽  
Analysis Method ◽  
Burgers Equations ◽  
Approximate Analytical Solutions ◽  
Homotopy Analysis ◽  
A New Technique ◽  
Exact Soliton Solutions

This article presents a comparative study of the accuracy between homotopy analysis method (HAM) and a new technique of homotopy analysis method (nHAM) for the Korteweg–de Vries (KdV) and Burgers’ equations. The resulted HAM and nHAM solutions at 8th-order and 6th-order approximations are then compared with that of the exact soliton solutions of KdV and Burgers’ equations, respectively. These results are shown to be in excellent agreement with the exact soliton solution. However, the result of HAM solution is ratified to be more accurate than the nHAM solution, which conforms to the existing finding. 

scholarly journals A Study on the Convergence of Series Solution of Non-Newtonian Third Grade Fluid with Variable Viscosity: By Means of Homotopy Analysis Method

2012 ◽  
Vol 2012 ◽  
pp. 1-11 ◽  
Author(s):  
R. Ellahi
Keyword(s):  
Homotopy Analysis Method ◽  
Variable Viscosity ◽  
Nonlinear Problems ◽  
Third Grade ◽  
Series Solutions ◽  
Analysis Method ◽  
Convergence Region ◽  
Homotopy Analysis ◽  
Grade Fluid ◽  
Physical Features

This work is concerned with the series solutions for the flow of third-grade non-Newtonian fluid with variable viscosity. Due to the nonlinear, coupled, and highly complicated nature of partial differential equations, finding an analytical solution is not an easy task. The homotopy analysis method (HAM) is employed for the presentation of series solutions. The HAM is accepted as an elegant tool for effective solutions for complicated nonlinear problems. The solutions of (Hayat et al., 2007) are developed, and their convergence has been discussed explicitly for two different models, namely, constant and variable viscosity. An error analysis is also described. In addition, the obtained results are illustrated graphically to depict the convergence region. The physical features of the pertinent parameters are presented in the form of numerical tables.

Analytic Solution of the Sharma-Tasso-Olver Equation by Homotopy Analysis Method

2010 ◽  
Vol 65 (4) ◽  
pp. 285-290 ◽  
Author(s):  
Saeid Abbasbandy ◽  
Mahnaz Ashtiani ◽  
Esmail Babolian
Keyword(s):  
Homotopy Analysis Method ◽  
Analytic Solution ◽  
Series Solutions ◽  
Analysis Method ◽  
Auxiliary Parameter ◽  
Convergence Region ◽  
Homotopy Analysis ◽  
Convergence Control ◽  
And Control ◽  
Convergence Control Parameter

An analytic technique, the homotopy analysis method (HAM), is applied to obtain the kink solution of the Sharma-Tasso-Olver equation. The homotopy analysis method is one of the analytic methods and provides us with a new way to obtain series solutions of such problems. HAM contains the auxiliary parameter ħwhich gives us a simple way to adjust and control the convergence region of series solution. “Due to this reason, it seems reasonable to rename ħthe convergence-control parameter” [1].

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 COMPARISON OF A GENERAL SERIES EXPANSION METHOD AND THE HOMOTOPY ANALYSIS METHOD

2010 ◽  
Vol 24 (15) ◽  
pp. 1699-1706 ◽  
Author(s):  
CHENG-SHI LIU ◽  
YANG LIU
Keyword(s):  
Series Expansion ◽  
Homotopy Analysis Method ◽  
Nonlinear Differential Equations ◽  
Expansion Method ◽  
Basis Functions ◽  
Analysis Method ◽  
Convergence Region ◽  
General Series ◽  
Homotopy Analysis ◽  
Series Expansion Method

A simple analytic tool, namely the general series expansion method, is proposed to find the solutions for nonlinear differential equations. A set of suitable basis functions [Formula: see text] is chosen such that the solution to the equation can be expressed by [Formula: see text]. In general, t0 can control and adjust the convergence region of the series solution such that our method has the same effect as the homotopy analysis method proposed by Liao, but our method is simpler and clearer. As a result, we show that the secret parameter h in the homotopy analysis methods can be explained by using our parameter t0. Therefore, our method reveals a key secret in the homotopy analysis method. For the purpose of comparison with the homotopy analysis method, a typical example is studied in detail.

Homotopy Analysis Method

2017 ◽  
pp. 173-226
Keyword(s):  
Homotopy Analysis Method ◽  
Series Solution ◽  
Nonlinear Partial Differential Equations ◽  
Analytic Method ◽  
Series Solutions ◽  
Analysis Method ◽  
Auxiliary Parameter ◽  
Convergence Region ◽  
Homotopy Analysis ◽  
And Control

In this chapter, the analytic solution of nonlinear partial differential equations arising in heat transfer is obtained using the newly developed analytic method, namely the Homotopy Analysis Method (HAM). The homotopy analysis method provides us with a new way to obtain series solutions of such problems. This method contains the auxiliary parameter provides us with a simple way to adjust and control the convergence region of series solution. By suitable choice of the auxiliary parameter, we can obtain reasonable solutions for large modulus.

Application of Optimal Homotopy Analysis Method for Solitary Wave Solutions of Kuramoto-Sivashinsky Equation

2011 ◽  
Vol 66 (1-2) ◽  
pp. 117-122
Author(s):  
Qi Wang
Keyword(s):  
Solitary Wave ◽  
Homotopy Analysis Method ◽  
Solitary Wave Solutions ◽  
Analysis Method ◽  
Convergence Region ◽  
Optimal Method ◽  
Optimal Homotopy Analysis Method ◽  
Wave Solutions ◽  
Homotopy Analysis ◽  
Sivashinsky Equation

In this paper, the optimal homotopy analysis method is applied to find the solitary wave solutions of the Kuramoto-Sivashinsky equation. With three auxiliary convergence-control parameters, whose possible optimal values can be obtained by minimizing the averaged residual error, the method used here provides us with a simple way to adjust and control the convergence region of the solution. Compared with the usual homotopy analysis method, the optimal method can be used to get much faster convergent series solutions.

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