scholarly journals Application of the homotopy analysis method to solving nonlinear evolution equations

2006 ◽  
Vol 55 (4) ◽  
pp. 1555
Author(s):  
Shi Yu-Ren ◽  
Xu Xin-Jian ◽  
Wu Zhi-Xi ◽  
Wang Ying-Hai ◽  
Yang Hong-Juan ◽  
...  
Keyword(s):  
Homotopy Analysis Method ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Nonlinear Evolution Equations ◽  
Analysis Method ◽  
Homotopy Analysis

New Applications of the Homotopy Analysis Method

2008 ◽  
Vol 63 (7-8) ◽  
pp. 385-392 ◽  
Author(s):  
Elsayed Abd Elaty Elwakil ◽  
Mohamed Aly Abdou
Keyword(s):  
Exact Solutions ◽  
Homotopy Analysis Method ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Nonlinear Problems ◽  
Nonlinear Evolution Equations ◽  
Analysis Method ◽  
Validity And Reliability ◽  
Analytical Technique ◽  
Homotopy Analysis

An analytical technique, namely the homotopy analysis method (HAM), is applied using a computerized symbolic computation to find the approximate and exact solutions of nonlinear evolution equations arising in mathematical physics. The HAM is a strong and easy to use analytic tool for nonlinear problems and does not need small parameters in the equations. The validity and reliability of the method is tested by application on three nonlinear problems, namely theWhitham-Broer-Kaup equations, coupled Korteweg-de Vries equation and coupled Burger’s equations. Comparisons are made between the results of the HAM with the exact solutions. The method is straightforward and concise, and it can also be applied to other nonlinear evolution equations in physics.

scholarly journals On the Bivariate Spectral Homotopy Analysis Method Approach for Solving Nonlinear Evolution Partial Differential Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
S. S. Motsa
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Homotopy Analysis Method ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Analysis Method ◽  
Partial Differential ◽  
New Approach ◽  
Homotopy Analysis ◽  
Spectral Homotopy Analysis Method

This paper presents a new application of the homotopy analysis method (HAM) for solving evolution equations described in terms of nonlinear partial differential equations (PDEs). The new approach, termed bivariate spectral homotopy analysis method (BISHAM), is based on the use of bivariate Lagrange interpolation in the so-called rule of solution expression of the HAM algorithm. The applicability of the new approach has been demonstrated by application on several examples of nonlinear evolution PDEs, namely, Fisher’s, Burgers-Fisher’s, Burger-Huxley’s, and Fitzhugh-Nagumo’s equations. Comparison with known exact results from literature has been used to confirm accuracy and effectiveness of the proposed method.

Traveling wave solutions for the Couple Boiti-Leon-Pempinelli System by using extended Jacobian elliptic function expansion method

2015 ◽  
Vol 11 (3) ◽  
pp. 3134-3138 ◽  
Author(s):  
Mostafa Khater ◽  
Mahmoud A.E. Abdelrahman
Keyword(s):  
Elliptic Function ◽  
Traveling Wave ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Traveling Wave Solutions ◽  
Expansion Method ◽  
Mathematical Physics ◽  
Nonlinear Evolution Equations ◽  
Jacobian Elliptic Function ◽  
Function Expansion

In this work, an extended Jacobian elliptic function expansion method is pro-posed for constructing the exact solutions of nonlinear evolution equations. The validity and reliability of the method are tested by its applications to the Couple Boiti-Leon-Pempinelli System which plays an important role in mathematical physics.

HOMOTOPY ANALYSIS METHOD TO SOLVE BOUSSINESQ EQUATIONS

2015 ◽  
Vol 10 (3) ◽  
pp. 2825-2833
Author(s):  
Achala Nargund ◽  
R Madhusudhan ◽  
S B Sathyanarayana
Keyword(s):  
Homotopy Analysis Method ◽  
Degrees Of Freedom ◽  
Initial Approximation ◽  
Boussinesq Equations ◽  
Convergent Series ◽  
Boussinesq System ◽  
Analysis Method ◽  
Analytical Technique ◽  
Homotopy Analysis ◽  
Region Of Convergence

In this paper, Homotopy analysis method is applied to the nonlinear coupleddifferential equations of classical Boussinesq system. We have applied Homotopy analysis method (HAM) for the application problems in [1, 2, 3, 4]. We have also plotted Domb-Sykes plot for the region of convergence. We have applied Pade for the HAM series to identify the singularity and reflect it in the graph. The HAM is a analytical technique which is used to solve non-linear problems to generate a convergent series. HAM gives complete freedom to choose the initial approximation of the solution, it is the auxiliary parameter h which gives us a convenient way to guarantee the convergence of homotopy series solution. It seems that moreartificial degrees of freedom implies larger possibility to gain better approximations by HAM.

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