Comparison Homotopy Analysis Method and Variational Iteration Method for KdV Equation

2008 ◽  
Author(s):  
H. Jafari ◽  
Mohammad A. Firouzjaei ◽  
Mohammad Saidy
Keyword(s):  
Homotopy Analysis Method ◽  
Iteration Method ◽  
Kdv Equation ◽  
Variational Iteration Method ◽  
Analysis Method ◽  
Variational Iteration ◽  
Homotopy Analysis

scholarly journals The Essence of the Variational Iteration Method and the Homotopy Analysis Method Is Different

2021 ◽  
Vol 64 (1) ◽  
pp. 47-63
Author(s):  
Mustafa Turkyilmazoglu ◽  
Keyword(s):  
Homotopy Analysis Method ◽  
Iteration Method ◽  
Mathematical Proof ◽  
Variational Iteration Method ◽  
Analysis Method ◽  
Variational Iteration ◽  
Homotopy Analysis ◽  
Published Paper ◽  
Special Case ◽  
Rigorous Mathematical Proof

The recently published paper “The variational iteration method is a special case of the homotopy analysis method” by Robert A. Van Gorder [1], weakly pointed out that the variational iteration method and all of its optimal analogues are specific cases of the more general homotopy analysis method. This assertion was not truly supported by a rigorous mathematical proof, nor by an accessible example from the attributed papers. In this brief, we refute the author's claim by supplementing three simple examples, which do not indicate that the variational iteration method is a special case of the homotopy analysis method. This is justified by a Theorem to compute the rate of convergence of both methods.

scholarly journals Constructing analytic solutions on the Tricomi equation

Open Physics ◽  
2018 ◽  
Vol 16 (1) ◽  
pp. 143-148 ◽  
Author(s):  
Emran Khoshrouye Ghiasi ◽  
Reza Saleh
Keyword(s):  
Homotopy Analysis Method ◽  
Series Solution ◽  
Variational Iteration Method ◽  
Approximate Solutions ◽  
Analytic Solutions ◽  
Analysis Method ◽  
Auxiliary Parameter ◽  
Tricomi Equation ◽  
Homotopy Analysis ◽  
Optimal Values

AbstractIn this paper, homotopy analysis method (HAM) and variational iteration method (VIM) are utilized to derive the approximate solutions of the Tricomi equation. Afterwards, the HAM is optimized to accelerate the convergence of the series solution by minimizing its square residual error at any order of the approximation. It is found that effect of the optimal values of auxiliary parameter on the convergence of the series solution is not negligible. Furthermore, the present results are found to agree well with those obtained through a closed-form equation available in the literature. To conclude, it is seen that the two are effective to achieve the solution of the partial differential equations.

scholarly journals A New Homotopy Analysis Method for Approximating the Analytic Solution of KdV Equation

2014 ◽  
Vol 7 (4) ◽  
pp. 826-831
Author(s):  
Vahid Barati ◽  
Mojtaba Nazari ◽  
Vincent Daniel David ◽  
Zainal Abdul Aziz
Keyword(s):  
Homotopy Analysis Method ◽  
Analytic Solution ◽  
Kdv Equation ◽  
Analysis Method ◽  
Homotopy Analysis

Application of the Homotopy Analysis Method for Solving Equal-Width Wave and Modified Equal-Width Wave Equations

2009 ◽  
Vol 64 (11) ◽  
pp. 685-690 ◽  
Author(s):  
Esmail Babolian ◽  
Jamshid Saeidian ◽  
Mahmood Paripour
Keyword(s):  
Homotopy Analysis Method ◽  
Wave Equations ◽  
Homotopy Perturbation Method ◽  
Adomian Decomposition Method ◽  
Variational Iteration Method ◽  
Analytic Method ◽  
Analysis Method ◽  
Homotopy Perturbation ◽  
Adomian Decomposition ◽  
Homotopy Analysis

Although the homotopy analysis method (HAM) is, by now, a well-known analytic method for handling functional equations, there is no general proof of its applicability to all kinds of equations. In this paper, by using this method to solve equal-width wave (EW) and modified equal-width wave (MEW) equations, we have made a new contribution to this field of research. Our goal is to emphasize on two points: one is the efficiency of HAM in handling these important family of equations and its superiority over other analytic methods like homotopy perturbation method (HPM), variational iteration method (VIM), and Adomian decomposition method (ADM). The other point is that although the considered two equations have different nonlinear terms, we have used the same auxiliary elements to solve them.

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