scholarly journals Comparison between Homotopy Analysis Method (HAM) and Variational Iteration Method (VIM) in Solving the Nonlinear Wave Propagation Equations in Shallow Water

2019 ◽  
Vol 2 (4) ◽  
pp. 37-46
Author(s):  
Mohsen Soltani ◽  
Rouhollah Amirabadi ◽  
◽  
Keyword(s):  
Wave Propagation ◽  
Shallow Water ◽  
Nonlinear Wave ◽  
Homotopy Analysis Method ◽  
Iteration Method ◽  
Variational Iteration Method ◽  
Analysis Method ◽  
Nonlinear Wave Propagation ◽  
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 Approximate Traveling Wave Solutions of Coupled Whitham-Broer-Kaup Shallow Water Equations by Homotopy Analysis Method

2008 ◽  
Vol 2008 ◽  
pp. 1-8 ◽  
Author(s):  
M. M. Rashidi ◽  
D. D. Ganji ◽  
S. Dinarvand
Keyword(s):  
Exact Solutions ◽  
Shallow Water ◽  
Homotopy Analysis Method ◽  
Shallow Water Equations ◽  
Traveling Wave ◽  
Nonlinear Partial Differential Equations ◽  
Traveling Wave Solutions ◽  
Analysis Method ◽  
Wave Solutions ◽  
Homotopy Analysis

The homotopy analysis method (HAM) is applied to obtain the approximate traveling wave solutions of the coupled Whitham-Broer-Kaup (WBK) equations in shallow water. Comparisons are made between the results of the proposed method and exact solutions. The results show that the homotopy analysis method is an attractive method in solving the systems of nonlinear partial differential equations.

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