scholarly journals Modified variational iteration and homotopy analysis method for solving variable coefficient variant boussinesq system

2020 ◽  
Vol 8 (1) ◽  
pp. 26-32
Author(s):  
Fadhil H. Easif ◽  
Saad A. Manaa ◽  
Ahmed J. Sabali
Keyword(s):  
Homotopy Analysis Method ◽  
Variable Coefficient ◽  
Boussinesq System ◽  
Analysis Method ◽  
Variational Iteration ◽  
Homotopy Analysis

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.

scholarly journals Application of the Homotopy Analysis Method for Solving the Variable Coefficient KdV-Burgers Equation

2014 ◽  
Vol 2014 ◽  
pp. 1-4 ◽  
Author(s):  
Dianchen Lu ◽  
Jie Liu
Keyword(s):  
Homotopy Analysis Method ◽  
Variable Coefficient ◽  
Burgers Equation ◽  
Approximate Solutions ◽  
Variable Coefficients ◽  
Hyperbolic Function ◽  
Analysis Method ◽  
Function Form ◽  
Transform Method ◽  
Homotopy Analysis

The homotopy analysis method is applied to solve the variable coefficient KdV-Burgers equation. With the aid of generalized elliptic method and Fourier’s transform method, the approximate solutions of double periodic form are obtained. These solutions may be degenerated into the approximate solutions of hyperbolic function form and the approximate solutions of trigonometric function form in the limit cases. The results indicate that this method is efficient for the nonlinear models with the dissipative terms and variable coefficients.

scholarly journals Series Solutions of Time-Fractional PDEs by Homotopy Analysis Method

2008 ◽  
Vol 2008 ◽  
pp. 1-16 ◽  
Author(s):  
O. Abdulaziz ◽  
I. Hashim ◽  
A. Saif
Keyword(s):  
Homotopy Analysis Method ◽  
Variable Coefficient ◽  
Series Solution ◽  
Fractional Derivatives ◽  
Hyperbolic Type ◽  
Series Solutions ◽  
Analysis Method ◽  
Fractional Partial Differential Equations ◽  
Fractional Pdes ◽  
Homotopy Analysis

The homotopy analysis method (HAM) is applied to solve linear and nonlinear fractional partial differential equations (fPDEs). The fractional derivatives are described by Caputo's sense. Series solutions of the fPDEs are obtained. A convergence theorem for the series solution is also given. The test examples, which include a variable coefficient, inhomogeneous and hyperbolic-type equations, demonstrate the capability of HAM for nonlinear fPDEs.

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.

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