scholarly journals A Comparative Approach to the Solution of the Zabolotskaya-Khokhlov Equation by Iteration Methods

2016 ◽  
Vol 2016 ◽  
pp. 1-8
Author(s):  
Saeed Ahmed ◽  
Muhammad Kalim
Keyword(s):  
Approximate Solution ◽  
Adomian Decomposition Method ◽  
Variational Iteration Method ◽  
Nonlinear Pdes ◽  
Comparative Approach ◽  
Comparison Study ◽  
Analysis Method ◽  
Adomian Decomposition ◽  
Homotopy Analysis ◽  
Iteration Methods

We employed different iteration methods like Homotopy Analysis Method (HAM), Adomian Decomposition Method (ADM), and Variational Iteration Method (VIM) to find the approximate solution to the Zabolotskaya-Khokhlov (ZK) equation. Iteration methods are used to solve linear and nonlinear PDEs whose classical methods are either very complex or too limited to apply. A comparison study has been made to see which of these methods converges to the approximate solution rapidly. The result revealed that, amongst these methods, ADM is more effective and simpler tool in its nature which does not require any transformation or linearization.

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.

scholarly journals Convergence of Iterative Methods Applied to Generalized Fisher Equation

2010 ◽  
Vol 2010 ◽  
pp. 1-16
Author(s):  
Sh. Sadigh Behzadi
Keyword(s):  
Homotopy Perturbation Method ◽  
Adomian Decomposition Method ◽  
Variational Iteration Method ◽  
Analysis Method ◽  
Approximation Solution ◽  
Homotopy Perturbation ◽  
Modified Adomian Decomposition Method ◽  
Adomian Decomposition ◽  
Homotopy Analysis ◽  
Generalized Fisher Equation

A generalized Fisher's equation is solved by using the modified Adomian decomposition method (MADM), variational iteration method (VIM), homotopy analysis method (HAM), and modified homotopy perturbation method (MHPM). The approximation solution of this equation is calculated in the form of series whose components are computed easily. The existence, uniqueness, and convergence of the proposed methods are proved. Numerical example is studied to demonstrate the accuracy of the present methods.

Analytical Solutions for Rumor Spreading Dynamical Model in a Social Network

2015 ◽  
Vol 4 (1) ◽  
Author(s):  
R. Fallahpour ◽  
S. Chakouvari ◽  
H. Askari
Keyword(s):  
Decomposition Method ◽  
Adomian Decomposition Method ◽  
Variational Iteration Method ◽  
Analysis Method ◽  
Rumor Spreading ◽  
Spreading Model ◽  
Adomian Decomposition ◽  
Homotopy Analysis ◽  
The Subject ◽  
Forgetting Mechanism

AbstractIn this paper, Laplace Adomian decomposition method is utilized for evaluating of spreading model of rumor. Firstly, a succinct review is constructed on the subject of using analytical methods such as Adomian decomposion method, Variational iteration method and Homotopy Analysis method for epidemic models and biomathematics. In continue a spreading model of rumor with consideration of forgetting mechanism is assumed and subsequently LADM is exerted for solving of it. By means of the aforementioned method, a general solution is achieved for this problem which can be readily employed for assessing of rumor model without exerting any computer program. In addition, obtained consequences for this problem are discussed for different cases and parameters. Furthermore, it is shown the method is so straightforward and fruitful for analyzing equations which have complicated terms same as rumor model. By employing numerical methods, it is revealed LADM is so powerful and accurate for eliciting solutions of this model. Eventually, it is concluded that this method is so appropriate for this problem and it can provide researchers a very powerful vehicle for scrutinizing rumor models in diverse kinds of social networks such as Facebook, YouTube, Flickr, LinkedIn and Tuitor.

scholarly journals The Time-Fractional Coupled-Korteweg-de-Vries Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Abdon Atangana ◽  
Aydin Secer
Keyword(s):  
Decomposition Method ◽  
Homotopy Perturbation Method ◽  
Numerical Solutions ◽  
Adomian Decomposition Method ◽  
Variational Iteration Method ◽  
Analytical Technique ◽  
Adomian Decomposition ◽  
Homotopy Decomposition ◽  
De Vries ◽  
Homotopy Analysis

We put into practice a relatively new analytical technique, the homotopy decomposition method, for solving the nonlinear fractional coupled-Korteweg-de-Vries equations. Numerical solutions are given, and some properties exhibit reasonable dependence on the fractional-order derivatives’ values. The fractional derivatives are described in the Caputo sense. The reliability of HDM and the reduction in computations give HDM a wider applicability. In addition, the calculations involved in HDM are very simple and straightforward. It is demonstrated that HDM is a powerful and efficient tool for FPDEs. It was also demonstrated that HDM is more efficient than the adomian decomposition method (ADM), variational iteration method (VIM), homotopy analysis method (HAM), and homotopy perturbation method (HPM).

scholarly journals Solution of the space-fractional telegraph equations by using HAM

2015 ◽  
Vol 37 ◽  
pp. 320
Author(s):  
Mehdi Abedi-Varaki ◽  
Shahram Rajabi ◽  
Vahid Ghorbani ◽  
Farzad Hosseinzadeh
Keyword(s):  
Homotopy Analysis Method ◽  
Decomposition Method ◽  
Adomian Decomposition Method ◽  
Approximate Solutions ◽  
Analysis Method ◽  
Space And Time ◽  
Adomian Decomposition ◽  
Homotopy Analysis ◽  
Telegraph Equations

In this study by using the Homotopy Analysis Method (HAM) obtained approximate solutions for the space and time-fractional telegraph equations. In Caputo sense (Yildirim, 2010)these equations considered. Examples are solved and the obtained results show to be more accurate than Adomian Decomposition Method (ADM) and are more efficient and commodious.

scholarly journals Numerical Solution of Seventh-Order Boundary Value Problems by a Novel Method

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
Mustafa Inc ◽  
Ali Akgül
Keyword(s):  
Boundary Value Problems ◽  
Reproducing Kernel ◽  
Reproducing Kernel Hilbert Space ◽  
Boundary Value ◽  
Adomian Decomposition Method ◽  
Variational Iteration Method ◽  
Variation Of Parameters ◽  
Adomian Decomposition ◽  
Homotopy Analysis ◽  
Seventh Order

We demonstrate the efficiency of reproducing kernel Hilbert space method on the seventh-order boundary value problems satisfying boundary conditions. These results have been compared with the results that are obtained by variational iteration method (VIM), homotopy perturbation method (HPM), Adomian decomposition method (ADM), variation of parameters method (VPM), and homotopy analysis method (HAM). Obtained results show that our method is very effective.

scholarly journals A NOVEL ANALYTIC METHOD FOR SOLVING LINEAR AND NON-LINEAR TELEGRAPH EQUATION

2020 ◽  
Vol 17 (35) ◽  
pp. 536-548
Author(s):  
Ahmed K. Al-JABERI ◽  
Ehsan M. HAMEED ◽  
Mohammed S. Abdul-WAHAB
Keyword(s):  
Nonlinear Partial Differential Equations ◽  
Adomian Decomposition Method ◽  
Telegraph Equation ◽  
Analytic Method ◽  
Analysis Method ◽  
Adomian Decomposition ◽  
Homotopy Analysis ◽  
Exact Resolution ◽  
Fundamental Equations ◽  
Better Than

The modeling of many phenomena in various fields such as mathematics, physics, chemistry, engineering, biology, and astronomy is done by the nonlinear partial differential equations (PDE). The hyperbolic telegraph equation is one of them, where it describes the vibrations of structures (e.g., buildings, beams, and machines) and are the basis for fundamental equations of atomic physics. There are several analytical and numerical methods are used to solve the telegraph equation. An analytical solution considers framing the problem in a well-understood form and calculating the exact resolution. It also helps to understand the answers to the problem in terms of accuracy and convergence. These analytic methods have limitations with accuracy and convergence. Therefore, a novel analytic approximate method is proposed to deal with constraints in this paper. This method uses the Taylors' series in its derivation. The proposed method has used for solving the secondorder, hyperbolic equation (Telegraph equation) with the initial condition. Three examples have presented to check the effectiveness, accuracy, and convergence of the method. The solutions of the proposed method also compared with those obtained by the Adomian decomposition method (ADM), and the Homotopy analysis method (HAM). The technique is easy to implement and produces accurate results. In particular, these results display that the proposed method is efficient and better than the other methods in terms of accuracy and convergence.

APPLICATION OF HOMOTOPY PERTURBATION AND VARIATIONAL ITERATION METHODS TO SIR EPIDEMIC MODEL

2011 ◽  
Vol 11 (01) ◽  
pp. 149-161 ◽  
Author(s):  
ABDOUL R. GHOTBI ◽  
A. BARARI ◽  
M. OMIDVAR ◽  
G. DOMAIRRY
Keyword(s):  
Vaccine Coverage ◽  
Adomian Decomposition Method ◽  
Variational Iteration Method ◽  
Sir Epidemic Model ◽  
Childhood Diseases ◽  
Variational Iteration ◽  
Homotopy Perturbation ◽  
Sir Epidemic ◽  
Adomian Decomposition ◽  
Iteration Methods

Children born are susceptible to various diseases such as mumps and chicken pox. These diseases are the most common form of infectious diseases. In recent years, scientists have been trying to devise strategies to fight against these diseases. Since vaccination is considered the most effective strategy against childhood diseases, the development of the framework that would predict the optimal vaccine coverage level needed to prevent the spread of diseases is crucial. The SIR model is a standard compartmental model that has been used to describe many epidemiological diseases. In this article, two methods namely homotopy perturbation method (HPM) and variational iteration method (VIM) are employed to compute an approximation to the solution of nonlinear system of differential equations governing the problem. The obtained results are compared with those obtained by Adomian decomposition method (ADM). This research reveals that although the obtained results are the same, HPM and VIM are much more robust, more convenient and efficient in comparison to ADM.

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