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.

scholarly journals On new computations of the fractional epidemic childhood disease model pertaining to the generalized fractional derivative with nonsingular kernel

2021 ◽  
Vol 7 (3) ◽  
pp. 4552-4573
Author(s):  
Saima Rashid ◽  
◽  
Fahd Jarad ◽  
Fatimah S. Bayones ◽  
◽  
...  
Keyword(s):  
Fractional Derivative ◽  
Decomposition Method ◽  
Disease Model ◽  
Adomian Decomposition Method ◽  
Approximate Solutions ◽  
Sir Epidemic Model ◽  
Childhood Diseases ◽  
Sir Epidemic ◽  
Adomian Decomposition ◽  
The Stability

<abstract><p>The present research investigates the Susceptible-Infected-Recovered (SIR) epidemic model of childhood diseases and its complications with the Atangana-Baleanu fractional derivative operator in the Caputo sense (ABC). With the aid of the Elzaki Adomian decomposition method (EADM), the approximate solutions of the aforesaid model are discussed by exerting the Adomian decomposition method. By employing the fixed point postulates and the Picard–Lindelöf approach, the stability, existence, and uniqueness consequences of the model are demonstrated. Furthermore, we illustrate the essential hypothesis for disease control in order to find the role of unaware infectives in the spread of childhood diseases. Besides that, simulation results and graphical illustrations are presented for various fractional-orders. A comparison analysis is shown with the previous findings. It is hoped that ABC fractional derivative and the projected algorithm will provide new venues in futuristic studies to manipulate and analyze several epidemiological models.</p></abstract>

scholarly journals VIM for Solving the Pollution Problem of a System of Lakes

2010 ◽  
Vol 2010 ◽  
pp. 1-6 ◽  
Author(s):  
J. Biazar ◽  
M. Shahbala ◽  
H. Ebrahimi
Keyword(s):  
Differential Equations ◽  
Iteration Method ◽  
Adomian Decomposition Method ◽  
Variational Iteration Method ◽  
Environment Monitoring ◽  
System Of Differential Equations ◽  
Variational Iteration ◽  
Pollution Problem ◽  
Adomian Decomposition ◽  
Different Types

Pollution has become a very serious threat to our environment. Monitoring pollution is the first step toward planning to save the environment. The use of differential equations of monitoring pollution has become possible. In this paper the pollution problem of three lakes with interconnecting channels has been studied. The variational iteration method has been applied to compute an approximate solution of the system of differential equations, governing on the problem. Three different types of input models: sinusoidal, impulse, and step will be considered for monitoring the pollution in the lakes. The results are compared with those obtained by Adomian decomposition method. This comparison reveals that the variational iteration method is easier to be implemented.

HPM AND VIM METHODS FOR FINDING THE EXACT SOLUTIONS OF THE NONLINEAR DISPERSIVE EQUATIONS AND SEVENTH-ORDER SAWADA–KOTERA EQUATION

2009 ◽  
Vol 23 (01) ◽  
pp. 39-52 ◽  
Author(s):  
D. D. GANJI ◽  
N. JAMSHIDI ◽  
Z. Z. GANJI
Keyword(s):  
Iteration Method ◽  
Decomposition Method ◽  
Homotopy Perturbation Method ◽  
Adomian Decomposition Method ◽  
Variational Iteration Method ◽  
Dispersive Equations ◽  
Nonlinear Dispersive Equations ◽  
Homotopy Perturbation ◽  
Adomian Decomposition ◽  
Seventh Order

In this paper, nonlinear dispersive equations and seventh-order Sawada–Kotera equation are solved using homotopy perturbation method (HPM) and variational iteration method (VIM). The results obtained by the proposed methods are then compared with that of Adomian decomposition method (ADM). The comparisons demonstrate that the two obtained solutions are in excellent agreement. The numerical results calculated show that the methods can be accurately implemented to these types of nonlinear equations. The results of HPM and VIM confirm the correctness of those obtained by Adomian decomposition method.

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.

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.

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