scholarly journals Convergence of the New Iterative Method

2011 ◽  
Vol 2011 ◽  
pp. 1-10 ◽  
Author(s):  
Sachin Bhalekar ◽  
Varsha Daftardar-Gejji
Keyword(s):  
Iterative Method ◽  
Functional Equations ◽  
Decomposition Method ◽  
Adomian Decomposition Method ◽  
Efficient Technique ◽  
Nonlinear Functional ◽  
Adomian Decomposition ◽  
New Iterative Method ◽  
Sufficiency Conditions

A new iterative method introduced by Daftardar-Gejji and Jafari (2006) (DJ Method) is an efficient technique to solve nonlinear functional equations. In the present paper, sufficiency conditions for convergence of DJM have been presented. Further equivalence of DJM and Adomian decomposition method is established.

scholarly journals Solving Fractional-Order Logistic Equation Using a New Iterative Method

2012 ◽  
Vol 2012 ◽  
pp. 1-12 ◽  
Author(s):  
Sachin Bhalekar ◽  
Varsha Daftardar-Gejji
Keyword(s):  
Iterative Method ◽  
Perturbation Method ◽  
Decomposition Method ◽  
Homotopy Perturbation Method ◽  
Adomian Decomposition Method ◽  
Logistic Equation ◽  
Series Solutions ◽  
Homotopy Perturbation ◽  
Adomian Decomposition ◽  
New Iterative Method

A fractional version of logistic equation is solved using new iterative method proposed by Daftardar-Gejji and Jafari (2006). Convergence of the series solutions obtained is discussed. The solutions obtained are compared with Adomian decomposition method and homotopy perturbation method.

scholarly journals SOR-Like New Iterative Method for Solving the Epidemic Model and the Prey and Predator Problem

2020 ◽  
Vol 2020 ◽  
pp. 1-13
Author(s):  
Atika Radid ◽  
Karim Rhofir
Keyword(s):  
Iterative Method ◽  
Adomian Decomposition Method ◽  
Time Interval ◽  
Time Intervals ◽  
Nonlinear Functional ◽  
Adomian Decomposition ◽  
Wide Range ◽  
Functional Problem ◽  
New Iterative Method ◽  
Variational Iterative Method

Our aim in this paper is to propose an SOR-like new iterative method by introducing a relaxation parameter ω to improve the new iterative method proposed by Daftardar-Gejji and Jafari (NIM) [J. Math. Anal. Appl. 316 (2006) 753–763] in order to solve two problems. The first one is the problem of the spread of a nonfatal disease in a population which is assumed to have constant size over the period of the epidemic, and the other one is the problem of prey and predator. The proposed method is not limited to these two problems but can be applicable to a wide range of systems of nonlinear functional problem. The results, for different values of ω, show that we found some known methods and our method compared to methods using the calculation of special polynomials and derivatives like the Adomian decomposition method (ADM), the calculation of the Lagrange multiplier as in the variational iterative method (VIM), or the construction of a homotopy as in the homotopy perturbation method (HPM) has several advantages, such as very effective and very simple to implement. Unfortunately, these methods do not guarantee a valid approximation in large time interval. To overcome this, we applied our method for approximating the solution of the problems in a sequence of time intervals as a multistage approach. Some numerical results are presented with plots according to the parameter ω.

scholarly journals A Comparison between Adomian’s Polynomials and He’s Polynomials for Nonlinear Functional Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-4 ◽  
Author(s):  
Hossein Jafari ◽  
Saber Ghasempoor ◽  
Chaudry Masood Khalique
Keyword(s):  
Functional Equations ◽  
Decomposition Method ◽  
Homotopy Perturbation Method ◽  
Adomian Decomposition Method ◽  
Adomian Polynomials ◽  
Nonlinear Functional ◽  
Homotopy Perturbation ◽  
He’S Polynomials ◽  
Adomian Decomposition ◽  
Adomian’S Polynomials

We will compare the standard Adomian decomposition method and the homotopy perturbation method applied to obtain the solution of nonlinear functional equations. We prove analytically that the two methods are equivalent for solving nonlinear functional equations. In Ghorbani (2009), Ghorbani presented a new definition which he called as He’s polynomials. In this paper, we also show that He’s polynomials are only the Adomian polynomials.

scholarly journals Numerical Simulation of Cubic-Quartic Optical Solitons with Perturbed Fokas–Lenells Equation Using Improved Adomian Decomposition Algorithm

Mathematics ◽  
2022 ◽  
Vol 10 (1) ◽  
pp. 138
Author(s):  
Alyaa A. Al-Qarni ◽  
Huda O. Bakodah ◽  
Aisha A. Alshaery ◽  
Anjan Biswas ◽  
Yakup Yıldırım ◽  
...  
Keyword(s):  
Numerical Simulation ◽  
Iterative Method ◽  
Decomposition Method ◽  
Adomian Decomposition Method ◽  
Optical Solitons ◽  
Numerical Results ◽  
Decomposition Algorithm ◽  
Adomian Decomposition ◽  
High Level ◽  
Do So

The current manuscript displays elegant numerical results for cubic-quartic optical solitons associated with the perturbed Fokas–Lenells equations. To do so, we devise a generalized iterative method for the model using the improved Adomian decomposition method (ADM) and further seek validation from certain well-known results in the literature. As proven, the proposed scheme is efficient and possess a high level of accuracy.

Numerical Solution of Time-Fractional Klein–Gordon Equation by Using the Decomposition Methods

2016 ◽  
Vol 11 (4) ◽  
Author(s):  
Hossein Jafari
Keyword(s):  
Iterative Method ◽  
Numerical Solution ◽  
Decomposition Method ◽  
Gordon Equation ◽  
Adomian Decomposition Method ◽  
Type Equation ◽  
Decomposition Methods ◽  
Adomian Decomposition ◽  
Klein Gordon ◽  
Klein Gordon Equation

In this paper, we apply two decomposition methods, the Adomian decomposition method (ADM) and a well-established iterative method, to solve time-fractional Klein–Gordon type equation. We compare these methods and discuss the convergence of them. The obtained results reveal that these methods are very accurate and effective.

COMPARATIVE ANALYSIS OF NATURAL TRANSFORM DECOMPOSITION METHOD AND NEW ITERATIVE METHOD FOR FRACTIONAL FOAM DRAINAGE PROBLEM AND FRACTIONAL ORDER MODIFIED REGULARIZED LONG-WAVE EQUATION

Fractals ◽  
2020 ◽  
Vol 28 (07) ◽  
pp. 2050124
Author(s):  
RASHID NAWAZ ◽  
NASIR ALI ◽  
LAIQ ZADA ◽  
ZAHIR SHAH ◽  
ASIFA TASSADDIQ ◽  
...  
Keyword(s):  
Wave Equation ◽  
Iterative Method ◽  
Fractional Order ◽  
Decomposition Method ◽  
Long Wave ◽  
Adomian Decomposition ◽  
Regularized Long Wave Equation ◽  
Drainage Problem ◽  
New Iterative Method ◽  
Foam Drainage

In this paper, a comparative study of natural transform decomposition method and new iterative method is presented. The proposed methods are tested upon nonlinear fractional order foam drainage problem and fractional order modified regularized long-wave equation. The solutions obtained by the proposed methods have been compared with the classical solutions and the solution obtained by Adomian decomposition method. Furthermore, the efficiency and reliability of the proposed methods are shown with the help of numerical and graphical results. The fractional order derivatives are defined in Caputo’s sense whose order belongs to the closed interval [0,1]. The results reveal that the methods are quickly convergent and yield encouraging results.

scholarly journals New Approximate Analytical Solutions of the Falkner-Skan Equation

2012 ◽  
Vol 2012 ◽  
pp. 1-12 ◽  
Author(s):  
Beong In Yun
Keyword(s):  
Iterative Method ◽  
Analytical Solutions ◽  
Decomposition Method ◽  
Adomian Decomposition Method ◽  
Approximate Solutions ◽  
Correction Method ◽  
Approximate Analytical Solutions ◽  
Adomian Decomposition ◽  
Number Of Iterations

We propose an iterative method for solving the Falkner-Skan equation. The method provides approximate analytical solutions which consist of coefficients of the previous iterate solution. By some examples, we show that the presented method with a small number of iterations is competitive with the existing method such as Adomian decomposition method. Furthermore, to improve the accuracy of the proposed method, we suggest an efficient correction method. In practice, for some examples one can observe that the correction method results in highly improved approximate solutions.

scholarly journals Approximate Solutions for Systems of Volterra Integro-differential Equations Using Laplace –Adomian Method

2020 ◽  
pp. 2655-2662
Author(s):  
Firas S. Ahmed
Keyword(s):  
Exact Solution ◽  
Differential Equations ◽  
Iterative Method ◽  
Decomposition Method ◽  
Adomian Decomposition Method ◽  
Approximate Solutions ◽  
Adomian Decomposition ◽  
Adomian Method ◽  
The Given

Some modified techniques are used in this article in order to have approximate solutions for systems of Volterra integro-differential equations. The suggested techniques are the so called Laplace-Adomian decomposition method and Laplace iterative method. The proposed methods are robust and accurate as can be seen from the given illustrative examples and from the comparison that are made with the exact solution.

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