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 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.

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.

scholarly journals Solving Heat and Wave-Like Equations Using He's Polynomials

2009 ◽  
Vol 2009 ◽  
pp. 1-12 ◽  
Author(s):  
Syed Tauseef Mohyud-Din
Keyword(s):  
Perturbation Method ◽  
Decomposition Method ◽  
Homotopy Perturbation Method ◽  
Iterative Scheme ◽  
Nonlinear Problems ◽  
Homotopy Perturbation ◽  
He’S Polynomials ◽  
Clear Advantage ◽  
Suggested Technique ◽  
Adomian’S Polynomials

We use He's polynomials which are calculated form homotopy perturbation method (HPM) for solving heat and wave-like equations. The proposed iterative scheme finds the solution without any discretization, linearization, or restrictive assumptions. Several examples are given to verify the reliability and efficiency of the method. The fact that suggested technique solves nonlinear problems without using Adomian's polynomials is a clear advantage of this algorithm over the decomposition method.

scholarly journals Solution of Temperature Distribution in a Radiating Fin Using Homotopy Perturbation Method

2009 ◽  
Vol 2009 ◽  
pp. 1-8 ◽  
Author(s):  
M. J. Hosseini ◽  
M. Gorji ◽  
M. Ghanbarpour
Keyword(s):  
Perturbation Method ◽  
Decomposition Method ◽  
Homotopy Perturbation Method ◽  
Adomian Decomposition Method ◽  
Analytic Approximation ◽  
Enhance Heat Transfer ◽  
Homotopy Perturbation ◽  
Adomian Decomposition ◽  
Extended Surfaces ◽  
Comparison Of The Results

Radiating extended surfaces are widely used to enhance heat transfer between primary surface and the environment. The present paper applies the homotopy perturbation to obtain analytic approximation of distribution of temperature in heat fin radiating, which is compared with the results obtained by Adomian decomposition method (ADM). Comparison of the results obtained by the method reveals that homotopy perturbation method (HPM) is more effective and easy to use.

scholarly journals Homotopy Perturbation Method for Fractional Gas Dynamics Equation Using Sumudu Transform

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Jagdev Singh ◽  
Devendra Kumar ◽  
A. Kılıçman
Keyword(s):  
Perturbation Method ◽  
Gas Dynamics ◽  
Homotopy Perturbation Method ◽  
Numerical Solutions ◽  
Adomian Decomposition Method ◽  
Homotopy Perturbation ◽  
He’S Polynomials ◽  
Adomian Decomposition ◽  
Sumudu Transform ◽  
Gas Dynamics Equation

A user friendly algorithm based on new homotopy perturbation Sumudu transform method (HPSTM) is proposed to solve nonlinear fractional gas dynamics equation. The fractional derivative is considered in the Caputo sense. Further, the same problem is solved by Adomian decomposition method (ADM). The results obtained by the two methods are in agreement and hence this technique may be considered an alternative and efficient method for finding approximate solutions of both linear and nonlinear fractional differential equations. The HPSTM is a combined form of Sumudu transform, homotopy perturbation method, and He’s polynomials. The nonlinear terms can be easily handled by the use of He’s polynomials. The numerical solutions obtained by the proposed method show that the approach is easy to implement and computationally very attractive.

scholarly journals Homotopy Perturbation Method to Obtain Positive Solutions of Nonlinear Boundary Value Problems of Fractional Order

2014 ◽  
Vol 2014 ◽  
pp. 1-5 ◽  
Author(s):  
Hossein Jafari ◽  
Khadijeh Bagherian ◽  
Seithuti P. Moshokoa
Keyword(s):  
Perturbation Method ◽  
Decomposition Method ◽  
Homotopy Perturbation Method ◽  
Nonlinear Boundary ◽  
Boundary Value ◽  
Adomian Decomposition Method ◽  
Type Problem ◽  
Nonlinear Boundary Value Problems ◽  
Homotopy Perturbation ◽  
Adomian Decomposition

We use the homotopy perturbation method for solving the fractional nonlinear two-point boundary value problem. The obtained results by the homotopy perturbation method are then compared with the Adomian decomposition method. We solve the fractional Bratu-type problem as an illustrative example.

Sign in / Sign up

Export Citation Format

Share Document