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Bougoffa L.,Al Imam University | Rach R.C.,316 S. Maple St. | El-Manouni S.,Al Imam University
International Journal of Computer Mathematics | Year: 2013

In this article, we study some fundamental results concerning the convergence of the Adomian decomposition method (ADM) for an abstract Cauchy problem of a system of first-order nonlinear differential equations. Under certain conditions, we obtain upper estimates for the norm of solutions of this system. We also obtain results about the error estimates for the approximate solutions by the ADM and discuss their applications. © 2013 Taylor & Francis. Source


Duan J.-S.,Shanghai Institute of Technology | Duan J.-S.,Zhaoqing University | Rach R.,316 S. Maple St. | Wazwaz A.-M.,Saint Xavier University
CMES - Computer Modeling in Engineering and Sciences | Year: 2013

In this paper, we propose a new modification of the Adomian decomposition method for solution of higher-order nonlinear initial value problems with variable system coefficients and solutions of systems of coupled nonlinear initial value problems. We consider various algorithms for the Adomian decomposition series and the series of Adomian polynomials to calculate the solutions of canonical first- and second-order nonlinear initial value problems in order to derive a systematic algorithm for the general case of higher-order nonlinear initial value problems and systems of coupled higher-order nonlinear initial value problems. Our new modified recursion scheme is designed to decelerate the Adomian decomposition series so as to always calculate the solution's Taylor expansion series using easy-to-integrate terms. The corresponding nonlinear recurrence relations for the solution coefficients are deduced. Next we consider convergence acceleration and error analysis for the sequence of solution approximations. Multistage decomposition and numeric algorithms are designed and we debut efficient MATHEMATICA routines PSSOL and NSOL that implement our new algorithms. Finally we investigate several expository examples in order to demonstrate the rapid convergence of our new approach. Copyright © 2013 Tech Science Press. Source


Wazwaz A.-M.,Saint Xavier University | Rach R.,316 S. Maple St. | Duan J.-S.,Zhaoqing University | Duan J.-S.,Shanghai Institute of Technology
Central European Journal of Engineering | Year: 2013

In this paper, we use the systematic modified Adomian decomposition method (ADM) and the phenomenon of the self-canceling "noise" terms for solving nonlinear weakly-singular Volterra, Fredholm, and Volterra-Fredholm integral equations. We show that the proposed approach minimizes the computation, when compared with other conventional schemes. Our results are validated by investigating several examples. © Versita sp. z o.o. Source


Bougoffa L.,Islamic University | Rach R.C.,316 S. Maple St.
Romanian Journal of Physics | Year: 2015

In this paper, we propose a direct method to obtain an exact solution of the Thomas-Fermi equation. An approximate analytic solution is also obtained, which demonstrates to be quite accurate by comparison with the Sommerfeld, numerical and variational solutions. © 2015, Editura Academiei Romane. All rights reserved. Source


Wazwaz A.-M.,Saint Xavier University | Rach R.,316 S. Maple St. | Bougoffa L.,Islamic University | Duan J.-S.,Shanghai Institute of Technology
CMES - Computer Modeling in Engineering and Sciences | Year: 2014

In this paper, we construct the Lane-Emden-Fowler type equations of higher orders. We study the linear and the nonlinear Lane-Emden-Fowler type equations of the third and fourth orders, where other forms can be treated in a similar manner. We use the systematic Adomian decomposition method to handle these types of equations with specified initial conditions. We confirm that the Adomian decomposition method provides an efficient algorithm for exact and approximate analytic solutions of these equations. We corroborate this study by investigating several numerical examples that emphasize initial value problems. Copyright © 2014 Tech Science Press Source

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