316 S. Maple St.

Hartford, MI, United States

316 S. Maple St.

Hartford, MI, United States

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Aly E.H.,King Abdulaziz University | Ebaid A.,University of Tabuk | Rach R.,316 S. Maple St.
Computers and Mathematics with Applications | Year: 2012

A new straightforward approach for solving ordinary and partial second-order boundary value problems with Neumann boundary conditions is introduced in this research. This approach depends mainly on the Adomian decomposition method with a new definition of the differential operator and its inverse, which has been modified for Neumann boundary conditions. The effectiveness of the proposed approach is verified by several linear and nonlinear examples. © 2012 Published by Elsevier Ltd.


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.


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.


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


Duan J.-S.,Shanghai Institute of Technology | Rach R.,316 S. Maple St. | Wazwaz A.-M.,Saint Xavier University
Journal of Mathematical Chemistry | Year: 2015

In this paper, we examine a system of two coupled nonlinear differential equations that relates the concentrations of carbon dioxide CO(Formula presented.) and phenyl glycidyl ether in solution. This system is subject to a set of Dirichlet boundary conditions and a mixed set of Neumann and Dirichlet boundary conditions. We apply the Adomian decomposition method combined with the Duan–Rach modified recursion scheme to analytically treat this system of coupled nonlinear boundary value problems. The rapid convergence of our analytic approximate solutions is demonstrated by graphs of the objective error analysis instead of comparison to an alternate solution technique alone. The Adomian decomposition method yields a rapidly convergent, easily computable, and readily verifiable sequence of analytic approximate solutions that is suitable for numerical parametric simulations. Thus our sequence of approximate solutions are shown to identically satisfy the original set of model equations as closely as we please. © 2014, Springer International Publishing Switzerland.


Rach R.,316 S. Maple St. | Duan J.-S.,Zhaoqing University | Duan J.-S.,Shanghai Institute of Technology | Wazwaz A.-M.,Saint Xavier University
Journal of Mathematical Chemistry | Year: 2014

In this paper, we consider the coupled Lane-Emden boundary value problems in catalytic diffusion reactions by the Adomian decomposition method. First, we utilize systems of Volterra integral forms of the Lane-Emden equations and derive the modified recursion scheme for the components of the decomposition series solutions. The numerical results display that the Adomian decomposition method gives reliable algorithm for analytic approximate solutions of these systems. The error analysis of the sequence of the analytic approximate solutions can be performed by using the error remainder functions and the maximal error remainder parameters, which demonstrate an approximate exponential rate of convergence. © 2013 Springer Science+Business Media New York.


Wazwaz A.-M.,Saint Xavier University | Rach R.,316 S. Maple St. | Duan J.-S.,Shanghai Institute of Technology
Match | Year: 2016

In this paper, we will extend our work in 1. on a system of two coupled nonlinear Lane-Emden differential equations, that governs the concentrations of oxygen and the carbon substrate. This mathematical model describes substrate removal, oxygen utilization and excess sludge production within a microbial floc particle as surrounded by a biodegradable substrate [1-5.. We had previously applied the Adomian decomposition method (ADM) combined with the Duan-Rach modified recursion scheme in [1., and for a comparative study, we will apply the variational iteration method (VIM) for analytical approximations of the concentrations of oxygen and the carbon substrate. The variational iteration method, as the Adomian decomposition method, yields a rapidly convergent, easily computable and readily verifiable sequence of analytic approximations that are convenient for parametric simulations.


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.


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.

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