The George Adomian Center for Applied Mathematics

South Rockwood, MI, United States

The George Adomian Center for Applied Mathematics

South Rockwood, MI, United States
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Ebaid A.,University of Tabuk | Rach R.,The George Adomian Center for Applied Mathematics | El-Zahar E.,University of the Humanities | El-Zahar E.,Menoufia University
Acta Astronautica | Year: 2017

In this paper, the Adomian decomposition method (ADM) is proposed to solve the hyperbolic Kepler equation which is often used to describe the eccentric anomaly of a comet of extrasolar origin in its hyperbolic trajectory past the Sun. A convenient method is therefore needed to solve this equation to accurately determine the radial distance and/or the Cartesian coordinates of the comet. It has been shown that Adomian's series using a few terms are sufficient to achieve extremely accurate numerical results even for much higher values of eccentricity than those in the literature. Besides, an exceptionally rapid rate of convergence of the sequence of the obtained approximate solutions has been demonstrated. Such approximate solutions possess the odd property in the mean anomaly which are illustrated through several plots. Moreover, the absolute remainder error, using only three components of Adomian's solution decreases across a specified domain, approaches zero as the eccentric anomaly tends to infinity. Also, the absolute remainder error decreases by increasing the number of components of the Adomian decomposition series. In view of the obtained results, the present method may be the most effective approach to treat the hyperbolic Kepler equation. © 2017 IAA

Bougoffa L.,Islamic University | Mziou S.,Islamic University | Rach R.C.,The George Adomian Center for Applied Mathematics
International Journal of Computational Methods in Engineering Science and Mechanics | Year: 2017

This article shows that the well-known nonlinear boundary value problem of the differential equation for temperature distribution of convective straight fins with temperature-dependent thermal conductivity is exactly solvable in an implicit form. Furthermore, an exact solution in an explicit form is derived. Also, an accurate analytic solution (series solution) is obtained by a new variation of the Adomian decomposition method. © 2017 Taylor & Francis Group, LLC

Rach R.C.,The George Adomian Center for Applied Mathematics | Bougoffa L.,Islamic University | Khanfer A.,Islamic University
Applied Mathematics and Information Sciences | Year: 2015

In this paper, we have considered the nonlinear coupled boundary-layer equations that describe the problem of injection or extraction of fluid along the surface of an inclined wall embedded in a saturated porous medium. We obtain very accurate approximate analytic solutions in closed-form by a direct method for the special cases λ = 0 and λ = 1. Furthermore, an accurate analytic series solution is obtained by the modified Adomian decomposition method for the two limiting cases of free and forced convection by setting Rak/Pex = 0 and Rak/Pex 6 ≠ 0, respectively. © 2015 NSP Natural Sciences Publishing Cor.

Fatoorehchi H.,University of Tehran | Rach R.,The George Adomian Center for Applied Mathematics | Abolghasemi H.,University of Tehran
Romanian Journal of Physics | Year: 2015

We develop a family of improved iterative formulas for computation of matrix inverses. Towards this purpose, we first consider a general class of scalar Newtontype root-finders, which have been improved by incorporating the Adomian decomposition method. Subsequently, we extend such scalar root-finders to their respective matrix analogs by means of an innovative computer screening program. Our formulas are shown, through numerical experiments, to surpass five well-known iterative schemes taken from the literature, both in terms of the CPU elapsed time and iteration count. According to these results, one of our new formulas saves more than 13% in CPU time in the worst case, when compared with all five previous iterative methods. In addition, the convergence order of a simple member of the family of our matrix inversion formulas was proven to be at least three to better elucidate our new approach. © 2015, Editura Academiei Romane. All rights reserved.

Fatoorehchi H.,University of Tehran | Fatoorehchi H.,Iran Liquefied Natural Gas Co. | Rach R.,The George Adomian Center for Applied Mathematics | Tavakoli O.,University of Tehran | Abolghasemi H.,University of Tehran
Chemical Engineering Communications | Year: 2015

In this study, an efficient iterative algorithm is devised to handle a nonlinear equation arising in estimation of thermodynamic properties at supercritical conditions. The approach is based on a synergistic combination of the classic Newton-Raphshon algorithm and the Adomian decomposition method. We demonstrate that the proposed method enjoys a higher degree of accuracy while requiring fewer iterations to reach a specific solution compared to that by the Newton-Raphson algorithm. To illustrate the efficiency of the aforementioned solution technique, several numerical examples are provided. The proposed method has been easily implemented in computer codes to provide parametric, not just numeric, solutions to the model equations. Consequently, one can derive other thermodynamic properties, which have not been treated parametrically to date, based on our new combined approach. © 2015, Copyright Taylor & Francis Group, LLC.

Fatoorehchi H.,University of Tehran | Abolghasemi H.,University of Tehran | Zarghami R.,University of Tehran | Rach R.,The George Adomian Center for Applied Mathematics | von Freeden S.,University of Stuttgart
Korean Journal of Chemical Engineering | Year: 2015

An efficient method based on the Faddeev-Leverrier algorithm combined with the Adomian decomposition method is devised to facilitate the stability analysis of multi-input multi-output control systems. In contrast to prior eigenvalue algorithms, our method affords all eigenvalues of the state matrix, either real or complex. Specifically, the calculation of the complex eigenvalues is made possible through special canonical forms, mainly involving square root operators, of the characteristic equation of the state matrix. Moreover, the proposed method does not require an initial guess, which is often a matter of concern since an inappropriate guess can cause failure in such available schemes. For the sake of illustration, a number of numerical examples, including chemical reaction processes, are also provided that demonstrate the efficiency of our new technique. © 2015, Korean Institute of Chemical Engineers, Seoul, Korea.

Fatoorehchi H.,University of Tehran | Abolghasemi H.,University of Tehran | Rach R.,The George Adomian Center for Applied Mathematics
Fluid Phase Equilibria | Year: 2015

We develop a robust algorithm for isothermal flash calculations of multi-component mixtures. The algorithm provides an explicit, parametric solution for a recent variation of the Rachford-Rice equation by employing a powerful analytical technique, namely the Adomian decomposition method. The computational speed of the algorithm is shown to be further enhanced by applying the iterated Shanks transformation. Unlike previous methods, our algorithm obviates the troublesome need for an initial guess, guarantees an excellent rate of convergence and is demonstrated to be more computationally economic than prior art. Several reliable examples from the literature, including the flash calculation for a mixture with nineteen components, as well as an extensive set of hypothetical, randomly-generated problems are presented to illustrate the exceptional efficacy of our new approach. © 2015 Elsevier B.V.

Fatoorehchi H.,University of Tehran | Abolghasemi H.,University of Tehran | Rach R.,The George Adomian Center for Applied Mathematics
Journal of Petroleum Science and Engineering | Year: 2014

The use of the Hankinson-Thomas-Phillips correlation for prediction of the natural gas compressibility factor is a common practice in natural gas engineering calculations. However, this equation suffers a serious deficiency from a computational viewpoint; in that it is not explicit with respect to the z-factor and hence is subject to time-consuming trial and error procedures. In this paper, we propose an explicit series expansion equivalent to the Hankinson-Thomas-Phillips equation by the aid of a powerful mathematical technique known as the Adomian decomposition method. Furthermore, we have enhanced our formula by a applying nonlinear convergence accelerator algorithm, namely the Shanks transform. The proposed equation is simple, easy to use, and is shown to be extremely accurate in reproducing the experimental PVT data of natural gases. Moreover, in contrast to the previous numerical algorithms such as the Newton-Raphson algorithm, the explicit nature of our formula obviates the need for any initial guess as an input for calculation of the z-factor. Such independence permits our formula to always quickly converge to the correct z-factor. © 2014 Elsevier B.V.

Fatoorehchi H.,University of Tehran | Abolghasemi H.,University of Tehran | Zarghami R.,University of Tehran | Rach R.,The George Adomian Center for Applied Mathematics
Canadian Journal of Chemical Engineering | Year: 2015

A number of control schemes including nonlinear feedback, dislocated feedback, and speed feedback have been proposed and assessed for a bromate-malonic acid-cerium Belousov-Zhabotinsky batch reaction process. The tuning parameters of the Oregonator model were firstly adjusted based on a UV-vis spectrophotometric analysis in the experimental part of the research. The adjusted Oregonator model successfully reproduced the innate induction time and periodicity of the BZ-batch system. Subsequently, the controllers were implemented and numerical simulations were carried out by employing the multi-stage Adomian decomposition method. The nominal analysis method was used to study the linear stability of each design. All the controlled systems were found to be linearly stable for certain continuous regions of controller gain. The performance of the proposed control laws was assessed and the dislocated feedback control strategy was shown to be able to drive the system states toward desired setpoints quickly. Furthermore, the validity of the dislocated feedback control design was doubly ensured by the sliding mode control theory. It was found that those feedback schemes which manipulate cerium ion concentration can be practically realized by means of electrochemical oxidation or oxygen aeration. Our results were confirmed by the Simulink software package and the block diagram representations are included in the paper. © 2015 Canadian Society for Chemical Engineering.

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