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Article
Publication date: 7 November 2016

Abdul-Majid Wazwaz, Randolph Rach and Lazhar Bougoffa

The purpose of this paper is to use the Adomian decomposition method (ADM) for solving boundary value problems with dual solutions.

Abstract

Purpose

The purpose of this paper is to use the Adomian decomposition method (ADM) for solving boundary value problems with dual solutions.

Design/methodology/approach

The ADM has been previously demonstrated to be eminently practical with widespread applicability to frontier problems arising in scientific applications. In this work, the authors seek to determine the relative merits of the ADM in the context of several important nonlinear boundary value models characterized by the existence of dual solutions.

Findings

The ADM is shown to readily solve specific nonlinear BVPs possessing more than one solution. The decomposition series solution of these models requires the calculation of the Adomian polynomials appropriate to the particular system nonlinearity. The authors show that the ADM solves these models for any analytic nonlinearity in a practical and straightforward manner. The conclusions are supported by several numerical examples arising in various scientific applications which admit dual solutions.

Originality/value

This paper presents an accurate work for solving nonlinear BVPs that possess dual solutions. The authors have demonstrated the widespread applicability of the ADM for solving various forms of these nonlinear equations.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 26 no. 8
Type: Research Article
ISSN: 0961-5539

Keywords

Article
Publication date: 7 November 2016

Lazhar Bougoffa, Jun-Sheng Duan and Randolph Rach

The purpose of this paper is to first deduce a new form of the exact analytic solution of the well-known nonlinear second-order differential equation subject to a set of…

Abstract

Purpose

The purpose of this paper is to first deduce a new form of the exact analytic solution of the well-known nonlinear second-order differential equation subject to a set of mixed nonlinear Robin and Neumann boundary conditions that model the thin film flows of fourth-grade fluids, and second to compare the approximate analytic solutions by the Adomian decomposition method (ADM) with the new exact analytic solution to validate its accuracy for parametric simulations of the thin film fluid flows, even for more complex models of non-Newtonian fluids in industrial applications.

Design/methodology/approach

The approach to calculating a new form of the exact analytic solution of thin film fluid flows rests upon a sequence of transformations including the modification of the classic technique due to Scipione del Ferro and Niccolò Fontana Tartaglia. Next the authors establish a lemma that justifies the new expression of the exact analytic solution for thin film fluid flows of fourth-grade fluids. Second, the authors apply a modification of the systematic ADM to quickly and easily calculate the sequence of analytic approximate solutions for this strongly nonlinear model of thin film flow of fourth-grade fluids. The ADM has been previously demonstrated to be eminently practical with widespread applicability to frontier problems arising in scientific and engineering applications. Herein, the authors seek to establish the relative merits of the ADM in the context of the thin film flows of fourth-grade fluids.

Findings

The ADM is shown to closely agree with the new expression of the exact analytic solution. The authors have calculated the error remainder functions and the maximal error remainder parameters in the error analysis to corroborate the solutions. The error analysis demonstrates the rapid rate of convergence and that we can approximate the exact solution as closely as we please; furthermore the rate of convergence is shown to be approximately exponential, and thus only a low-stage approximation will be adequate for engineering simulations as previously documented in the literature.

Originality/value

This paper presents an accurate work for solving thin film flows of fourth-grade fluids. The authors have compared the approximate analytic solutions by the ADM with the new expression of the exact analytic solution for this strongly nonlinear model. The authors commend this technique for more complex thin film fluid flow models.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 26 no. 8
Type: Research Article
ISSN: 0961-5539

Keywords

Content available

Abstract

Details

Kybernetes, vol. 41 no. 7/8
Type: Research Article
ISSN: 0368-492X

Article
Publication date: 17 September 2008

Randolph C. Rach

To provide a new proof of convergence of the Adomian decomposition series for solving nonlinear ordinary and partial differential equations based upon a thorough…

1268

Abstract

Purpose

To provide a new proof of convergence of the Adomian decomposition series for solving nonlinear ordinary and partial differential equations based upon a thorough examination of the historical milieu preceding the Adomian decomposition method.

Design/methodology/approach

Develops a theoretical background of the Adomian decomposition method under the auspices of the Cauchy‐Kovalevskaya theorem of existence and uniqueness for solution of differential equations. Beginning from the concepts of a parametrized Taylor expansion series as previously introduced in the Murray‐Miller theorem based on analytic parameters, and the Banach‐space analog of the Taylor expansion series about a function instead of a constant as briefly discussed by Cherruault et al., the Adomian decompositions series and the series of Adomian polynomials are found to be a uniformly convergent series of analytic functions for the solution u and the nonlinear composite function f(u). To derive the unifying formula for the family of classes of Adomian polynomials, the author develops the novel notion of a sequence of parametrized partial sums as defined by truncation operators, acting upon infinite series, which induce these parametrized sums for simple discard rules and appropriate decomposition parameters. Thus, the defining algorithm of the Adomian polynomials is the difference of these consecutive parametrized partial sums.

Findings

The four classes of Adomian polynomials are shown to belong to a common family of decomposition series, which admit solution by recursion, and are derived from one unifying formula. The series of Adomian polynomials and hence the solution as computed as an Adomian decomposition series are shown to be uniformly convergent. Furthermore, the limiting value of the mth Adomian polynomial approaches zero as the index m approaches infinity for the prerequisites of the Cauchy‐Kovalevskaya theorem. The novel truncation operators as governed by discard rules are analogous to an ideal low‐pass filter, where the decomposition parameters represent the cut‐off frequency for rearranging a uniformly convergent series so as to induce the parametrized partial sums.

Originality/value

This paper unifies the notion of the family of Adomian polynomials for solving nonlinear differential equations. Further it presents the new notion of parametrized partial sums as a tool for rearranging a uniformly convergent series. It offers a deeper understanding of the elegant and powerful Adomian decomposition method for solving nonlinear ordinary and partial differential equations, which are of paramount importance in modeling natural phenomena and man‐made device performance parameters.

Details

Kybernetes, vol. 37 no. 7
Type: Research Article
ISSN: 0368-492X

Keywords

Article
Publication date: 1 February 2013

Randolph Rach, Abdul‐Majid Wazwaz and Jun‐Sheng Duan

The purpose of this paper is to propose a new modification of the Adomian decomposition method for resolution of higher‐order inhomogeneous nonlinear initial value problems.

Abstract

Purpose

The purpose of this paper is to propose a new modification of the Adomian decomposition method for resolution of higher‐order inhomogeneous nonlinear initial value problems.

Design/methodology/approach

First the authors review the standard Adomian decomposition scheme and the Adomian polynomials for solving nonlinear differential equations. Next, the advantages of Duan's new algorithms and subroutines for fast generation of the Adomian polynomials to high orders are discussed. Then algorithms are considered for the solution of a sequence of first‐, second‐, third‐ and fourth‐order inhomogeneous nonlinear initial value problems with constant system coefficients by the new modified recursion scheme in order to derive a systematic algorithm for the general case of higher‐order inhomogeneous nonlinear initial value problems.

Findings

The authors investigate seven expository examples of inhomogeneous nonlinear initial value problems: the exact solution was known in advance, in order to demonstrate the rapid convergence of the new approach, including first‐ through sixth‐order derivatives and quadratic, cubic, quartic and exponential nonlinear terms in the solution and a sextic nonlinearity in the first‐order derivative. The key difference between the various modified recursion schemes is the choice of the initial solution component, using different choices to partition and delay the subsequent parts through the recursion steps. The authors' new approach extends this concept.

Originality/value

The new modified decomposition method provides a significant advantage for computing the solution's Taylor expansion series, both systematically and rapidly, as demonstrated in the various expository examples.

Article
Publication date: 18 October 2011

Abdul‐Majid Wazwaz and Randolph Rach

The purpose of this paper is to provide a comparison of the Adomian decomposition method (ADM) with the variational iteration method (VIM) for solving the Lane‐Emden…

524

Abstract

Purpose

The purpose of this paper is to provide a comparison of the Adomian decomposition method (ADM) with the variational iteration method (VIM) for solving the Lane‐Emden equations of the first and second kinds.

Design/methodology/approach

The paper examines the theoretical framework of the Adomian decomposition method and compares it with the variational iteration method. The paper seeks to determine the relative merits and computational benefits of both the Adomian decomposition method and the variational iteration method in the context of the important physical models of the Lane‐Emden equations of the first and second kinds.

Findings

The Adomian decomposition method is shown to readily solve the Lane‐Emden equations of both the first and second kinds for all positive real values of the system coefficient α and for all positive real values of the nonlinear exponent m. The decomposition series solution of these nonlinear differential equations requires the calculation of the Adomian polynomials appropriate to the particular system nonlinearity. The paper shows that the variational iteration method works effectively to solve the Lane‐Emden equation of the first kind for system coefficient values α=1, 2, 3, 4 but only for positive integer values of the nonlinear exponent m. The successive approximations of the solution of these nonlinear differential equations require the determination of the appropriate Lagrange multipliers, which are established in this paper. These two methodologies overcome the singular behavior at the origin x=0. The paper shows that the variational iteration method is impractical for solving either the Lane‐Emden equation of the first kind for non‐integer values of the system exponent m or the Lane‐Emden equations of the second kind. Indeed the Adomian decomposition method is shown to solve even the generalized Lane‐Emden equation for any analytic nonlinearity and all positive values of the system coefficient α in a practical and straightforward manner. The conclusions are supported by several numerical examples.

Originality/value

This paper presents an accurate comparison of the Adomian decomposition method with the variational iteration method for solving the Lane‐Emden equations of the first and second kinds. The paper presents a new solution algorithm for the generalized Lane‐Emden equation for any analytic system nonlinearity and for any model geometry as characterized by all possible positive real values of the system coefficient α.

Article
Publication date: 19 April 2013

Lazhar Bougoffa and Randolph C. Rach

The purpose of this paper is to present a new approach to solve nonlocal boundary value problems of linear and nonlinear first‐ and second‐order differential equations…

Abstract

Purpose

The purpose of this paper is to present a new approach to solve nonlocal boundary value problems of linear and nonlinear first‐ and second‐order differential equations subject to nonlocal conditions of integral type.

Design/methodology/approach

The authors first transform the given nonlocal boundary value problems of first‐ and second‐order differential equations into local boundary value problems of second‐ and third‐order differential equations, respectively. Then a modified Adomian decomposition method is applied, which permits convenient resolution of these equations.

Findings

The new technique, as presented in this paper in extending the applicability of the Adomian decomposition method, has been shown to be very efficient for solving nonlocal boundary value problems of linear and nonlinear first‐ and second‐order differential equations subject to nonlocal conditions of integral type.

Originality/value

The paper presents a new solution algorithm for the nonlocal boundary value problems of linear and nonlinear first‐ and second‐order differential equations subject to nonlocal conditions of integral type.

Content available
625

Abstract

Details

Kybernetes, vol. 41 no. 7/8
Type: Research Article
ISSN: 0368-492X

Article
Publication date: 5 May 2015

Lazhar Bougoffa, Randolph Rach, Abdul-Majid Wazwaz and Jun-Sheng Duan

The purpose of this paper is concerned with a reliable treatment of the classical Stephan problem. The Adomian decomposition method (ADM) is used to carry out the…

Abstract

Purpose

The purpose of this paper is concerned with a reliable treatment of the classical Stephan problem. The Adomian decomposition method (ADM) is used to carry out the analysis, Moreover, the authors extend the work to examine the Stefan problem with variable latent heat. The study confirms the accuracy and efficiency of the employed method.

Design/methodology/approach

The new technique, as presented in this paper in extending the applicability of the ADM, has been shown to be very efficient for solving the Stefan problem.

Findings

The Stefan problem with variable latent heat was examined as well. The ADM was effectively used for analytic treatment of the Stefan problem with and without variable latent heat.

Originality/value

The paper presents a new solution algorithm for the Stefan problem.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 25 no. 4
Type: Research Article
ISSN: 0961-5539

Keywords

Article
Publication date: 2 March 2012

Lazhar Bougoffa, Manal Al‐Haqbani and Randolph C. Rach

In this paper, Fredholm integral equations of the first kind, the Schlomilch integral equation, and a class of related integral equations of the first kind with constant…

334

Abstract

Purpose

In this paper, Fredholm integral equations of the first kind, the Schlomilch integral equation, and a class of related integral equations of the first kind with constant limits of integration are transformed in such a manner that the Adomian decomposition method (ADM) can be applied. Some examples with closed‐form solutions are studied in detail to further illustrate the proposed technique, and the results obtained indicate this approach is indeed practical and efficient. The purpose of this paper is to develop a new iterative procedure where the integral equations of the first kind are recast into a canonical form suitable for the ADM. Hence it examines how this new procedure provides the exact solution.

Design/methodology/approach

The new technique, as presented in this paper in extending the applicability of the ADM, has been shown to be very efficient for solving Fredholm integral equations of the first kind, the Schlomilch integral equation and a related class of nonlinear integral equations with constant limits of integration.

Findings

By using the new proposed technique, the ADM can be easily used to solve the integral equations of the first kind, the Schlomilch integral equation, and a class of related integral equations of the first kind with constant limits of integration.

Originality/value

The paper shows that this new technique is easy to implement and produces accurate results.

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