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Article
Publication date: 1 June 2000

S. Guellal, Y. Cherruault, M.J. Pujol and P. Grimalt

In some papers G. Adomian has presented a decomposition technique in order to solve different non‐linear equations. The solution is found as an infinite series quickly…

Abstract

In some papers G. Adomian has presented a decomposition technique in order to solve different non‐linear equations. The solution is found as an infinite series quickly converging to accurate solutions. The method is well‐suited for physical problems and it avoids linearization, perturbation and other restrictions, methods and assumptions which may change the problem being solved – sometimes seriously – unnecessarily. Proofs of convergence are given by Cherruault and co‐authors. Many numerical studies for physical phenomena, such as Fisher’s equation, Lorentz’s equation and Edem’s equation are given and solved. In this work, the general equation given by ∂ p \overt = (∇ ⋅(q(x)⋅ ∇p)) + f(x, t) is solved by using decomposition methods, and is compared to other techniques. This equation can be used to describe the motion of a fluid flow in the so‐called reservoir region, where p(x, t) represents the pressure distribution, f(x, t) describes the withdrawal or injection of the fluid, and q(x) is the transmissibility in the reservoir region.

Details

Kybernetes, vol. 29 no. 4
Type: Research Article
ISSN: 0368-492X

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Article
Publication date: 1 June 2003

M.J. Pujol and P. Grimalt

This paper describes a non‐linear reaction‐diffusion equation, which models how a substance spreads in the surface of the cortex so as to avoid a massive destruction of…

Abstract

This paper describes a non‐linear reaction‐diffusion equation, which models how a substance spreads in the surface of the cortex so as to avoid a massive destruction of neurones when cerebral tissue is not oxygenated correctly. For the explicit finite differences method, the necessary stability condition is provided by a reaction‐diffusion equation with non‐linearity given by a decreasing function. The solution to the non‐linear reaction‐diffusion equation of the model can be obtained via one of the two methods: the finite differences (explicit schema) method and the Adomian method.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 13 no. 4
Type: Research Article
ISSN: 0961-5539

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Article
Publication date: 1 December 2001

K. Abbaoui, M.J. Pujol, Y. Cherruault, N. Himoun and P. Grimalt

A new approach of the decomposition method (Adomian) in which the Adomian scheme is obtained in a more natural way than in the classical presentation, is given. A new…

Abstract

A new approach of the decomposition method (Adomian) in which the Adomian scheme is obtained in a more natural way than in the classical presentation, is given. A new condition for obtaining convergence of the decomposition series is also included.

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Kybernetes, vol. 30 no. 9/10
Type: Research Article
ISSN: 0368-492X

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Article
Publication date: 1 June 2003

Cèsar Bordehore, Angela Pascual, Maria J. Pujol, Julio Escolano, Inmaculada Manchón and Pedro Grimalt

Faltung equations (closed cycle type) have a wide range of biological applications, nonetheless, they are poorly studied. We use a Volterra‐Kostitzin model (which is a…

Abstract

Faltung equations (closed cycle type) have a wide range of biological applications, nonetheless, they are poorly studied. We use a Volterra‐Kostitzin model (which is a Faltung equation) to study the dynamics of a certain species, where the integral term represents a residual action. The complexity of resolution of this non‐linear equation using classical numeric methods is here solved with the Adomian decomposition method. Our method provides the same graphic solution as others do, such as the numeric method Miladie. However, the decomposition method of Adomian has the advantage that neither time nor space are considered discontinuous and that it gives an analytical solution with a reliable approximation.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 13 no. 4
Type: Research Article
ISSN: 0961-5539

Keywords

Content available

Abstract

Details

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

Content available

Abstract

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Kybernetes, vol. 27 no. 1
Type: Research Article
ISSN: 0368-492X

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Article
Publication date: 2 August 2013

María José Pujol, Francisco A. Pujol, Fidel Aznar, Mar Pujol and Ramón Rizo

In this paper the authors aim to show the advantages of using the decomposition method introduced by Adomian to solve Emden's equation, a classical non‐linear equation…

Abstract

Purpose

In this paper the authors aim to show the advantages of using the decomposition method introduced by Adomian to solve Emden's equation, a classical non‐linear equation that appears in the study of the thermal behaviour of a spherical cloud and of the gravitational potential of a polytropic fluid at hydrostatic equilibrium.

Design/methodology/approach

In their work, the authors first review Emden's equation and its possible solutions using the Frobenius and power series methods; then, Adomian polynomials are introduced. Afterwards, Emden's equation is solved using Adomian's decomposition method and, finally, they conclude with a comparison of the solution given by Adomian's method with the solution obtained by the other methods, for certain cases where the exact solution is known.

Findings

Solving Emden's equation for n in the interval [0, 5] is very interesting for several scientific applications, such as astronomy. However, the exact solution is known only for n=0, n=1 and n=5. The experiments show that Adomian's method achieves an approximate solution which overlaps with the exact solution when n=0, and that coincides with the Taylor expansion of the exact solutions for n=1 and n=5. As a result, the authors obtained quite satisfactory results from their proposal.

Originality/value

The main classical methods for obtaining approximate solutions of Emden's equation have serious computational drawbacks. The authors make a new, efficient numerical implementation for solving this equation, constructing iteratively the Adomian polynomials, which leads to a solution of Emden's equation that extends the range of variation of parameter n compared to the solutions given by both the Frobenius and the power series methods.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 23 no. 6
Type: Research Article
ISSN: 0961-5539

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Article
Publication date: 1 August 2004

Mohamed Benabidallah and Yves Cherruault

In this study, the aim is to solve a class of boundary systems, using the Adomian decomposition method. This method gave very good results for solving algebraic…

Abstract

In this study, the aim is to solve a class of boundary systems, using the Adomian decomposition method. This method gave very good results for solving algebraic, differential integral and partial differential equations.

Details

Kybernetes, vol. 33 no. 7
Type: Research Article
ISSN: 0368-492X

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Article
Publication date: 1 August 2005

Benjamin Mampassi, Bisso Saley, Blaise Somé and Yves Cherruault

To compute an optimal control of non‐linear reaction diffusion equations that are modelling inhibitor problems in the brain.

Abstract

Purpose

To compute an optimal control of non‐linear reaction diffusion equations that are modelling inhibitor problems in the brain.

Design/methodology/approach

A new numerical approach that combines a spectral method in time and the Adomian decomposition method in space. The coupling of these two methods is used to solve an optimal control problem in cancer research.

Findings

The main conclusion is that the numerical approach we have developed leads to a new way for solving such problems.

Research limitations/implications

Focused research on computing control optimal in non‐linear diffusion reaction equations. The main idea that is developed lies in the approximation of the control space in view of the spectral expansion in the Legendre basis.

Practical implications

Through this work we are convinced that one way to derive efficient numerical optimal control is to associate the Legendre expansion in time and Runge Kutta approximation. We expect to obtain general results from optimal control associated with non‐linear parabolic problem in higher dimension.

Originality/value

Coupling of methods provides a numerical solution of an optical control problem in Cancer research.

Details

Kybernetes, vol. 34 no. 7/8
Type: Research Article
ISSN: 0368-492X

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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.

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