bluuebunny/crossref_metadata_2025_split huggingface dataset:
New methods for applying the Adomian method to partial differential equations with boundary conditions
Authors: Benneouala Titem , Cherruault Yves , Abbaoui Karim 
Published: January 2007

PurposeTo find methods for solving non‐linear partial differential equations. The decomposition method may be applied, but a difficulty arises when applied to non‐linear partial differential equations with initial and boundary conditions. In this work, two methods are described that take into account the boundary conditions.Design/methodology/approachThe decomposition method whilst being a powerful tool for solving non‐linear functional equations encounters difficulties in finding solutions of partial differential equations with boundary conditions. In this paper, two methods are introduced which consist of setting boundary conditions to the equations so that the decomposition methods can be applied.FindingsBy using the two proposed methods the decomposition method can then be easily used. In this work the two methods taking account of the boundary conditions were found to be efficient and allows a solution to be found using the Adomian decomposition method.Research limitations/implicationsThe two new methods provide solutions by the application of the decomposition method of George Adomian as extended by other researchers. Both are efficient: the first giving interesting results for linear and non‐linear problems; the second one is also efficient, but difficulties could arise from the calculations of the required series.Practical implicationsThe research provides two efficient methods. The first method gives the demonstrated results for linear and non‐linear problems due to the use of symbolic software such as Mathematica or Maple.Originality/valueBoth methods illustrate the powerful use of the decomposition techniques pioneered by Adomian and as a result of their application may be applied to the solving of non‐linear functional equations of any kind. This paper tackles the problems by introducing new methods of applying the Adomian techniques to partial differential equations with boundary conditions.
A different approach for solving singular two‐point boundary value problems
Authors: Inc M. , Ergüt M. , Cherruault Y. 
Published: January 2007

PurposeAims to solve singular two‐point boundary value problems, linear and non‐linear by using modified and standard decomposition methods, respectively.Design/methodology/approachThe approximate solution of this problem is calculated in the form of series with easily computable components.FindingsThe accuracy of the presented numerical method was examined in comparison with others and was found to be superior.Research limitations/implicationsThe decomposition method provided a reliable technique that required less work when compared with the traditional techniques. The method does not require unjustified assumptions linearisation discretizations or perturbation.Practical implicationsThe Adomian decomposition method was found to be very easy to apply to both differential equations, higher‐order boundary value problems and linear or non‐differential systems.Originality/valueThe technique is both innovative and efficient and an original approach for solving singular two‐point boundary value problems.
Geometrical interpretation and approximate solution of non‐linear KdV equation
Authors: Bektas M. , Inc M. , Cherruault Y. 
Published: January 2007

PurposeThe purpose is to study an analytical solution of non‐linear Korteweg‐de Vries (KdV) equation by using the Adomian decomposition method (ADM).Design/methodology/approachThe solution is calculated in the form of a series with easily computable components. The non‐linear KdV equation has been considered and the analytic solution is compared with its numerical solution by using the ADM and Mathematica software program.FindingsThis approach to the non‐linear evolution equation was found to be valuable as a tool for scientists and applied mathematicians, because it provides immediate and visible symbolic terms of analytical solution as well as its numerical approximate solution to both linear and non‐linear problems without linearization or discretization.Research limitations/implicationsThis geometrical interpretation and the produced approximate solution of the non‐linear KdV equation illustrates the use of the ADM. Research using ADM is ongoing but already the numerical results obtained in this paper justify the advantages of this methodology, even in a few terms of approximation.Practical implicationsUsing the Mathematica software package the ADM was implemented for homogenous KdV equation as an illustrative example which has distinct applications for scientists and applied mathematicians.Originality/valueThis is an original study of the use of ADM for the solution of the non‐linear KdV equation. It also shows how the Mathematica software package can be used in such studies.
Direct identification of general linear compartmental systems by means of (n−2) compartments measures
Authors: Hebri B. , Cherruault Y. 
Published: January 2007

PurposeAims to study direct identification of general linear compartmental systems by means of (n−2) compartmental measures. This is based on two main results.Design/methodology/approachThe first result presented is related to the existence and uniqueness of identification exchange parameters in linear compartmental systems by using a direct method with less restrictive assumptions. A second result given, permits us to show that (n−2) observations are sufficient to identify the compartmental systems.FindingsThis research study describes a method which shows that in an open linear compartmental systems there exists an energy dissipation from compartmental 1 to the systems exterior. An explicit relationship between the dissipated energy and the exchange parameters was established. The results are probably perfectible and are optimal for n=3, where only an observable compartment is needed.Practical implicationsThe identification of exchange parameters is easily obtained by using the matrix of the elementary masses and by solving a linear algebraic system. Among the open problems in compartmental analysis is the problem of minimizing the observable compartments which is studied in this paper.Originality/valueThe study is based on the original work of Yves Cherruault who has already presented methods for proving that a bicompartmental systems is uniquely identified. He has generalised his method for n‐compartments.
Solving inequalities by α‐dense curves. Application to global optimization
Authors: Mora G. , Cherruault Y. , Ubeda J.I. 
Published: January 2007

PurposeTo introduce an algorithm to solve inequalities defined by real functions on a certain compact set D of a general metric space (E, D). The device is based on α‐dense curves.Design/methodology/approachThe solution of inequalities using α‐dense curves and also an approach to a global optimization technique, similarly obtained to that of the inequalities.FindingsA new method is presented. The algorithm for solving inequalities is described which is based on a proven result. Inequalities of n‐variable dependence are reduced by transformation using α‐dense curves.Research limitations/implicationsThe research is a continuation of studies that resulted in a new method called Alienor for solving global optimization problems associated with multi‐variable functions.Originality/valueBased on earlier research by the authors α‐dense curves have been used which allow the transformation of a n‐variables global optimization problem into a one‐variable global one. This paper gives a new method for quickly solving the one‐variable problem.
Adomian method for solving adaptive control problem
Authors: Manseur S. , Cherruault Y. 
Published: January 2007

PurposeAims to propose methodology to solve adaptive control problems by the use of the Adomian decomposition method (ADM).Design/methodology/approachThe approach to this problem is through the ADM which consists in finding a solution of nonlinear state equations as a convergent series that depends on the unknown parameters of the system.FindingsIt was shown that a first approach for solving adaptive control problems could be tackled by using the ADM. The solution of the systems was obtained in a series forms as a function of the unknown parameters. The objective function became a function that explicitly depends on parameters. This, it was shown, can be minimised by classical or nonclassical global optimization methods.Research limitations/implicationsThe methods presented depends on ADM. It is proposed that future work involves the study of an adaptive control problem associated to a nonlinear compartmental systems of Michaels‐Menten type.Originality/valuePresents an applications of the developed ADM using it, as adapted to control problems. It contributes to mathematical modelling studied concerning most biological, physical phenomena which are described by nonlinear systems: differential, partial differential in form and for which there is little hope of finding an exact solution.
Global optimization via α‐dense curves
Authors: Benabidallah A. , Cherruault Y. 
Published: January 2007

PurposeTo study constrained or unconstrained global optimization problems in a cube of Rd where d is a positive integer.Design/methodology/approachα‐dense curves are initially used to transform this problem into a global optimization problem of a single variable. The optimization of the one variable is then treated by means of the Legendre‐Fenchel Transform. This discrete convex envelope of the one variable function obtained previously, can then be computed.FindingsGlobal optimization problems of this nature have already been extensively studied by the authors. In this paper they have coupled the Alienor method with Legendre‐Fenchel Tranform to compute a discrete convex envelope of the function to minimize. A fast algorithm was successfully used to do this.Research limitations/implicationsThis approach to global optimization is based on α‐dense curves and numerical tests performed on a Pentium IV (1,700 MHz) computer used with Mathematica 4 software.Practical implicationsGives the solutions illustrated in the numerous examples provided that show the practicality of the methodology.Originality/valueA new approach based on extensive research into global optimization via α‐dense curves.
Identification of closed Michaelis‐Menten system by a linear approach
Authors: Hebri B. , Cherruault Y. 
Published: January 2007

PurposeThis paper aims at identifying the closed Michaelis‐Menten (M‐M) system. The system governs many phenomena that are based on “enzyme reactions”.Design/methodology/approachA linearization method is used. The system, however is not homogeneous and consequently earlier identification results obtained by the author cannot be applied. The approach is to show that the closed Michaelis‐Mentan system can be approximated by a sequence of open M‐M systems. The M‐M parameters are then obtained by continuity from the Cauchy's problem solution.FindingsThe main results – the identification of a closed M‐M system is described. The linearization of the closed M‐M compartmental system led to a linear nonhomogeneous system. Earlier classical methods could not applied but the “Reasonable Biological Hypothesis” of the author allowed further study.Research limitations/implicationsThis study takes a preferred theoretical aspect. A generalization to n, n≥3 is considered to be interesting because we can consider an open compartmental system having n compartments as a closed system with n+1 compartments. Further work on this generalization is contemplated.Practical implicationsThis theoretical study will further increase the use compartmental analysis to modelling any phenomena depending only on time.Originality/valueProvides a methodology for identification of M‐M system using a linear strategy.
A new decomposition method based on infinite products
Authors: Abbasbandy S. 
Published: January 2007

PurposeThe purpose is to apply a new decomposition method for solving non‐linear functional equations.Design/methodology/approachThe method used is similar to the Adomian decomposition method (ADM). In this approach, Adomian's original solution for solving non‐linear functional equations and Cherruault's proposed new definition of ADM are replaced by a new decomposition technique.FindingsAdomian gives the solution using his decomposition method as an infinite series usually converging to an accurate solution. Cherruault's new definition also proved the convergence under suitable conditions and a reasonable hypothesis. This paper has developed a new decomposition technique based on infinite products. It was found that it then became possible to prove the convergence under some given conditions.Research limitations/implicationsThe new method has its convergence proved and a number of research examples presented. The method proved efficient but there remains many difficulties in this area. Further research is needed to see whether this method can be used for solving initial value problems.Originality/valueProduces another tool and furthers the development of ADM by introducing a method based on infinite products.
A reliable method for obtaining approximate solutions of linear and nonlinear Volterra‐Fredholm integro‐differential equations
Authors: Inc M. , Cherruault Y. 
Published: January 2007

PurposeBased on the original methods of Adomian a decomposition method has been developed to find the analytic approximation of the linear and nonlinear Volterra‐Fredholm (V‐F) integro‐differential equations under the initial or boundary conditions.Design/methodology/approachDesigned around the methods of Adomian and later researchers. The methodology to obtain numerical solutions of the V‐F integro‐differential equations is one whose essential features is its rapid convergence and high degree of accuracy which it approximates. This is achieved in only a few terms of its iterative scheme which is devised to avoid linearization, perturbation and any transformation in order to find solutions to given problems.FindingsThe scheme was shown to have many advantages over the traditional methods. In particular it provided discretization and provided an efficient numerical solution with high accuracy, minimal calculations as well as an avoidance of physical unrealistic assumptions.Research limitations/implicationsA reliable method for obtaining approximate solutions of linear and nonlinear V‐F integro‐differential using the decomposition method which avoids the tedious work needed by traditional techniques has been developed. Exact solutions were easily obtained.Practical implicationsThe new method had most of its symbolic and numerical computations performed using the Computer Algebra Systems‐Mathematica. Numerical results from selected examples were presented.Originality/valueA new effective and accurate methodology has been developed and demonstrated.
A global optimization method for a large number of variables (variant of Alienor method)
Authors: Konfé Balira O. , Cherruault Yves , Benneouala Titem 
Published: January 2007

PurposeTo use α‐dense curves to allow the transform of a multiple function into a single variable function in order to solve global optimization problems.Design/methodology/approachUse is made of the established Alienor method which has already been applied to biological and industrial processes. The problems tackled have a number of variables and the chosen optimization method is a variant of the Alienor method.FindingsA new method for solving global optimization problem, called the Alienor method is now the subject of many variants. In this paper, it was found that a new reducing transformation α‐dense in Rn was successful in solving this type of problem when associated to a functional depending on a large number of variables. The reducing transformation is very efficient and accurate.Research limitations/implicationsThis is a variant of the proven Alienor Method which has improved the resolution of global optimization problems. It showed that the reducing transformation has the advantage that a small calculation time is obtained even when the relevant series are slowly increasing. Further development of the method is anticipated.Practical implicationsProved very effective for obtaining the global optimum with good precision and very short calculation time for large numbers of variables. Can be performed on micro‐calculators.Originality/valueNew variant of proven method. Of interest in solution of concrete problems in biology and industry.
A new reducing transformation for global optimization (with Alienor method)
Authors: Cherruault Yves 
Published: January 2007

PurposeTo propose a new reducing transformation that allows the calculation of the absolute minimum of a function that depends on a large number of variables to be performed quickly.Design/methodology/approachThe methodology depends on the building of α‐dense curve in a compact of Rn that allows the approximate at any point of this compact with a desired precision. This approach allows global optimization problems that depend on a large number of variables, be tackled quickly and precisely.FindingsIt was found that this new method for densifying a space Rn (compact) by means of simple parametric curve (a space filling curve) could be used to deal with global optimization problems of some several hundreds or thousands of variables in some seconds or minutes. The technique is being based on the cosine function.Research limitations/implicationsThe results depend on the use of a computer system or “micro‐calculator”.Practical implicationsThis is an “economic” method and the technique which uses the cosine function allows the reduction of the calculation time and avoids calculus errors. It has the practical advantages that coupled with a transformation eliminating local minima, it permits the solution of global optimization problems of more than 1,000 variables in less than 1 min.Originality/valueThe method is innovative and shown to be accurate and fast even with a function of a large number of variables.
Alienor method for global optimization with a large number of variables
Authors: Benneouala T. , Cherruault Y. 
Published: January 2007

PurposeTo show the usefulness of the Alienor method when applied to the global optimization problems that depend on large number of variables.Design/methodology/approachThe approach is to use reducing transformations. The first is due to Cherruault and the second to Mora.FindingsIt was found that the Alienor method was very efficient and reliable in solving global optimization problems of many variables. Results produced to confirm this conclusion.Research limitations/implicationsThe numerical results presented showed that the Alienor method was suitable for finding global minimum even in the case of a very large number of variables. The research provides a new methodology for solving such problems.Practical implicationsNo other method, we believe, can obtain such results in so short a time for hundreds or even thousands of variables.Originality/valueThe new approach relies on the originality of both the Cherruault and the Mora transformations and their earlier invention of the Alienor method.
A new “Optimization‐Preserving‐Operator” applied to global optimization
Authors: Konfe Balira O. , Cherruault Yves , Some Blaise , Benneouala Titem 
Published: January 2007

PurposeTo introduce Optimization‐Preserving‐Operators (O‐P‐Os), which are operators that are defined on classes of real functions that depend on a single variable, and allow us to eliminate local optima and to preserve global optima.Design/methodology/approachOutline a new method to build O‐P‐Os. These are introduced as O‐P‐O* and lead to a new approach for solving global optimization problems.FindingsIt was found that classical discretization methods for obtaining optimum of one variable function was too time‐consuming. The simple method introduced provided solutions to the test functions chosen as examples. The solutions were provided in a short time.Research limitations/implicationsProvides new tools for mathematical programming and in particular the global optimization problems. The O‐P‐O* introduced innovative technique for solving such problems.Practical implicationsO‐P‐O* produces solutions to global optimization problems in a much improved time. The algorithm derived, and the steps for its operation proved on implementation, the efficiency of the new method. This was demonstrated by numerical results for selected functions obtained using microcomputer systems.Originality/valueProvides new way of solving global optimization problems.
Optimal control of chemotherapy of HIV model using the coupled Adomian/Alienor methods
Authors: Manseur S. , Attalah K. , Cherruault Y. 
Published: January 2007

PurposeThis paper aims at applying the proven Adomian and Alienor methods to solve the problem of optimal chemotherapy of HIV model.Design/methodology/approachThe combination of the Adomian decomposition method and the Alienor reduction method allows us to solve the control problem as if it were a classical one‐dimensional minimization problem. The methodology is applied to a HIV model and simulation results given.FindingsA general abstract framework for the control of a non‐linear evolution system has been developed. It was shown that it is possible to control a system by using the powerful techniques of Adomian and Alienor, and produce results comparable with those obtained by classical methods where other cost functions are used.Research limitations/implicationsThe benefit of this work is based on the CD4 healthy cells being maximized and the cost based on a drug dose being minimized. The new methodology could be used after further research for solving many control problems in biology and in other areas such as those involving industrial processes.Practical implicationsNew methodology is cost‐effective in controlling the drug dose affecting rate of infection of cells by HIV virus.Originality/valueNew application of proven methodology will solve many control problems in biocybernetics/biomedicine/industry and other fields.
Second‐order cybernetics and enactive perception
Authors: Bishop J.M. , Nasuto J.S. 
Published: January 2007

PurposeTo present an account of cognition integrating second‐order cybernetics (SOC) together with enactive perception and dynamic systems theory.Design/methodology/approachThe paper presents a brief critique of classical models of cognition then outlines how integration of SOC, enactive perception and dynamic systems theory can overcome some weaknesses of the classical paradigm.FindingsPresents the critique of evolutionary robotics showing how the issues of teleology and autonomy are left unresolved by this paradigm although their solution fits within the proposed framework.Research limitations/implicationsThe paper highlights the importance of genuine autonomy in the development of artificial cognitive systems. It sets out a framework within which the robotic research of cognitive systems could succeed.Practical implicationsThere are no immediate practical implications but see research implications.Originality/valueIt joins the discussion on the fundamental nature of cognitive systems and emphasise the importance of autonomy and embodiment.