Modeling of electromagnetic fields in rectangular waveguides by coupled finite–infinite elements

The paper describes the 3D infinite element for modeling of stationary harmonic electromagnetic fields in waveguides. The proposed approximation is a straightforward modification of the analogous approach developed for analysis of scattering problems in unbounded 2D and 3D domains [Comp. Meth. Appl. Mech. Engrg. 188 (2000) 625; Int. J. Num. Meth. Engrg. 57 (2003) 899]. Exponential shape functions in the longitudinal direction are used similarly as in [Comp. Meth. Appl. Mech. Engrg. 140 (1997) 221]. However, arbitrary order of approximation in the tangential direction may be selected due to compatibility with the hp-adaptive edge FE element discretization for Maxwell’s equations in bounded domains reported in [Comp. Meth. Appl. Mech. Engrg. 152 (1998) 103; Int. J. Num. Meth. Engrg. 53 (2002) 147; Comp. Meth. Appl. Mech. Engrg. 169 (1999) 331].     

Node-pancyclicity and edge-pancyclicity of hypercube variants

Twisted cubes, crossed cubes, Möbius cubes, and locally twisted cubes are some of the widely studied hypercube variants. The 4-pancyclicity of twisted cubes, crossed cubes, Möbius cubes, locally twisted cubes and the 4-edge-pancyclicity of twisted cubes, crossed cubes, Möbius cubes are proven in [C.P. Chang, J.N. Wang, L.H. Hsu, Topological properties of twisted cube, Inform. Sci. 113 (1999) 147-167; C.P. Chang, T.Y. Sung, L.H. Hsu, Edge congestion and topological properties of crossed cubes, IEEE Trans. Parall. Distr. 11 (1) (2000) 64-80; J. Fan, Hamilton-connectivity and cycle embedding of the Möbius cubes, Inform. Process. Lett. 82 (2002) 113-117; X. Yang, G.M. Megson, D.J. Evans, Locally twisted cubes are 4-pancyclic, Appl. Math. Lett. 17 (2004) 919-925; J. Fan, N. Yu, X. Jia, X. Lin, Embedding of cycles in twisted cubes with edge-pancyclic, Algorithmica, submitted for publication; J. Fan, X. Lin, X. Jia, Node-pancyclic and edge-pancyclic of crossed cubes, Inform. Process. Lett. 93 (2005) 133-138; M. Xu, J.M. Xu, Edge-pancyclicity of Möbius cubes, Inform. Process. Lett. 96 (2005) 136-140], respectively. It should be noted that 4-edge-pancyclicity implies 4-node-pancyclicity which further implies 4-pancyclicity. In this paper, we outline an approach to prove the 4-edge-pancyclicity of some hypercube variants and we prove in particular that Möbius cubes and locally twisted cubes are 4-edge-pancyclic.     

On local convergence of a Newton-type method in Banach space

In this study we are concerned with the local convergence of a Newton-type method introduced by us [I.K. Argyros and D. Chen, On the midpoint iterative method for solving nonlinear equations in Banach spaces, Appl. Math. Lett. 5 (1992), pp. 7–9.] for approximating a solution of a nonlinear equation in a Banach space setting. This method has also been studied by Homeier [H.H.H. Homeier, A modified Newton method for rootfinding with cubic convergence, J. Comput. Appl. Math. 157 (2003), pp. 227–230.] and Özban [A.Y. Özban, Some new variants of Newton's method, Appl. Math. Lett. 17 (2004), pp. 677–682.] in real or complex space. The benefits of using this method over other methods using the same information have been explained in [I.K. Argyros, Computational theory of iterative methods, in Studies in Computational Mathematics, Vol. 15, C.K. Chui and L. Wuytack, eds., Elsevier Science Inc., New York, USA, 2007.; I.K. Argyros and D. Chen, On the midpoint iterative method for solving nonlinear equations in Banach spaces, Appl. Math. Lett. 5 (1992), pp. 7–9.; H.H.H. Homeier, A modified Newton method for rootfinding with cubic convergence, J. Comput. Appl. Math. 157 (2003), pp. 227–230.; A.Y. Özban, Some new variants of Newton's method, Appl. Math. Lett. 17 (2004), pp. 677–682.]. Here, we give the convergence radii for this method under a type of weak Lipschitz conditions proven to be fruitful by Wang in the case of Newton's method [X. Wang, Convergence of Newton's method and inverse function in Banach space, Math. Comput. 68 (1999), pp. 169–186 and X. Wang, Convergence of Newton's method and uniqueness of the solution of equations in Banach space, IMA J. Numer. Anal. 20 (2000), pp. 123–134.]. Numerical examples are also provided.     

An improved convergence analysis for the Newton–Kantorovich method using recurrence relations

We generate a sequence using the Newton–Kantorovich method in order to approximate a locally unique solution of an operator equation on a Banach space under Hölder continuity conditions. Using recurrence relations, Hölder as well as centre-Hölder continuity assumptions on the operator involved, we provide a semilocal convergence analysis with the following advantages over the elegant work by Hernánde? in (The Newton method for operators with Hölder continuous first derivative, J. Optim. Theory Appl. 109(3) (2001), pp. 631–648.) (under the same computational cost): finer error bounds on the distances involved, and a more precise information on the location of the solution. Our results also compare favourably with recent and relevant ones in (I.K. Argyros, Concerning the “terra incognita” between convergence regions of two Newton methods, Nonlinear Anal. 62 (2005), pp. 179–194; I.K. Argyros, Computational Theory of Iterative Methods, in Studies in Computational Mathematics, Vol. 15, C.K. Chui and L. Wuytack, eds., Elsevier Publ. Co., New York, USA, 2007; I.K. Argyros, On the gap between the semilocal convergence domain of two Newton methods, Appl. Math. 34(2) (2007), pp. 193–204; I.K. Argyros, On the convergence region of Newton's method under Hölder continuity conditions, submitted for publication; I.K. Argyros, Estimates on majorizing sequences in the Newton–Kantorovich method, submitted for publication; F. Cianciaruso and E. DePascale, Newton–Kantorovich approximations when the derivative is Hölderian: Old and new results, Numer. Funct. Anal. Optim. 24 (2003), pp. 713–723; F. Cianciaruso and E. DePascale, Estimates of majorizing sequences in the Newton–Kantorovich method, Numer. Funct. Anal. Optim. 27(5–6) (2006), pp. 529–538; F. Cianciaruso and E. DePascale, Estimates of majorizing sequences in the Newton–Kanorovich method: A further improvement, J. Math. Anal. Appl. 322 (2006), pp. 329–335; N.T. Demidovich, P.P. Zabreiko, and Ju.V. Lysenko, Some remarks on the Newton–Kantorovich mehtod for nonlinear equations with Hölder continuous linearizations, Izv. Akad. Nauk Belorus 3 (1993), pp. 22–26 (in Russian). (E. DePascale and P.P. Zabreiko, The convergence of the Newton–Kantorovich method under Vertgeim conditions, A new improvement, Z. Anal. Anwendvugen 17 (1998), pp. 271–280.) and (L.V. Kantorovich and G.P. Akilov, Functional Analysis in Normed Spaces, Pergamon Press, Oxford, 1982; J.V. Lysenko, Conditions for the convergence of the Newton–Kantorovich method for nonlinear equations with Hölder linearizations, Dokl. Akad. Nauk BSSR 38 (1994), pp. 20–24 (in Russian); B.A. Vertgeim, On some methods for the approximate solution of nonlinear functional equations in Banach spaces, Uspekhi Mat. Nauk 12 (1957), pp. 166–169 (in Russian); Amer. Math. Soc. Transl. 16 (1960), pp. 378–382. (English Trans.).)     

A note on variable threshold concept lattices: Threshold-based operators are reducible to classical concept-forming operators

The present paper deals with formal concept analysis of data with fuzzy attributes. We clarify several points of a new approach of [S.Q. Fan, W.X. Zhang, Variable threshold concept lattice, Inf. Sci., accepted for publication] which is based on using thresholds in concept-forming operators. We show that the extent- and intent-forming operators from [S.Q. Fan, W.X. Zhang, Inf. Sci., accepted for publication] can be defined in terms of basic fuzzy set operations and the original operators as introduced and studied e.g. in [R. Belohlavek, Fuzzy Galois connections, Math. Logic Quarterly 45 (4) (1999) 497-504; R. Belohlavek, Concept lattices and order in fuzzy logic, Ann. Pure Appl. Logic 128 (2004) 277-298; S. Pollandt, Fuzzy Begriffe, Springer-Verlag, Berlin/Heidelberg, 1997]. As a consequence, main properties of the new operators from [S.Q. Fan, W.X. Zhang, Inf. Sci., accepted for publication], including the properties studied in [S.Q. Fan, W.X. Zhang, Inf. Sci., accepted for publication], can be obtained as consequences of the original operators from [R. Belohlavek, 1999; R. Belohlavek, 2004; S. Pollandt, 1997].     

Applications of Euclidean constructions to computer graphics

This paper presents an interactive graphics system called L.E.G.O. The purpose of L.E.G.O. is to model two- and three-dimensional objects using Euclidean geometry constructions. L.E.G.O. has a layered structure which makes it convenient to use, both for the experienced programmer and the novice. The programs may be written in a compiled language (C), written interactively in an interpreted language (LISP) or developed using a graphical interface in a multiple-window environment. Applications of L.E.G.O. include computer-assisted instruction of geometry and computer graphics, geometric modeling, and kinematic analysis. The use of imperative constructions and the powerful interface based on the idea of graphical programming are the most distinctive features of the system.     

Generation of templates for low-resolution text recognition using a hypothesis graph

We propose a recognition method of character-string images captured by portable digital cameras. A challenging task in character-string recognition is the segmentation of characters. In the proposed method, a hypothesis graph is used for recognition-based segmentation of the character-string images. The hypothesis graph is constructed by the subspace method, using eigenvectors as conditionally elastic templates. To obtain these templates, a generation-based approach is introduced in the training stage. Various templates are generated to cope with low-resolution. We have experimentally proved that the proposed scheme achieves high recognition performance even for low-resolution character-string images. The text was submitted by the authors in English. Hiroyuki Ishida. Received his B.S. and M.S. degrees from the Department of Information Engineering and from the Graduate School of Information Science, respectively, at Nagoya University. He is currently pursuing a Ph.D. in Information Science at Nagoya University. Ichiro Ide. Received his B.S. degree from the Department of Electronic Engineering, his M.S. degree from the Department of Information Engineering, and his Ph.D. from the Department of Electrical Engineering at the University of Tokyo. He is currently an Associate Professor in the Graduate School of Information Science at Nagoya University. Tomokazu Takahashi. Received his B.S. degree from the Department of Information Engineering at Ibaraki University, and his M.S. and Ph.D. from the Graduate School of Science and Engineering at Ibaraki University. His research interests include computer graphics and image recognition. Hiroshi Murase. Received his B.S., M.S., and Ph.D. degrees from the Graduate School of Electrical Engineering at Nagoya University. He is currently a Professor in the Graduate School of Information Science at Nagoya University. He received the Ministry Award from the Ministry of Education, Culture, Sports, Science and Technology in Japan in 2003. He is a Fellow of the IEEE.     

Correction to “Antiwindup Design With Guaranteed Regions of Stability: An LMI-Based Approach”

In the footnote of J.M. Gomes da Silva, Jr. and S. Tarbouriech (see ibid., vol.50, no.1, p.110, 2005), we have stated that the value of 61.29 obtained by Y.Y. Cao et al. (2002) was apparently incorrect. In fact, the different values come from the fact that the example considered by Y.Y. Cao is slightly different from the one by J.M. Gomes da Silva, Jr. and S. Tarbouriech (2005). Hence, the result presented by Y.Y. Cao is indeed correct and we apologize to the authors for the misunderstanding     

Tauberian theorems for statistically convergent double sequences

In [F. Moricz, Tauberian theorems for Cesàro summable double sequences, Studia Math. 110 (1994) 83-96], Moricz has proved some Tauberian theorems for Cesàro summable double sequences and deduced the Tauberian theorems of Landau [E. Landau, Über die Bedeutung einiger neuerer Grenzwertsätze der Herren Hardy and Axer, Prac. Mat.-Fiz. 21 (1910) 97-177] and Hardy [G.H. Hardy, Divergent Series, Univ. Press, Oxford 1956] type. In [J.A. Fridy, M.K. Khan, Statistical extension of some classical Tauberian theorems, Proc. Amer. Math. Soc. 128 (2000) 2347-2355], Fridy and Khan have given statistical extensions of some classical Tauberian theorems. The concept of statistical convergence for double sequences has recently been introduced by Mursaleen and Edely [M. Mursaleen, Osama H. H. Edely, Statistical convergence of double sequences, J. Math. Anal. Appl. 288 (2003) 223-231] and by Moricz [F. Moricz, Statistical convergence of multiple sequences, Arch. Math. 81 (2003) 82-89] independently. In this paper we give some Tauberian theorems for statistically convergent double sequences. We have also provided some examples including an example to Problem 1 of Moricz [F. Moricz, Tauberian theorems for Cesàro summable double sequences, Studia Math. 110 (1994) 83-96].     

Reviews

MARK FRESKO CONSULTANCY. Sources of digital information. British Library R&D Report 6102. London: British Library Research and Development Department, 1994. No ISBN given. No price indicated. 260 pp.

CHRIS CLARE and GORDON STUTELEY. Information systems — strategy to design. London: Chapman and Hall, 1995. ISBN 0 412 576708. £16.99.

POPE, IVAN. Internet UK. Hemel Hempstead: Prentice Hall International, 294 pp. ISBN 013 190950, £19.95. SCHOFIELD, SUE. UK Internet book. Wokingham: Addison‐Wesley, 301 pp. ISBN 0201 42766 4, £19.95.

RIMMER, STEVE. Planet Internet. New York: Windcrest, 1995. ISBN 0 8306 24724. £22.95 (pbk).

JILL H. ELLSWORTH and MATTHEW V. ELLSWORTH. The Internet business book. London: John Wiley & Sons, 1994. $22.95, 376pp. ISBN 0 471058092.

JOHN S. QUARTERMAN and SMOOT CARL‐MITCHELL. The e‐mail companion: communicate effectively via the Internet and other global networks. Addison‐Wesley, 1994. 318 pp. ISBN 0 201 40658 6. $19.95.

McCLURE, C.R., MOEN, W.E. and RYAN, J. Libraries and the Internet/NREN: perspectives, issues and challenges. London: Mecklermedia 1994. $35.00. ISBN 0 89736 824 7. McCLURE, C.R., BERTOT, J.C., and ZWEIZIG, D.L. Public libraries and the Internet: study results, policy issues and recommendations. Washington: National Commission on Libraries and Information Science, 1994. No price or ISBN given.

FRANCES BLOMELEY. Networks and network services: a user's guide. Immediate Publishing, 1994. ISBN 1–89831–00–03. 246 pp. £14.95.

NEIL SMITH (ed) ibraries, networks and Europe: a European networking study. British Library Research and Development Department, 1994. (LIR Series 101) 91 pp. ISBN 0 7123 3295 2. £25. (Distributed by Turpin Distribution Services Ltd, Blackhorse Road, Letchworth, Herts. SG6 1HN).

ALAN BRYANT. Creating successful bulletin board systems. Addison‐Wesley, 1994. ISBN 0–201–62668–3. $39.95.

Directory of electronic journals, newsletters and academic discussion lists. 4th ed. Compiled by Lisabeth A. King and Diane Kovacs, edited by Ann Okerson. Washington, DC: Association of Research Libraries, 1994. 575 pp. ISSN 1057–1337. $54 (paperback), $33 (ARL members).

THE BRITISH LIBRARY and ELECTRONIC PUBLISHING SERVICES LTD. Electronic publishing practice in the UK: LIR Report 95. University Press, Cambridge, 1994. 185 pp. ISBN 0 7123 3280 4. £30.00. (Distributed by Turpin Distribution Services, Blackhorse Road, Letchworth S96 1HN).

INTERNET WORLD'S On Internet 94: an international guide to electronic journals, newsletters, texts, discussion lists, and other resources on the Internet. edited by Tony Abbott with a Preface by Daniel P. Dem. Westport, London: Mecklermedia, 1994. £29.50 $45.00. ISBN 0–88736–929–4.

S. BANG. The Internet unleashed. Indianapolis: SAMS Publishing, 1994. $44.95. ISBN 0 672 30466 X.

GAIL K. DICKINSON, Selection and evaluation of electronic resources. Libraries Unlimited, 1994. ISBN 1 56308 098 2. £22.50.     

Increasing the efficiency of combinatorial logical data analysis in recognition and classification problems

Problems of increasing the efficiency of combinatorial logical data analysis in recognition problems are examined. A technique for correct conversion of initial information for reduction of its dimensionality is proposed. Results of testing this technique for problems of real medical prognoses are given. Djukova Elena V. Born 1945. Graduated from Moscow State University in 1967. Candidate’s degree in Physics and Mathematics in 1979. Doctoral degree in Physics and Mathematics in 1997. Dorodnicyn Computing Center, Russian Academy of Sciences, leading researcher. Moscow State University, lecturer. Moscow Pedagogical University, lecturer. Scientific interests: discrete mathematics and mathematical method of pattern recognition. Author of 70 papers. Peskov Nikolai V. Born 1978. Graduated from Moscow State University in 2000. Candidate’s degree in 2004. Dorodnicyn Computing Center, Russian Academy of Sciences, junior researcher. Scientific interests: discrete mathematics and mathematical methods of pattern recognition. Author of ten papers. Inyakin Andrey S. Born 1978. Graduated from Moscow State University in 2000. Dorodnicyn Computing Center, Russian Academy of Sciences, junior researcher. Scientific interests: discrete mathematics and mathematical methods of pattern recognition. Author of ten papers. Sakharov Aleksei A. Born 1980. Graduated from Moscow State University in 2003. Moscow Pedagogical University, graduate student. Scientific interests: discrete mathematics and mathematical method of pattern recognition. Author of three papers.     

The Grey-Rankin bound for nonbinary codes

The Grey-Rankin bound for nonbinary codes is obtained. Examples of codes meeting this bound are given. Original Russian Text Ş L.A. Bassalygo, S.M. Dodunekov, V.A. Zinoviev, T. Helleseth, 2006, published in Problemy Peredachi Informatsii, 2006, Vol. 42, No. 3, pp. 37–44. Supported in part by the Russian Foundation for Basic Research, project no. 06-01-00226.     

Progress in the Development of National Knowledge Infrastructure

This paper presents the recent process in a long-term research project,called National Knowledge Infrastructure(or NKI).Initiated in the early 2000,the project aims to develop a multi-domain shareable knowledge base for knowledge-intensive applications.To develop NKI,we have used domain-specific ontologies as a solid basis,and have built more than 600 ontologies.Using these ontologies and our knowledge acquisition methods,we have extracted about 1.1 millions of domain assertions.For users to access our NKI knowledge,we have developed a uniform multi-modal human-knowledge interface.We have also implemented a knowledge application programming interface for various applications to share the NKI knowledge.     

Direct numerical and large eddy simulation of longitudinal flow along triangular array of rods using the lattice Boltzmann method

Results of direct numerical (DNS) and large eddy simulation (LES) of turbulent longitudinal flow in rod bundles are presented using the lattice Boltzmann method with the Bhatnagar–Gross–Krook collision operator [P.L. Bhatnagar, E.P. Gross, M. Krook, A model for collision processes in gases. I. Small amplitude processes in charged and neutral one-component systems, Phys. Rev. 94 (1954) 511; Y.H. Qian, d’Humiéres, P. Lallemand, Lattice BGK models for Navier-Stokes equation, Europhys. Lett. 17 (1992) 479] as a computational framework. The problem requires the accurate modeling of curved walls, to which the method proposed by Yu et al. [D. Yu, M.R. Luo, W. Shyy, Viscous flow computations with the method of lattice Boltzmann equation, Prog. Aerospace Sci. 39 (2003) 329] has been applied. The computational domain is a regular hexagonal prism around the rod. Opposite sides of the prism are coupled periodically. In the longitudinal direction periodical boundary conditions are applied and the flow is driven by a body force. Simulations were carried out using two three-dimensional lattices. It has been found that the application of the model with 19 velocities (D3Q19) gives qualitatively false result. However, we have found that the application of the model with 27 links (D3Q27) can provide the proper mean axial velocity profile, and it also predicts the secondary flow patterns deduced from measurements [A.C. Trupp, R.S. Azad, The structure of turbulent flow in triangular array rod bundles, Nucl. Eng. Des. 32 (1975) 47]. Flow pulsation phenomenon is also observed in our simulations just like in some recent measurements of Krauss and Meyer [T. Krauss, L. Meyer, Experimental investigation of turbulent transport of momentum and energy in heated rod bundle, Nucl. Eng. Des. 180 (1998) 185].     

Stabilization control for biped follow walking

This paper proposes a follow-walking motion for biped humanoid robots based on a stabilization control and a complete walking-motion pattern. To follow human motion, the unit patterns of the trunk, the waist and the lower limbs are generated and synthesized. During the follow motion, the biped robots are balanced by the stabilization control that calculates the combined motion of the trunk and the waist that compensates for the moments produced by the motion of the lower limbs. For confirmation of the follow-walking motion, we have developed a life-sized humanoid robot, WABIAN-RII (WAseda BIped humANoid robot-Revised II). It has a total of 43 mechanical d.o.f.; two 6-d.o.f. legs, two 10-d.o.f. arms, a 4-d.o.f. neck, 4-d.o.f. in the eyes and a torso with a 3-d.o.f. waist.     

Encrypting the global information infrastructure

Information. Global conncetivity. Electronic commerce. Competition. Economic espionage. Global organized crime. Chemical, biological, and nuclear weapons. Terrorism. Conflict. Economic and social instability. Violations of privacy and human rights. Erosion of trust. These are some of the global realities we live with today. They explain why cryptography must be an integral part of the Global Information Infrastructure to protect privacy, intellectual property, and financial assets, and to provide a foundation of trust for electronic commerce. They also explain why we ought to proceed thoughtfully in the deployment of this technology.     

Generating rooted triangulations without repetitions

We use the reverse search technique to give algorithms for generating all graphs onn points that are 2- and 3-connected planar triangulations withr points on the outer face. The triangulations are rooted, which means the outer face has a fixed labelling. The triangulations are produced without duplications inO(n ²) time per triangulation. The algorithms useO(n) space. A program for generating all 3-connected rooted triangulations based on this algorithm is available by ftp.This research was supported by N.S.E.R.C. Grant Number A3013, F.C.A.R. Grant Number EQ1678, and a bilateral exchange from J.S.P.S./N.S.E.R.C.     

Double hashing with passbits

Double hashing with bucket capacity one is augmented with multiple passbits to obtain significant reduction to unsuccessful search lengths. This improves the analysis of Martini et al. [P.M. Martini, W.A. Burkhard, Double hashing with multiple passbits, Internat. J. Found. Theoret. Comput. Sci. 14 (6) (2003) 1165-1188] by providing a closed form expression for the expected unsuccessful search lengths.     

Automatic computation of the travelling wave solutions to nonlinear PDEs

Various extensions of the tanh-function method and their implementations for finding explicit travelling wave solutions to nonlinear partial differential equations (PDEs) have been reported in the literature. However, some solutions are often missed by these packages. In this paper, a new algorithm and its implementation called TWS for solving single nonlinear PDEs are presented. TWS is implemented in Maple 10. It turns out that, for PDEs whose balancing numbers are not positive integers, TWS works much better than existing packages. Furthermore, TWS obtains more solutions than existing packages for most cases.

Program summary

Program title:TWSCatalogue identifier:AEAM_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEAM_v1_0.htmlProgram obtainable from:CPC Program Library, Queen's University, Belfast, N. IrelandLicensing provisions:Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.htmlNo. of lines in distributed program, including test data, etc.:1250No. of bytes in distributed program, including test data, etc.:78 101Distribution format:tar.gzProgramming language:Maple 10Computer:A laptop with 1.6 GHz Pentium CPUOperating system:Windows XP ProfessionalRAM:760 MbytesClassification:5Nature of problem:Finding the travelling wave solutions to single nonlinear PDEs.Solution method:Based on tanh-function method.Restrictions:The current version of this package can only deal with single autonomous PDEs or ODEs, not systems of PDEs or ODEs. However, the PDEs can have any finite number of independent space variables in addition to time t.Unusual features:For PDEs whose balancing numbers are not positive integers, TWS works much better than existing packages. Furthermore, TWS obtains more solutions than existing packages for most cases.Additional comments:It is easy to use.Running time:Less than 20 seconds for most cases, between 20 to 100 seconds for some cases, over 100 seconds for few cases.References:[1] E.S. Cheb-Terrab, K. von Bulow, Comput. Phys. Comm. 90 (1995) 102.[2] S.A. Elwakil, S.K. El-Labany, M.A. Zahran, R. Sabry, Phys. Lett. A 299 (2002) 179.[3] E. Fan, Phys. Lett. 277 (2000) 212.[4] W. Malfliet, Amer. J. Phys. 60 (1992) 650.[5] W. Malfliet, W. Hereman, Phys. Scripta 54 (1996) 563.[6] E.J. Parkes, B.R. Duffy, Comput. Phys. Comm. 98 (1996) 288.