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Version du 8 mai 2011 à 13:28

Les Polynômes de Boubaker sont les élément d’une série polynomiale à coefficients entiers :^[1]:

{\begin{aligned}B_{0}(x)&{}=1\\B_{1}(x)&{}=x\\B_{2}(x)&{}=x^{2}+2\\B_{3}(x)&{}=x^{3}+x\\B_{4}(x)&{}=x^{4}-2\\B_{5}(x)&{}=x^{5}-x^{3}-3x\\B_{6}(x)&{}=x^{6}-2x^{4}-3x^{2}+2\\B_{7}(x)&{}=x^{7}-3x^{5}-2x^{3}+5x\\B_{8}(x)&{}=x^{8}-4x^{6}+8x^{2}-2\\B_{9}(x)&{}=x^{9}-5x^{7}+3x^{5}+10x^{3}-7x\\&{}\,\,\,\vdots \end{aligned}}

Les Polynômes de Boubaker sont aussi définis par la formule récursive :

{\begin{aligned}B_{0}(x)&=1,\\B_{1}(x)&=x,\\B_{2}(x)&=x^{2}+2,\\B_{m}(x)&=xB_{m-1}(x)-B_{m-2}(x)\quad {\text{for }}m>2.\end{aligned}}

Une autre définition possible de ces polynômes est :

B_{n}(x)=\sum _{p=0}^{\lfloor n/2\rfloor }{\frac {n-4p}{n-p}}{\binom {n-p}{p}}(-1)^{p}x^{n-2p}

Par ailleurs, les Polynômes de Boubaker peuvent être définis à travers leur équation différentielle caractéristique ;

{\begin{aligned}(x^{2}-1)(3nx^{2}+n-2)y{''}+3x(nx^{2}+3n-2)y{'}-n(3n^{2}x^{2}+n^{2}-6n+8)y=0\,\end{aligned}}

Les Polynômes de Boubaker sont à la base de plusieurs suites entières publiées dans l’Encyclopédie des suites entières On-Line Encyclopedia of Integer Sequences (OEIS)^[2] et PlanetMath.

Applications

Polynômes de Boubaker ont des applications dans maintes domaines scientifiques :

Références

↑ On the earliest definition of the Boubaker polynomials http://runners.ritsumei.ac.jp/cgi-bin/swets/hold-query-e?mode=1&key=&idxno=29124246
↑ Sequences A135929 , A135936 par Neil J. A. Sloane, A137276 par Roger L. Bagula et Gary Adamson,A138476 , par A. Bannour, A137289, A136256, A136255 par R. L. Bagula à On-Line Encyclopedia of Integer Sequences
↑ Book:Cryogenics: Theory, Processes and Applications, Chapter 8: Cryogenics Vessels Thermal Profilng Using the Boubaker Polynomials Expansion Scheme Investigation , Editor: Allyson E.Hayes
↑ A solution to the accelerated-predator-satiety Lotka–Volterra predator–prey problem using Boubaker polynomial expansion scheme
↑ The stability of the Boubaker polynomials expansion scheme (BPES)-based solution to Lotka–Volterra problem
↑ The Comparative Boubaker Polynomials Expansion Scheme (BPES) and Homotopy Perturbation Method (HPM) for solving a standard nonlinear second-order boundary value problem
↑ [ http://web3.cas.usf.edu/main/depts/mth/7thde/data/Abstracts-7thDEDS-Tampa.pdf The boubaker polynomials expansion scheme for solving nonlinear science problems]
↑ Paul Barry, Aoife Hennessy, Meixner-Type Results for Riordan Arrays and Associated Integer Sequences, Chapter 6: The Boubaker polynomials
↑ [ http://www.springerlink.com/content/d78h761823628gl2/ H. Koçak, Z. Dahong, A. Yildirimn, A range-free method to determine antoine vapor-pressure heat transfer-related equation coefficients using the Boubaker polynomials expansion scheme ]
↑ H. Koçak, Z. Dahong, A. Yildirim, Analytical expression to temperature-dependent Kirkwood-Fröhlich dipole orientation parameter using the Boubaker Polynomials Expansion Scheme (BPES)
↑ [ http://pdf.aiaa.org/jaPreview/JTHT/2009/PVJA41850.pdf A. Belhadj, O. F. Onyango and N. Rozibaeva, Boubaker Polynomials Expansion Scheme-Related Heat Transfer Investigation Inside Keyhole Model]
↑ [ http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6TJ2-51J36F8-1&_user=10&_coverDate=03%2F31%2F2011&_rdoc=1&_fmt=high&_orig=gateway&_origin=gateway&_sort=d&_docanchor=&view=c&_searchStrId=1736224281&_rerunOrigin=google&_acct=C000050221&_version=1&_urlVersion=0&_userid=10&md5=a7eda39bb611bdf91b105cb6d9f7e420&searchtype=a D. H. Zhang, Study of a non-linear mechanical system using Boubaker polynomials expansion scheme BPES]
↑ Emna Gargouri-Ellouze, Noreen Sher Akbar, Sohail Nadeem,Modelling Nonlinear Bivariate Dependence Using the Boubaker Polynomials Copula The Boubaker polynomials
↑ W. X. Yue, H. Koçak, D. H. Zhang , A. Yıldırım, A second attempt to establish an analytical expression to steam-water dipole orientation parameter using the Boubaker polynomials expansion scheme
↑ [ http://www.mathem.pub.ro/apps/v12/A12-zh.pdf D. H. Zhang, L. Naing| title = The Boubaker polynomials expansion scheme BPES for solving a standard boundary value problem]
↑ A. Belhadj, J. Bessrour, M. Bouhafs and L. Barrallier, Experimental and theoretical cooling velocity profile inside laser welded metals using keyhole approximation and Boubaker polynomials expansion
↑ S. Amir Hossein A. E. Tabatabaei, T. Gang Z., O. Bamidele A. and Folorunsho O. Moses, Cut-off cooling velocity profiling inside a keyhole model using the Boubaker polynomials expansion scheme| url=http://www.citeulike.org/article/4834321
↑ S. Fridjine and M. Amlouk, A NEW PARAMETER-ABACUS FOR OTIMIZING PV-T HYBRID SOLAR DEVICES FUNCTIONAL MATERIALS USING BOUBAKER POLYNOMIALS EXPANSION SCHEME
↑ T. G. Zhao, Y. X. Wang , K. B. Ben Mahmoud, Limit and uniqueness of the Boubaker-Zhao polynomials single imaginary root sequence
↑ RECURRENCE RELATIONS FOR POLYNOMIAL SEQUENCES VIA RIORDAN MATRICES, Pages 24-25: BOUBAKER POLYNOMIALS associated Riordan matrix

Liens additionnels

Encyclopedia of Physics Research: [1]
NASA: USA-Physics Abstract Service Database

[2] and [3]

La Presse

[=http://www.lapresse.tn/index.php?opt=15&categ=4&news=63764], tunisie7arts.com

[4]

John Wiley & Sons

[5]

AIAA American Institute of Aeronautics and Astronautics, Inc.

[6]

ASME American Society of Mechanical Engineers, Inc.

[7]

WS World Scientific Publishing Co Pte Ltd

[8]

Publications académiques

[=http://www.diogenes.bg/ijam/]

Articles

[9]

[10]

Articlies à Sciendedirect.com:

[11], [12],[13],[14], [15],[16],[17],[18],[19],[20]

ENEA Ente Nazionale per le Energie Alternative

[21]

Taylor and Francis

[22]

Université de San Pietroburg

[23]

[24]

[1] On the earliest definition of the Boubaker polynomials http://runners.ritsumei.ac.jp/cgi-bin/swets/hold-query-e?mode=1&key=&idxno=29124246

[2] Sequences A135929 , A135936 par Neil J. A. Sloane, A137276 par Roger L. Bagula et Gary Adamson,A138476 , par A. Bannour, A137289, A136256, A136255 par R. L. Bagula à On-Line Encyclopedia of Integer Sequences

[3] Book:Cryogenics: Theory, Processes and Applications, Chapter 8: Cryogenics Vessels Thermal Profilng Using the Boubaker Polynomials Expansion Scheme Investigation , Editor: Allyson E.Hayes

[4] A solution to the accelerated-predator-satiety Lotka–Volterra predator–prey problem using Boubaker polynomial expansion scheme

[5] The stability of the Boubaker polynomials expansion scheme (BPES)-based solution to Lotka–Volterra problem

[6] The Comparative Boubaker Polynomials Expansion Scheme (BPES) and Homotopy Perturbation Method (HPM) for solving a standard nonlinear second-order boundary value problem

[7] [ http://web3.cas.usf.edu/main/depts/mth/7thde/data/Abstracts-7thDEDS-Tampa.pdf The boubaker polynomials expansion scheme for solving nonlinear science problems]

[8] Paul Barry, Aoife Hennessy, Meixner-Type Results for Riordan Arrays and Associated Integer Sequences, Chapter 6: The Boubaker polynomials

[9] [ http://www.springerlink.com/content/d78h761823628gl2/ H. Koçak, Z. Dahong, A. Yildirimn, A range-free method to determine antoine vapor-pressure heat transfer-related equation coefficients using the Boubaker polynomials expansion scheme ]

[10] H. Koçak, Z. Dahong, A. Yildirim, Analytical expression to temperature-dependent Kirkwood-Fröhlich dipole orientation parameter using the Boubaker Polynomials Expansion Scheme (BPES)

[11] [ http://pdf.aiaa.org/jaPreview/JTHT/2009/PVJA41850.pdf A. Belhadj, O. F. Onyango and N. Rozibaeva, Boubaker Polynomials Expansion Scheme-Related Heat Transfer Investigation Inside Keyhole Model]

[12] [ http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6TJ2-51J36F8-1&_user=10&_coverDate=03%2F31%2F2011&_rdoc=1&_fmt=high&_orig=gateway&_origin=gateway&_sort=d&_docanchor=&view=c&_searchStrId=1736224281&_rerunOrigin=google&_acct=C000050221&_version=1&_urlVersion=0&_userid=10&md5=a7eda39bb611bdf91b105cb6d9f7e420&searchtype=a D. H. Zhang, Study of a non-linear mechanical system using Boubaker polynomials expansion scheme BPES]

[13] Emna Gargouri-Ellouze, Noreen Sher Akbar, Sohail Nadeem,Modelling Nonlinear Bivariate Dependence Using the Boubaker Polynomials Copula The Boubaker polynomials

[14] W. X. Yue, H. Koçak, D. H. Zhang , A. Yıldırım, A second attempt to establish an analytical expression to steam-water dipole orientation parameter using the Boubaker polynomials expansion scheme

[15] [ http://www.mathem.pub.ro/apps/v12/A12-zh.pdf D. H. Zhang, L. Naing| title = The Boubaker polynomials expansion scheme BPES for solving a standard boundary value problem]

[16] A. Belhadj, J. Bessrour, M. Bouhafs and L. Barrallier, Experimental and theoretical cooling velocity profile inside laser welded metals using keyhole approximation and Boubaker polynomials expansion

[17] S. Amir Hossein A. E. Tabatabaei, T. Gang Z., O. Bamidele A. and Folorunsho O. Moses, Cut-off cooling velocity profiling inside a keyhole model using the Boubaker polynomials expansion scheme| url=http://www.citeulike.org/article/4834321

[18] S. Fridjine and M. Amlouk, A NEW PARAMETER-ABACUS FOR OTIMIZING PV-T HYBRID SOLAR DEVICES FUNCTIONAL MATERIALS USING BOUBAKER POLYNOMIALS EXPANSION SCHEME

[19] T. G. Zhao, Y. X. Wang , K. B. Ben Mahmoud, Limit and uniqueness of the Boubaker-Zhao polynomials single imaginary root sequence

[20] RECURRENCE RELATIONS FOR POLYNOMIAL SEQUENCES VIA RIORDAN MATRICES, Pages 24-25: BOUBAKER POLYNOMIALS associated Riordan matrix

[1]

[2]

[3]

[4]

[5]

[6]

[7]

[8]

[9]

[10]

[11]

[12]

[13]

[14]

[15]

[16]

[17]

[18]

[19]

[20]

Version du 8 mai 2011 à 13:25 modifier JackPotte (discussion \| contributions) Administrateurs d’interface, Administrateurs 28 512 modifications →Applications ← Modification précédente		Version du 8 mai 2011 à 13:28 modifier annuler JackPotte (discussion \| contributions) Administrateurs d’interface, Administrateurs 28 512 modifications mAucun résumé des modifications Modification suivante →
Ligne 1 :		Ligne 1 :
	[[File:Boubaker Polynomials (n=0-6), x=(-2,2).svg\|thumb\|right\|300px\|~~Polinomi~~ di Boubaker ''B''<sub>n</sub>(''x''), ~~con~~ ''n'' da 0 a 6.]]		[[File:Boubaker Polynomials (n=0-6), x=(-2,2).svg\|thumb\|right\|300px\|Polynômes de Boubaker ''B''<sub>n</sub>(''x''), avec ''n'' de 0 à 6.]]

	Les '''Polynômes de Boubaker''' sont les élément d’une série polynomiale à coefficients entiers :<ref>On the earliest definition of the '''Boubaker polynomials''' http://runners.ritsumei.ac.jp/cgi-bin/swets/hold-query-e?mode=1&key=&idxno=29124246</ref>:		Les '''Polynômes de Boubaker''' sont les élément d’une série polynomiale à coefficients entiers :<ref>On the earliest definition of the '''Boubaker polynomials''' http://runners.ritsumei.ac.jp/cgi-bin/swets/hold-query-e?mode=1&key=&idxno=29124246</ref>: