Définition

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

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

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

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

Une autre définition possible de ces polynômes est :

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

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

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

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 :

• cryogenique^[3]
• biologie^[4]
• Systèmes Dynamiques^[5]
• Systèmes Non-Lineaires^[6],[7]
• Théorie de l’Approximation^[8]
• thermodynamique^{[9],[10],[11]}
• mécanique^[12]
• Hydrologie^[13]
• Dynamique Moleculaire^[14]
• Manifolds^[15]
• calorimetrie^[16]
• biophysique^[17]
• Photovoltaique^[18]
• Analyse Complexe^[19]
• Analyse Matricielle^[20]

Références

1. ↑ O. D. Oyodum, O. B. Awojoyogbe, M. Dada, K. B. Ben Mahmoud and J. Magnuson, "Comment on 'Enhancement of pyrolysis spray disposal performance using thermal time-response to precursor uniform deposition' by K. Boubaker, A. Chaouachi, M. Amlouk and H. Bouzouita. On the earliest definition of the Boubaker polynomials, Eur. Phys. J. Appl. Phys., 46 2 (2009) 21201 DOI: http://dx.doi.org/10.1051/epjap/2009036
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

Liens additionnels

• Encyclopedia of Physics Research:
• NASA: USA-Physics Abstract Service Database

and

• La Presse

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

• John Wiley & Sons

• AIAA American Institute of Aeronautics and Astronautics, Inc.

• ASME American Society of Mechanical Engineers, Inc.

• WS World Scientific Publishing Co Pte Ltd

• Publications académiques

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

• Articles

Articlies à Sciendedirect.com:

, ,,, ,,,,,

• ENEA Ente Nazionale per le Energie Alternative

• Taylor and Francis

• Université de San Pietroburg

• China National Publication Corp. ,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,, et .