Boubaker Polynomials

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Definitions and recurrence relations[edit]

The Boubaker polynomials are the components of a polynomial sequence [1][2]:

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

The Boubaker polynomials are also defined in general mode through the recurrence relation:

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.

Note that the first three polynomials are explicitly defined, and that the formula can only be used for m > 2. Another definition of Boubaker polynomials is:

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

The Boubaker polynomials can be defined through the differential equation:

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


The Boubaker polynomials have generated many integer sequences in the On-Line Encyclopedia of Integer Sequences ([3] and PlanetMath.


Several times, last time in 2009, Wikipedia chose not to host an article on the subject of Boubakr polynomials, see w:Wikipedia:Articles for deletion/Boubaker polynomials (3rd nomination). This resource is about the polynomials and applications. However, the history of Wikipedia treatment of this topic and users involved with this topic may be studied and discussed at /Wikipedia.

" The Boubaker polynomials were established for the first by Boubaker (2006) as a guide for solving a one-dimensional formulation of heat transfer equation...
\frac{\partial^2 f(x,t)}{\partial x^2}=k\frac{\partial f}{\partial x}      (on the domain -H<x<0 and t>0) " [4]

The importance of this heat equation in applied mathematics is uncontroversial, as is illustrated in the next section.


The Boubaker polynomials have been used in different scientific fields:


  1. O.D. Oyodum, O.B. Awojoyogbe, M.K. Dada, J.N. Magnuson, Eur. Phys. J. Appl. Phys. Volume 46, pages 2120-21202, On the earliest definition of the Boubaker polynomials , (behind pay wall)
  2. Milovanovic, Gradimir V., et al. "Some properties of Boubaker polynomials and applications." AIP Conference Proceedings-American Institute of Physics. Vol. 1479. No. 1. 2012.
  3. Sequences A135929 , A135936 by Neil J. A. Sloane, A137276 by Roger L. Bagula et Gary Adamson,A138476 , by A. Bannour, A137289, A136256, A136255 by R. L. Bagula à On-Line Encyclopedia of Integer Sequences
  4. This is a direct quote from: Boubaker, K. (2007). On modified Boubaker polynomials: some differential and analytical properties of the new polynomials issued from an attempt for solving bi-varied heat equation. Trends in Applied Sciences Research, 2(6), 540-544. There is no 2006 reference in this article, and the reference cited as "accepted" in 2007 cannot be found on Google Scholar.
  5. citation|title= Book:Cryogenics: Theory, Processes and Applications, Chapter 8: Cryogenics Vessels Thermal Profilng Using the Boubaker Polynomials Expansion Scheme Investigation , Editor: Allyson E.Hayes
  6. Journal of Theoretical Biology (Elsevier)|id=doi:10.1016/j.jtbi.2010.12.002 B. Dubey, T.G. Zhao, M. Jonsson, H. Rahmanov,A solution to the accelerated-predator-satiety Lotka–Volterra predator–prey problem using Boubaker polynomial expansion scheme,
  7. Journal of Theoretical Biology (Elsevier)|id=doi:10.1016/j.jtbi.2010.01.026 A. Milgeam|title = The stability of the Boubaker polynomials expansion scheme (BPES)-based solution to Lotka–Volterra problem |
  8. Mathematical and Computer Modelling(Elsevier)|iddoi:10.1016/j.mcm.2011.02.031 H. Koçak, A. Yıldırım, D.H. Zhang, S.T. Mohyud-Din,The Comparative Boubaker Polynomials Expansion Scheme (BPES) and Homotopy Perturbation Method (HPM) for solving a standard nonlinear second-order boundary value problem,
  9. The 7th International Conference on Differential Equations and Dynamic Systems, University of South Florida, Tampa, Fmorida USA, 15-18 December 2010 <Page 40 > A. Yildirim,The boubaker polynomials expansion scheme for solving nonlinear science problems,
  10. Journal of Integer Sequences (JIS)Paul Barry, Aoife Hennessy,Meixner-Type Results for Riordan Arrays and Associated Integer Sequences, Chapter 6: The Boubaker polynomials
  11. Russian Journal of Physical Chemistry A, Focus on Chemistry (Springer) H. Koçak, Z. Dahong, A. Yildirim,A range-free method to determine antoine vapor-pressure heat transfer-related equation coefficients using the Boubaker polynomials expansion scheme
  12. Indian Journal of Physics(Springer) H. Koçak, Z. Dahong, A. Yildirim,Analytical expression to temperature-dependent Kirkwood-Fröhlich dipole orientation parameter using the Boubaker Polynomials Expansion Scheme (BPES)
  13. Jornal of Thermophysics and Heat Transfer (American Institute of Aeronautics and Astronautics) AIAA)A. Belhadj, O. F. Onyango and N. Rozibaeva,Boubaker Polynomials Expansion Scheme-Related Heat Transfer Investigation Inside Keyhole Model|
  14. D.H. Zhang, "Study of a non-linear mechanical system using Boubaker polynomials expansion scheme BPES," International Journal of Non-Linear Mechanics Volume 46, Issue 2, March 2011, Pages 443–445.[1]
  15. Studies in Nonlinear Sciences (SNS)Emna Gargouri-Ellouze, Noreen Sher Akbar, Sohail Nadeem,Modelling Nonlinear Bivariate Dependence Using the Boubaker Polynomials Copula The Boubaker polynomials
  16. Journal of Structural Chemistry (Springer) 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
  17. Applied Sciences,(Balkan Society of Geometers, Geometry Balkan Press) D. H. Zhang, L. Naing,The Boubaker polynomials expansion scheme BPES for solving a standard boundary value problem
  18. Journal of Thermal Analysis and Calorimetry(Akadémiai Kiadó, Springer Science & Kluwer Academic Publishers B.V.)|id=doi:10.1007/s10973-009-0094-4 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|
  19. Heat and Mass Transfer(Springer Berlin / Heidelberg)|id= Volume 45, Number 10 / août 2009, pages:1247-1251 doi:10.1007/s00231-009-0493-x 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| |
  21. citation T. G. Zhao, Y. X. Wang and K. B. Ben Mahmoud| title=Limit and uniqueness of the Boubaker-Zhao polynomials single imaginary root sequence | journal=International Journal of Mathematics and Computation | volume=1 |number=08 | ISSN=0974-5718 |
  22. A. Luzon , M. | last2=Moron | | title=RECURRENCE RELATIONS FOR POLYNOMIAL SEQUENCES VIA RIORDAN MATRICES, Pages 24-25: BOUBAKER POLYNOMIALS associated Riordan matrix |
  23. M. Agida , A. S. . | last2=Kumar |title=A Boubaker Polynomials Expansion Scheme Solution to Random Love’s Equation in the Case of a Rational Kernel || journal=El. Journal of theretical physics ( EJTP) |
  24. A. S. Kumar , An analytical solution to applied mathematics-related Love's equation using the ‘’’Boubaker polynomials’’’ expansion scheme| journal=International Journal of the Franklin Institute (elsevier) |
  25. B. Tirimula Rao, P. Srinivsu, C. Anantha Rao, K. Satya Vivek Vardhan , Jami Vidyadhari ,Page 8 : Boubaker polynomials ,
  26. Kiliç Bülent, Erdal Bas, Page 7, Citation 27: Boubaker polynomials ,

External links[edit]

citation|title=a New Parameter:. AN Abacus for Optimizing Pv-T Hybrid Solar Device Functional Materials Using the Boubaker Polynomials Expansion Scheme and citation|title=
citation|title=De Khawarizmi à Euler|journal=La Presse Magazine|date=January 9, 2008 aussi
citation|title=Le polynôme de Boubaker|journal=La Presse Magazine|date=April 22, 2007|issue=1019|page=6 fr
citation|title=A new polynomial sequence... The Boubaker Polynomials|journal=Numerical Methods for Partial Differential Equations NMPDE
citation|title=Solution to Heat Equation Using Boubaker Polynomials|journal=J. of Thermophysics and Heat Transfer
citation|title=Boubaker Polynomials Weak Solutions to a Robin Boundary Conditioned Dynamic-State Heat Transfer Problem |journal=International Journal of Heat Transfer
  • WS World Scientific Publishing Co Pte Ltd
citation|title=AMLOUK–BOUBAKER EXPANSIVITY..USING BOUBAKER POLYNOMIALS |journal=Functional Materials Letter
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citation|title=A new polynomial sequence ...The Boubaker Polynomials|journal=International Journal of Applied Mathematics
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citation|title=An attempt... using Boubaker Polynomials|journal= International Journal of Heat and Technology
citation|title=Numerical Distribution of Temperature During Welding Using Boubaker Polynomials |journal= Numerical Heat Transfer, Part A, Applications
citation|title=Establishment of an Ordinary Generating Function and a Christoffel-Darboux Type First-Order Differential Equation for the Heat Equation Related Boubaker-Turki Polynomials|journal= Journal of Differential Equations and C.P.
citation|title=Some new properties of the applied-physics related Boubaker polynomials
citation 3, Page 97 C.P.|=
Meixner-type results for Riordan arrays and associated integer sequences, Chap. 6 Boubaker Polynomials
An analytical solution to applied mathematics-related Love’s equation using the Boubaker polynomials expansion scheme
Some new features of the Boubaker polynomials expansion scheme
The Boubaker polynomials expansion scheme BPES for solving a standard boundary value problem