Authors:
Nazeer Ahmed Khoso,Muhammad Mujtaba Shaikh,Abdul Wasim Shaikh,DOI NO:
https://doi.org/10.26782/jmcms.2020.11.00011Keywords:
Orthogonalization,Boubaker polynomials,zeros,Recurrence relation,Gram-Schmidt process,Sturm-Liouville form,Abstract
In this work, we explore some unknown properties of the Boubaker polynomials. The orthogonalization of the Boubaker polynomials has not been discussed in the literature. Since most of the application areas of such polynomial sequences demand orthogonal polynomials, the orthogonality of the Boubaker polynomials will help extend its theareas of application. We investigate orthogonality of classical Boubaker polynomials using Sturm-Liouville form and then apply the Gram-Schmidt orthogonalization process to develop modified Boubaker polynomials which are also orthogonal. Some classical properties, like orthogonality and orthonormality relation and zeros, of the modified Boubaker polynomials, have been proved. The contributions from this study have an impact on the further application of modified Boubaker polynomials to not only the cases where classical polynomials could be used but also in cases where the classical ones could not be used due to orthogonality issue.Refference:
I. Abramowitz, M. and Stegun, I. A. (Eds.). “Orthogonal Polynomials.” Ch. 22 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 771-802, 1972.
II. Ahmed, I. N. (2020). Numerical Solution for Solving Two-Points Boundary Value Problems Using Orthogonal Boubaker Polynomials. Emirates Journal for Engineering Research, 25(2), 4.
III. Amlouk, A., Boubaker, K., & Amlouk, M. (2010). SnO2 thin films morphological and optical properties in terms of the Boubaker Polynomials Expansion Scheme BPES-related Opto-Thermal Expansivity ψAB. Journal of Alloys and Compounds, 490(1-2), 602-604.
IV. Andrews, G. E.; Askey, R.; and Roy, R. “Jacobi Polynomials and Gram Determinants” and “Generating Functions for Jacobi Polynomials.” §6.3 and 6.4 in a Functions. Cambridge, England: Cambridge University Press, pp. 293-306, 1999.
V. Andrews, G. E.; Askey, R.; and Roy, R. “Laguerre Polynomials.” §6.2 in Special Functions. Cambridge, England: Cambridge University Press, pp. 282-293, 1999
VI. Barry, P. (2013). On the connection coefficients of the Chebyshev-Boubaker polynomials. The Scientific World Journal, 2013.
VII. 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
VIII. Boubaker, K. (2011). Boubaker polynomials expansion scheme (BPES) solution to Boltzmann diffusion equation in the case of strongly anisotropic neutral particles forward–backward scattering. Annals of Nuclear Energy, 38(8), 1715–1717.
IX. Boubaker, K., Chaouachi, A., Amlouk, M., & Bouzouita, H. (2007). Enhancement of pyrolysis spray disposal performance using thermal time-response to precursor uniform deposition. The European Physical Journal-Applied Physics, 37(1), 105-109.
X. Carlitz, L. “A Note on the Bessel Polynomials.” Duke Math. J. 24, 151-162, 1957.
XI. Chew, W. C., & Kong, J. A. (1981, March). Asymptotic formula for the capacitance of two oppositely charged discs. In Mathematical Proceedings of the Cambridge Philosophical Society (Vol. 89, No. 2, pp. 373-384). Cambridge University Press
XII. Dada, M., Awojoy ogbe, O. B., Hasler, M. F., Mahmoud, K. B. B., & Bannour, A. (2008). Establishment of a Chebyshev-dependent inhomogeneous second order differential equation for the applied physics-related Boubaker-Turki polynomials. Applications and Applied Mathematics: An International Journal, 3(2), 329-336.
XIII. Dubey, B., Zhao, T.G., Jonsson, M., Rahmanov, H., 2010. A solution to the accelerated-predator-satiety Lotka–Volterra predator–prey problem using Boubaker polynomial expansion scheme. J. Theor. Biol. 264 (1), 154–160.
XIV. Labiadh, H., & Boubaker, K. (2007). A Sturm-Liouville shaped characteristic differential equation as a guide to establish a quasi-polynomial expression to the Boubaker polynomials. Дифференциальные уравнения и процессы управления, (2), 117-133.
XV. Milovanović, G. V., & Joksimović, D. (2012). Some properties of Boubaker polynomials and applications. doi:10.1063/1.4756326
XVI. Milgram, A. (2011). The stability of the Boubaker polynomials expansion scheme (BPES)-based solution to Lotka–Volterra problem. Journal of Theoretical Biology, 271(1), 157–158. doi:10.1016/j.jtbi.2010.12.002
XVII. Ouda, E. H., Ibraheem, S. F., & Fahmi, I. N. A. (2016). Indirect Method for Optimal Control Problem Using Boubaker Polynomial. Baghdad Science Journal, 13, 1.
XVIII. Shaikh, M. M., & Boubaker, K. (2016). An efficient numerical method for computation of the number of complex zeros of real polynomials inside the open unit disk. Journal of the Association of Arab Universities for Basic and Applied Sciences, 21(1), 86–91.
XIX. Slama, S., Bessrour, J., Boubaker, K., Bouhafs, M., 2008b. A dynamical model for investigation of A3 point maximal spatial evolution during resistance spot welding using Boubaker polynomials. Eur. Phys. J. Appl. Phys. 44 (03), 317–322.
XX. Yücel, U. (2010). The Boubaker Polynomials Expansion Scheme for Solving Applied-physics Nonlinear high-order Differential Equations. Studies in Nonlinear Science, 1(1), 1-7.
XXI. Zhang, D. H., & Li, F. W. (2010). Boubaker Polynomials Expansion Scheme (BPES) optimisation of copper tin sulfide ternary materials precursor’s ratio-related properties. Materials Letters, 64(6), 778-780.