10.57647/j.jtap.2025.1902.22

Application of a finite class of orthogonal polynomials related to T-student distribution in spectral methods

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  • Amir Hossein Salehi Shayegan1,
  1. Mathematics Department, Faculty of Basic Science, Khatam-ol-Anbia (PBU) University, Tehran, Iran.

Received: 2025-01-12

Revised: 2025-03-11

Accepted: 2025-04-06

Published 2025-04-10

Copyright (c) 2025 Amir Hossein Salehi Shayegan (Author)

Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 International License.

How to Cite

1.
Salehi Shayegan AH. Application of a finite class of orthogonal polynomials related to T-student distribution in spectral methods. J Theor Appl phys. 2025 Apr. 10;19(2 (April 2025):1-15. Available from: https://oiccpress.com/jtap/article/view/16592
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Abstract

Classical orthogonal polynomials play an important role in science and engineering. These polynomials
are divided into two parts; the infinite and finite sequences of orthogonal polynomials. In this paper, we
present a sequence of orthogonal polynomials (I(p)
n (x)) which is finitely orthogonal with respect to T-student
distribution on infinite interval (−∞,+∞). In doing so, in the first part of the paper, general properties of this
sequence such as orthogonality relation, Rodrigues type formula, recurrence relations and also some of its
applications such as Gauss quadrature formulas and so on are indicated. In addition, in the second part, we
show how one can apply I(p)
n (x) in approximations and particularly in spectral methods. Error analysis and
convergence of the method are thoroughly investigated. At the end, two numerical examples are given for the
efficiency and accuracy of the proposed method. In conclusion, the finite class of orthogonal polynomials
related to T-student distribution provides an efficient spectral method on unbounded domain.

Keywords

  • Spectral methods,
  • Finite class of orthogonal polynomials,
  • T-student distribution
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References

  1. H. S. Radeaf, B. M. Mahmmod, S. H. Abdulhussain, and D. Al-Jumaeily. “A steganography based on orthogonal moments. ”. Proceedings of the International Conference on Information and Communication Technology, pages 147–153, 2019. doi: 10.1145/3321289.3321324.
  2. F. Sabbane and H. Tairi. “Watermarking approach based on Hermite transform and a sliding window algorithm. ”. Multimedia Tools and Applications, 82:47635–47667, 2007. doi: 10.1007/s11042-023-15324-x.
  3. M. H. Annaby, J. Prestin H. A. Ayad, and M. A. Rushdi. “Multiparameter discrete transforms based on discrete orthogonal polynomials and their application to image watermarking.”. Signal Processing: Image Communication, 99(8):116434, 2021. doi: 10.1016/j.image.2021.116434.
  4. A. F. Nikiforov and V. B. Uvarov. “Special functions of mathematical physics.”. Birkhauser, Basel-Boston, 1988.
  5. S. A. Chihara. “An introduction to orthogonal polynomials. ”. Gordon and Breach, New York, 1978.
  6. G. Szego. “Orthogonal polynomials, 4th ed.”. Amer. Math. Soc., 1975.
  7. W. Koepf and M. Masjed-Jamei. “Two classes of special functions using Fourier transforms of some finite classes of classical orthogonal polynomials. ”. Proceedings of the American Mathematical Society, 135:3599–3606, 2007. doi: 10.1090/s0002-9939-07-08889-2.
  8. J. Shen, T. Tang, and L. Wang. “Spectral methods: algorithms, analysis and applications”. Springer, 2011.
  9. S. V. Tsynkov. “Numerical solution of problems on unbounded domains. A review. ”. Applied Numerical Mathematics, 27:465–532, 1998. doi: 10.1016/S0168-9274(98)00025-7.
  10. M. Masjedjamei. “Three finite classes of hypergeometric orthogonal polynomials and their application in functions approximation.”. Integral Transforms and Special Functions, 13:169–191, 2002. doi: 10.1080/10652460212898.
  11. P. Malik and A. Swaminathan. “Derivatives of a finite class of orthogonal polynomials related to inverse gamma distribution. ”. Applied Mathematics and Computation, 218:6251–6262, 2012. doi: 10.1016/j.amc.2011.11.078.
  12. B. Guo and J. Shen. “Laguerre-Galerkin method for nonlinear partial differential equations on a semi-infinite interval.”. Numerische Mathemati, 86:635–654, 2000. doi: 10.1007/pl00005413.
  13. J. Shen. “Stable and efficient spectral methods in unbounded domains using Laguerre functions. ”. SIAM Journal on Numerical Analysis, 38:1113–1133, 2000. doi: 10.1137/s0036142999362936.
  14. F. Liu, H. Li, and Z. Wang. “Spectral methods using generalized Laguerre functions for second and fourth order problems.”. Numerical Algorithms, 75:1005–1040, 2017. doi: 10.1007/s11075-016-0228-2.
  15. S. Moazzezi, A. H. Salehi Shayegan, and A. Zakeri. “A spectral element method for solving backward parabolic problems. ”. International Journal for Computational Methods in Engineering Science and Mechanics, 19:324–339, 2018. doi: 10.1080/15502287.2018.1502838.
  16. M. Masjed-Jamei and A. H. Salehi Shayegan. “A numerical method for solving Riccati differential equations.”. Iranian Journal of Mathematical Sciences and Informatics, 12:51–71, 2017. doi: 10.7508/ijmsi.2017.2.004.
  17. A. H. Salehi Shayegan, R. Bayat Tajvar, A. Ghanbari, and A. Safaie. “Inverse source problem in a space fractional diffusion equation from the final overdetermination.”. Applications of Mathematics, 64:469–484, 2019. doi: 10.21136/am.2019.0251-18.