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Computational and theoretical aspects of Romanovski-Bessel polynomials and their applications in spectral approximations

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Abstract

Our concern in this paper is with the essential properties of a finite class of orthogonal polynomials with respect to a weight function related to the probability density function of the inverse gamma distribution over the positive real line. We present some basic properties of the Romanovski-Bessel polynomials, the Romanovski-Bessel-Gauss-type quadrature formulae and the associated interpolation, discrete transforms, spectral differentiation and integration techniques in the physical and frequency spaces, and basic approximation results for the weighted projection operator in the nonuniformly weighted Sobolev space. We discuss the relationship between such kinds of finite orthogonal polynomials and other classes of finite and infinite orthogonal polynomials. Moreover, we propose adaptive spectral tau and collocation methods based on Romanovski-Bessel polynomial basis for solving linear and nonlinear high-order differential equations, respectively.

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Acknowledgements

We would like to dedicate this paper to our co-author Howayda Abo-Gabal, who unexpectedly and tragically passed away during the review process. Howayda Abo-Gabal was an energetic early-career master’s student who inspired those around her with her passion for discovery, grasp of complexity, and ability to communicate research. She was also a wonderful friend and colleague who will be missed dearly. The first author (MAZ) wishes to acknowledge the partial support of the Science Committee of the Ministry of Education and Science of the Republic of Kazakhstan (Grant ”Dynamical Analysis and Synchronization of Complex Neural Networks with Its Applications”).

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Authors and Affiliations

  1. Department of Applied Mathematics, Physics Division, National Research Centre, Dokki, Cairo, 12622, Egypt

    Mahmoud A. Zaky

  2. Institute of Mathematics and Mathematical Modeling, Almaty, Kazakhstan

    Mahmoud A. Zaky

  3. Department of Mathematics, Faculty of Science, Cairo University, Giza, 12613, Egypt

    Howayda Abo-Gabal & Eid H. Doha

  4. Faculty of Education, Matrouh University, Matrouh, Egypt

    Ramy M. Hafez

Authors
  1. Mahmoud A. Zaky
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  2. Howayda Abo-Gabal
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  3. Ramy M. Hafez
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  4. Eid H. Doha
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Correspondence to Mahmoud A. Zaky.

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Zaky, M.A., Abo-Gabal, H., Hafez, R.M. et al. Computational and theoretical aspects of Romanovski-Bessel polynomials and their applications in spectral approximations. Numer Algor 89, 1567–1601 (2022). https://doi.org/10.1007/s11075-021-01165-y

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  • DOI: https://doi.org/10.1007/s11075-021-01165-y

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