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Spectral Treatment of One and Two Dimensional Second-Order BVPs via Certain Modified Shifted Chebyshev Polynomials

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Abstract

In this article, we construct new modified shifted third- and fourth-kinds Chebyshev polynomials. Two new operational matrices of derivatives of these polynomials are established. These modified Chebyshev polynomials are employed as basis functions to treat multi-dimensional second-order boundary value problems (BVPs). We propose two numerical algorithms for solving the linear and non-linear second-order two-point BVPs in one dimension governed by the homogeneous and non-homogeneous boundary conditions. The Petrov–Galerkin method (PGM) is employed to treat the linear differential equations, whereas the collocation approach is used for treating the non-linear ones. The operational matrices of derivatives of the modified Chebyshev polynomials are utilized along with suitable spectral methods for transforming the differential equations governed by their conditions into linear/non-linear systems of algebraic equations that can be numerically solved. The PGM is extended to be capable of treating the two-dimensional second-order BVPs. The convergence of the modified Chebyshev expansions is investigated. Furthermore, some numerical tests are presented to assess the efficiency and applicability of the proposed algorithms. Besides, comparisons with some other methods are presented.

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Acknowledgements

The authors would like to express their gratitude to the referees for their critical reading of the article as well as their insightful remarks, which significantly improved the manuscript.

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Authors and Affiliations

  1. Department of Mathematics, Faculty of Education, Ain Shams University, Roxy, Cairo, Egypt

    Heba Ashry & G. M. Moatimid

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

    W. M. Abd-Elhameed & Y. H. Youssri

  3. Department of Mathematics, College of Science, University of Jeddah, Jeddah, 23218, Saudi Arabia

    W. M. Abd-Elhameed

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  1. Heba Ashry
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  2. W. M. Abd-Elhameed
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  3. G. M. Moatimid
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  4. Y. H. Youssri
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Ashry, H., Abd-Elhameed, W.M., Moatimid, G.M. et al. Spectral Treatment of One and Two Dimensional Second-Order BVPs via Certain Modified Shifted Chebyshev Polynomials. Int. J. Appl. Comput. Math 7, 248 (2021). https://doi.org/10.1007/s40819-021-01186-0

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