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Chebyshev spectral methods for multi-order fractional neutral pantograph equations

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Abstract

This paper is concerned with the application of the spectral tau and collocation methods to delay multi-order fractional differential equations with vanishing delay \(rx\ (0<r<1)\). The fractional derivatives are described in the Caputo sense. The model solution is expanded in terms of Chebyshev polynomials. The convergence of the proposed approaches is investigated in the weighted \(L^2\)-norm. Numerical examples are provided to highlight the convergence rate and the flexibility of this approach. Our results confirm that nonlocal numerical methods are best suited to discretize fractional differential equations as they naturally take the global behavior of the solution into account.

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Acknowledgements

The authors would like to express special thanks to the referees for carefully reading, constructive comments and valuable remarks which significantly improved the quality of this paper. The authors extend their appreciation to Distinguished Scientist Fellowship Program (DSFP) at King Saud University (Saudi Arabia).

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Authors and Affiliations

  1. Jiangsu Key Laboratory for NSLSCS, School of Mathematical Sciences, Nanjing Normal University, Nanjing, 210023, China

    S. S. Ezz-Eldien & Y. Wang

  2. College of Education, King Saud University, Riyadh, Saudi Arabia

    S. S. Ezz-Eldien & A. A. Aldraiweesh

  3. Department of Mathematics, Faculty of Science, New Valley University, El-Kharga, 72511, Egypt

    S. S. Ezz-Eldien

  4. Department of Mathematics and Statistics, College of Science, Al-Imam Mohammad Ibn Saud Islamic University, Riyadh, Saudi Arabia

    M. A. Abdelkawy

  5. Department of Mathematics, Faculty of Science, Beni-Suef University, Beni Suef, Egypt

    M. A. Abdelkawy

  6. Department of Applied Mathematics, National Research Centre, Dokki, Giza, 12622, Egypt

    M. A. Zaky

  7. Department of Electrical Engineering, Institute of Engineering, Polytechnic of Porto, Rua Dr. António Bernardino de Almeida, 431, 4249-015, Porto, Portugal

    J. Tenreiro Machado

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  1. S. S. Ezz-Eldien
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  2. Y. Wang
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  3. M. A. Abdelkawy
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  4. M. A. Zaky
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  5. A. A. Aldraiweesh
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Correspondence to S. S. Ezz-Eldien.

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Ezz-Eldien, S.S., Wang, Y., Abdelkawy, M.A. et al. Chebyshev spectral methods for multi-order fractional neutral pantograph equations. Nonlinear Dyn 100, 3785–3797 (2020). https://doi.org/10.1007/s11071-020-05728-x

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