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Numerical algorithm for the variable-order Caputo fractional functional differential equation

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Abstract

While several high-order methods have been extensively developed for fixed-order fractional differential equations (FDEs), there are no such methods for variable-order FDEs. In this paper, we propose an accurate and robust approach to approximate the solution of functional Dirichlet boundary value problem with a type of variable-order Caputo fractional derivative. The proposed method is principally based on the shifted Chebyshev polynomials as basis functions and the matrix representation of variable-order fractional derivative of such polynomials. The underline variable-order FDE is then reduced to a system of algebraic equations, which greatly simplifies the solution process. Through numerical results, we confirm that the proposed scheme is very efficient and accurate for handling such problem.

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Acknowledgments

The authors are very grateful to the reviewers for carefully reading the paper and for their comments and suggestions which have improved the paper.

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Authors and Affiliations

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

    A. H. Bhrawy

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

    M. A. Zaky

  3. University of Science and Technology at Zewail City, 6th of October City, Giza, 12588, Egypt

    M. A. Zaky

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  1. A. H. Bhrawy
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Correspondence to A. H. Bhrawy.

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Bhrawy, A.H., Zaky, M.A. Numerical algorithm for the variable-order Caputo fractional functional differential equation. Nonlinear Dyn 85, 1815–1823 (2016). https://doi.org/10.1007/s11071-016-2797-y

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  • DOI: https://doi.org/10.1007/s11071-016-2797-y

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