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A review of operational matrices and spectral techniques for fractional calculus

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Abstract

Recently, operational matrices were adapted for solving several kinds of fractional differential equations (FDEs). The use of numerical techniques in conjunction with operational matrices of some orthogonal polynomials, for the solution of FDEs on finite and infinite intervals, produced highly accurate solutions for such equations. This article discusses spectral techniques based on operational matrices of fractional derivatives and integrals for solving several kinds of linear and nonlinear FDEs. More precisely, we present the operational matrices of fractional derivatives and integrals, for several polynomials on bounded domains, such as the Legendre, Chebyshev, Jacobi and Bernstein polynomials, and we use them with different spectral techniques for solving the aforementioned equations on bounded domains. The operational matrices of fractional derivatives and integrals are also presented for orthogonal Laguerre and modified generalized Laguerre polynomials, and their use with numerical techniques for solving FDEs on a semi-infinite interval is discussed. Several examples are presented to illustrate the numerical and theoretical properties of various spectral techniques for solving FDEs on finite and semi-infinite intervals.

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

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Authors and Affiliations

  1. Department of Mathematics, Faculty of Science, King Abdulaziz University, Jidda, Saudi Arabia

    Ali H. Bhrawy

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

    Ali H. Bhrawy & Taha M. Taha

  3. Department of Electrical Engineering, Institute of Engineering, Polytechnic of Porto, Rua Dr. Antonio Bernardino de Almeida, 431, 4200-072, Porto, Portugal

    José A. Tenreiro Machado

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  1. Ali H. Bhrawy
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Bhrawy, A.H., Taha, T.M. & Machado, J.A.T. A review of operational matrices and spectral techniques for fractional calculus. Nonlinear Dyn 81, 1023–1052 (2015). https://doi.org/10.1007/s11071-015-2087-0

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