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Nonstandard Gauss—Lobatto quadrature approximation to fractional derivatives

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Abstract

A family of nonstandard Gauss-Jacobi-Lobatto quadratures for numerical calculating integrals of the form ∫ 1-1 f′(x)(1-x)α dx, α > -1, is derived and applied to approximation of the usual fractional derivative. A software implementation of such quadratures was done by the recent Mathematica package OrthogonalPolynomials (cf. [A.S. Cvetković, G.V. Milovanović, Facta Univ. Ser. Math. Inform. 19 (2004), 17–36] and [G.V. Milovanović, A.S. Cvetković, Math. Balkanica 26 (2012), 169–184]). Several numerical examples are presented and they show the effectiveness of the proposed approach.

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Authors and Affiliations

  1. Department of Applied Mathematics, University of Kurdistan, P.O. Box 416, Sanandaj, Iran

    Shahrokh Esmaeili

  2. Mathematical Institute, Serbian Academy of Sciences and Arts, Kneza Mihaila 36, 11001, Belgrade, Serbia

    Gradimir V. Milovanović

Authors
  1. Shahrokh Esmaeili
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  2. Gradimir V. Milovanović
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Correspondence to Shahrokh Esmaeili.

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Dedicated to Professor Ivan Dimovski on the occasion of his 80th anniversary

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Esmaeili, S., Milovanović, G.V. Nonstandard Gauss—Lobatto quadrature approximation to fractional derivatives. Fract Calc Appl Anal 17, 1075–1099 (2014). https://doi.org/10.2478/s13540-014-0215-z

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