Nonstandard Gauss—Lobatto quadrature approximation to fractional derivatives

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• 作者：Shahrokh Esmaeili ; Gradimir V. Milovanovi?
• 关键词：Primary 65D30 ; Secondary 33C45 ; 41A55 ; 65D32 ; fractional derivative ; quadrature rule ; Gaussian quadrature ; nodes ; weights ; software implementation ; Mathematica package
• 刊名：Fractional Calculus and Applied Analysis
• 出版年：2014
• 出版时间：December 2014
• 年：2014
• 卷：17
• 期：4
• 页码：1075-1099
• 全文大小：962 KB
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• 作者单位：Shahrokh Esmaeili (1)
  Gradimir V. Milovanovi? (2)
  
  1. Department of Applied Mathematics, University of Kurdistan, P.O. Box 416, Sanandaj, Iran
  2. Mathematical Institute, Serbian Academy of Sciences and Arts, Kneza Mihaila 36, 11001, Belgrade, Serbia
  

文摘

A family of nonstandard Gauss-Jacobi-Lobatto quadratures for numerical calculating integrals of the form ?span class="a-plus-plus stack"> -1 1 f-/em>(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. Cvetkovi?, G.V. Milovanovi?, Facta Univ. Ser. Math. Inform. 19 (2004), 17-6] and [G.V. Milovanovi?, A.S. Cvetkovi?, Math. Balkanica 26 (2012), 169-84]). Several numerical examples are presented and they show the effectiveness of the proposed approach.