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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. Cvetkovi´c, G.V. Milovanovi´c, Facta Univ. Ser. Math. Inform. 19 (2004), 17–36] and [G.V. Milovanovi´c, A.S. Cvetkovi´c, Math. Balkanica 26 (2012), 169–184]). Several numerical examples are presented and they show the effectiveness of the proposed approach.
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