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Nonstandard Gauss-Lobatto quadrature approximation to fractional derivatives. (English) Zbl 1314.65037

Summary: A family of nonstandard Gauss-Jacobi-Lobatto quadratures for numerical calculating integrals of the form \(\int_{-1}^1 f'(x)(1-x)^\alpha \mathrm{d}x\), \(\alpha > -1\), 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 and G. V. Milovanović, Facta Univ., Ser. Math. Inf. 19, 17–36 (2004; Zbl 1081.33001); Math. Balk., New Ser. 26, No. 1–2, 169–184 (2012; Zbl 1272.33013)]). Several numerical examples are presented and they show the effectiveness of the proposed approach.

MSC:

65D30 Numerical integration
33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
41A55 Approximate quadratures
65D32 Numerical quadrature and cubature formulas
26A33 Fractional derivatives and integrals
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