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BY-NC-ND 3.0 license Open Access Published by De Gruyter September 3, 2014

Nonstandard Gauss—Lobatto quadrature approximation to fractional derivatives

Shahrokh Esmaeili and Gradimir Milovanović


A family of nonstandard Gauss-Jacobi-Lobatto quadratures for numerical calculating integrals of the form ∫-11 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.

[1] G.E. Andrews, R. Askey, R. Roy, Special Functions. Encyclopedia of Mathematics and its Applications, 71, Cambridge University Press, Cambridge (1999). in Google Scholar

[2] T.M. Atanacković, S. Pilipović, B. Stanković, D. Zorica, Fractional Calculus with Applications in Mechanics: Vibrations and Diffusion Processes. ISTE, London — Wiley, Hoboken (2014). in Google Scholar

[3] D. Baleanu, K. Diethelm, E. Scalas, J. J. Trujillo, Fractional Calculus: Models and Numerical Methods. World Scientific, Singapore (2012). 10.1142/8180Search in Google Scholar

[4] H. Brass, K. Petras, Quadrature Theory: The Theory of Numerical Integration on a Compact Interval. American Mathematical Soc., Providence, RI. (2011). in Google Scholar

[5] P.L. Butzer, U. Westphal, An introduction to fractional calculus, In: Applications of Fractional Calculus in Physics. World Sci. Publ., River Edge, NJ (2000), 1–85. in Google Scholar

[6] M. Caputo, Linear models of dissipation whose Q is almost frequency independent — II. Geophysical Journal of the Royal Astronomical Society 13 (1967), 529–539. in Google Scholar

[7] A.S. Cvetković, G.V. Milovanović, The Mathematica package ”OrthogonalPolynomials”. Facta Univ. Ser. Math. Inform. 9 (2014), 17–36. Search in Google Scholar

[8] K. Diethelm, The Analysis of Fractional Differential Equations. Springer-Verlag, Berlin, 2010. in Google Scholar

[9] K. Diethelm, Error bounds for the numerical integration of functions with limited smoothness. SIAM J. Numer. Anal. 52, No 2 (2014), 877–879. in Google Scholar

[10] S. Esmaeili, M. Shamsi, M. Dehghan, Numerical solution of fractional differential equations via a Volterra integral equation approach. Centr. Eur. J. Phys. 11 (2013), 1470–1481. in Google Scholar

[11] S. Esmaeili, M. Shamsi, Y. Luchko, Numerical solution of fractional differential equations with a collocation method based on Müntz polynomials. Comput. Math. Appl. 62 (2011), 918–929. in Google Scholar

[12] D. Funaro, Polynomial Approximation of Differential Equations. Springer-Verlag, Berlin (1992). 10.1007/978-3-540-46783-0Search in Google Scholar

[13] R. Garrappa, M. Popolizio, Evaluation of generalized Mittag-Leffler functions on the real line. Adv. Comput. Math. 39 (2013), 205–225. in Google Scholar

[14] W. Gautschi, Orthogonal Polynomials: Computation and Approximation. Oxford University Press, New York (2004). 10.1093/oso/9780198506720.001.0001Search in Google Scholar

[15] G.H. Golub, J.H. Welsch, Calculation of Gauss quadrature rules. Math. Comp. 23 (1969), 221–230. in Google Scholar

[16] R. Gorenflo, J. Loutchko, Y. Luchko, Computation of the Mittag-Leffler function and its derivatives, Fract. Calc. Appl. Anal. 5 (2002), 491–518. Search in Google Scholar

[17] R. Gorenflo, F. Mainardi, Fractional calculus: integral and differential equations of fractional order. In: Fractals and Fractional Calculus in Continuum Mechanics (Udine, 1996), CISMCourses and Lectures, 378, Springer, Vienna (1997), 223–276. in Google Scholar

[18] N. Hale, A. Townsend, Fast and accurate computation of Gauss-Legendre and Gauss-Jacobi quadrature nodes and weights. SIAM J. Sci. Comput. 35 (2013), 652–674. in Google Scholar

[19] N. Hale, L.N. Trefethen, Chebfun and numerical quadrature. Sci. China Ser. A 55 (2012), 1749–1760. in Google Scholar

[20] T. Hasegawa, H. Sugiura, Uniform approximation to fractional derivatives of functions of algebraic singularity. J. Comput. Appl. Math. 228 (2009), 247–253. in Google Scholar

[21] R. Herrmann, Fractional Calculus: An Introduction for Physicists. World Scientific, Singapore (2011). in Google Scholar

[22] A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations. North-Holland Mathematics Studies, 204, Elsevier Science B.V., Amsterdam (2006). in Google Scholar

[23] V. Kiryakova, Generalized Fractional Calculus and Applications. Pitman Research Notes in Math. Series, 301, Longman Scientific & Technical, Harlow; copubl. by John Wiley & Sons, Inc., New York (1994). Search in Google Scholar

[24] C. Li, F. Zheng, F. Liu, Spectral approximations to the fractional integral and derivative. Fract. Calc. Appl. Anal. 15 (2012), 383–406; DOI: 10.2478/s13540-012-0028-x; in Google Scholar

[25] F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity. An Introduction to Mathematical Models. Imperial College Press, London (2010). in Google Scholar

[26] F. Mainardi, Fractional calculus: some basic problems in continuum and statistical mechanics. In: Fractals and Fractional Calculus in Continuum Mechanics (Udine, 1996), CISM Courses and Lectures, 378, Springer, Vienna (1997), 291–348. in Google Scholar

[27] F. Mainardi, Y. Luchko, G. Pagnini, The fundamental solution of the space-time fractional diffusion equation. Fract. Calc. Appl. Anal. 4, No 2 (2001), 153–192. Search in Google Scholar

[28] G. Mastroianni, G.V. Milovanović, Interpolation Processes: Basic Theory and Applications. Springer-Verlag, Berlin (2008). in Google Scholar

[29] K.S. Miller, B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations. A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York (1993). Search in Google Scholar

[30] G.V. Milovanović, Müntz orthogonal polynomials and their numerical evaluation. In: Applications and Computation of Orthogonal Polynomials (W. Gautschi, G.H. Golub, and G. Opfer, Eds.), ISNM, Vol. 131, Birkhäuser, Basel (1999), 179–202. in Google Scholar

[31] G.V. Milovanović, Chapter 23: Computer algorithms and software packages. In: Walter Gautschi: Selected Works and Commentaries, Volume 3 (C. Brezinski, A. Sameh, Eds.), Birkhäuser, Basel (2014), 9–10. in Google Scholar

[32] G.V. Milovanović, A.S. Cvetković, Gaussian type quadrature rules for Müntz systems. SIAM J. Sci. Comput. 27 (2005), 893–913. in Google Scholar

[33] G.V. Milovanović, A.S. Cvetković, Nonstandard Gaussian quadrature formulae based on operator values. Adv. Comput. Math. 32 (2010), 431–486. in Google Scholar

[34] G.V. Milovanović, A.S. Cvetković, Special classes of orthogonal polynomials and corresponding quadratures of Gaussian type. Math. Balkanica 26, No 1–2 (2012), 169–184. Search in Google Scholar

[35] P. Novati, Numerical approximation to the fractional derivative operator. Numer. Math. 127 (2014), 539–566. in Google Scholar

[36] I. Podlubny, Fractional Differential Equations. Academic Press, San Diego, CA (1999). Search in Google Scholar

[37] I. Podlubny, M. Kacenak, The Matlab mlf code. MATLAB Central File Exchange (2001–2012), File ID: 8738. Search in Google Scholar

[38] S.G. Samko, A.A. Kilbas, O.I. Marichev, Fractional Integrals and Derivatives. Theory and Applications. Edited and with a foreword by S.M. Nikol’skiĭ, Transl. from the 1987 Russian original, Revised by the authors. Gordon and Breach Science Publishers, Yverdon (1993). Search in Google Scholar

[39] H. Sugiura, T. Hasegawa, Quadrature rule for Abel’s equations: uniformly approximating fractional derivatives. J. Comput. Appl. Math. 223 (2009), 459–468. in Google Scholar

[40] L.N. Trefethen, Approximation Theory and Approximation Practice, SIAM, Philadelphia, PA (2013). Search in Google Scholar

[41] D. Valério, J.J. Trujillo, M. Rivero, J.A.T. Machado, D. Baleanu, Fractional calculus: A survey of useful formulas. Eur. Phys. J. Special Topics 222 (2013), 1827–1846. in Google Scholar

Published Online: 2014-9-3
Published in Print: 2014-12-1

© 2014 Diogenes Co., Sofia

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

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