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Approximation of fractional derivatives via Gauss integration

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Abstract

This paper considers the approximations of three classes of fractional derivatives (FD) using modified Gauss integration (MGI) and Gauss-Laguerre integration (GLI). The main solutions of these fractional derivatives depend on the inverse of Laplace transforms, which are handled by these procedures. In the modified form of the integration, the weights and nodes are obtained by means of a difference equation that, gives a proper approximation form for the inverse of Laplace transform and hence the fractional derivatives. Theorems are established to indicate the degree of exactness and boundary of the error of the solutions. Numerical examples are given to illuminate the results of the application of these methods.

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References

  1. Hashemiparast, S.M., Fallahgoul, H.: Approximation of Laplace transform of fractional derivatives via Clenshaw-Curtis integration. Int. Jour. Comp. Math. 78, 1224–1238 (2011). doi:10.1080/00207160.2010.499935

    Article  Google Scholar 

  2. Hashemiparast S.M.: Numerical integration using local Taylor expansions in nodes. Appl. Math. Comp. 192, 332–336 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  3. Hashemiparast S.M., Eslahchi M.R., Dehghan M.: Determination of nodes in numerical integration rules using difference equations. Appl. Math. Comp. 176, 117–122 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  4. Ross B.: Fractional Caculus and its Application. Springer, Berlin (1974)

    Google Scholar 

  5. Samko S., Kilbas A., Marichev O.: Fractional Integral and Derivative-Theory and Applications. Gordon and Breach, New York (1993)

    Google Scholar 

  6. Podlubny I.: Fractional Differential Equations. Academic Press, London (1999)

    MATH  Google Scholar 

  7. Szego G.: Orthogonal Polynomials, vol. 23, 4th edn. AMS Coll. Publ., New York (1975)

    Google Scholar 

  8. Shen, J., Tang, T., Wang, L.: Spectral Methods Algorithem, Analysis and Applications

  9. Atkinson K., Han W.: Theoritical Numerical Analysis, 2nd edn. Springer, Berlin (2005)

    Book  Google Scholar 

  10. Temme N.M.: Gamma Functions: An Introduction to the Classical Functions of Mathematical Physics. Wiley, London (1996)

    Google Scholar 

  11. Cafagna, D.: Fractional Calculus: A Mathematical tool from the past for present engineers. IEEE Industrial Electronics Magazine (Summer) 35–40 (2007)

  12. Odibat Z.: Approximations of fractional integrals and Caputo fractional drivatives. App. Math. Compu. 178, 527–533 (2006)

    MathSciNet  MATH  Google Scholar 

  13. Gelfand, I.M., Shilov, G.E.: Generalized Functions. Nauka, Moscow 1 (1959)

  14. Anastasio T.J.: The fractional-order dynamics of brainstem vestibulo-oculomotor neurons. Biol. Cybernet 72, 69–79 (1994)

    Article  Google Scholar 

  15. Mainardi F.: Fractional relaxation-oscillation and fractional diffusion-wave phenomena. Chaos Solitons Fract. 9, 146–177 (1996)

    Google Scholar 

  16. Maria da Graca M., Duarte F.B.M., Tenreiro Machado J.A.: Fractional dynamics in the trajectory control of redundant manipulators. Commun. Nonlinear Sci. Numer. Simulat. 13, 1836–1844 (2008)

    Article  Google Scholar 

  17. Westerlund S.: Dead Matter has Memory Causal Consulting. Kalmar, Sweden (2002)

    Google Scholar 

  18. Oldham K., Spanier J.: The Fractional Calculus: Theory and Application of Differentiation and Integration to Arbitrary Order. Academic Press, London (1974)

    Google Scholar 

  19. Miller K., Ross B.: An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley, New York (1993)

    MATH  Google Scholar 

  20. Hasegawa, T., Sugiura, H.: Uniform approximation to fractional derivatives of functions of algebric singularity. J. Comp. Appl. Math. doi:10.1016/j.cam2008.09.018

  21. Hasegawa T., Sugiura H.: Quadrature rule for Abel‘s equations: uniformly approximation fractional derivatives. J. Comp. Appl. Math. 223, 459–468 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  22. Tenreiro Machado J.A.: Fractional derivatives: probability interpretation and frequency response of rational approximations. Commun Nonlinear Sci. Numer Simulat. 14, 3492–3497 (2009)

    Article  Google Scholar 

  23. Tenreiro Machado J., Kiryakova V., Mainardi F.: Recent history of fractional calculus. Commun. Nonlinear. Sci. Numer. Simulat. 16, 1140–1153 (2011)

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Faculty of Science, K. N. Toosi University of Technology, P. O. Box 16315-1618, Tehran, Iran

    S. M. Hashemiparast & H. Fallahgoul

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  1. S. M. Hashemiparast
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  2. H. Fallahgoul
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Correspondence to H. Fallahgoul.

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Hashemiparast, S.M., Fallahgoul, H. Approximation of fractional derivatives via Gauss integration. Ann Univ Ferrara 57, 67–87 (2011). https://doi.org/10.1007/s11565-011-0120-x

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  • DOI: https://doi.org/10.1007/s11565-011-0120-x

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