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.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
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
Hashemiparast S.M.: Numerical integration using local Taylor expansions in nodes. Appl. Math. Comp. 192, 332–336 (2007)
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)
Ross B.: Fractional Caculus and its Application. Springer, Berlin (1974)
Samko S., Kilbas A., Marichev O.: Fractional Integral and Derivative-Theory and Applications. Gordon and Breach, New York (1993)
Podlubny I.: Fractional Differential Equations. Academic Press, London (1999)
Szego G.: Orthogonal Polynomials, vol. 23, 4th edn. AMS Coll. Publ., New York (1975)
Shen, J., Tang, T., Wang, L.: Spectral Methods Algorithem, Analysis and Applications
Atkinson K., Han W.: Theoritical Numerical Analysis, 2nd edn. Springer, Berlin (2005)
Temme N.M.: Gamma Functions: An Introduction to the Classical Functions of Mathematical Physics. Wiley, London (1996)
Cafagna, D.: Fractional Calculus: A Mathematical tool from the past for present engineers. IEEE Industrial Electronics Magazine (Summer) 35–40 (2007)
Odibat Z.: Approximations of fractional integrals and Caputo fractional drivatives. App. Math. Compu. 178, 527–533 (2006)
Gelfand, I.M., Shilov, G.E.: Generalized Functions. Nauka, Moscow 1 (1959)
Anastasio T.J.: The fractional-order dynamics of brainstem vestibulo-oculomotor neurons. Biol. Cybernet 72, 69–79 (1994)
Mainardi F.: Fractional relaxation-oscillation and fractional diffusion-wave phenomena. Chaos Solitons Fract. 9, 146–177 (1996)
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)
Westerlund S.: Dead Matter has Memory Causal Consulting. Kalmar, Sweden (2002)
Oldham K., Spanier J.: The Fractional Calculus: Theory and Application of Differentiation and Integration to Arbitrary Order. Academic Press, London (1974)
Miller K., Ross B.: An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley, New York (1993)
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
Hasegawa T., Sugiura H.: Quadrature rule for Abel‘s equations: uniformly approximation fractional derivatives. J. Comp. Appl. Math. 223, 459–468 (2009)
Tenreiro Machado J.A.: Fractional derivatives: probability interpretation and frequency response of rational approximations. Commun Nonlinear Sci. Numer Simulat. 14, 3492–3497 (2009)
Tenreiro Machado J., Kiryakova V., Mainardi F.: Recent history of fractional calculus. Commun. Nonlinear. Sci. Numer. Simulat. 16, 1140–1153 (2011)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
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
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11565-011-0120-x