Abstract
New fractional derivatives, termed henceforth generalized fractional derivatives (GFDs), are introduced. Their definition is based on the concept that fractional derivatives (FDs) interpolate the integer- order derivatives. This idea generates infinite classes of FDs. The new FDs provide, beside the fractional order, any number of free parameters to better calibrate the response of a physical system or procedure. Their usefulness and consequences are subject of further investigation. Like the Caputo FD, the GFDs allow the application of initial conditions having direct physical significance. A numerical method is also developed for the solution of differential equations involving GFDs. Mechanical systems including fractional oscillators, viscoelastic plane bodies and plates described by such equations are analyzed.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Atanackovic T.M., Pilipovic S., Stankovic B., Zorica D.: Fractional Calculus with Applications in Mechanics: Vibrations and Diffusion Processes. Wiley, London (2014)
Atanackovic T.M., Pilipovic S., Stankovic B., Zorica D.: Fractional Calculus with Applications in Mechanics: Impact and Variational Principles. Wiley, London (2014)
Mainardi F.: Fractional Calculus and Waves in Linear Viscoelasticity. Imperial College Press, London (2010)
Kilbas A.A., Srivastava H.M., Trujillo J.J.: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam (2006)
Podlubny I.: Fractional Differential Equations. Academic Press, New York (1999)
Capelas de Oliveira, E., Machado, J.T.: A review of definitions for fractional derivatives and integrals. Math. Probl. Eng. 2014 (2014). doi:10.1155/2014/23845
Caputo M.: Linear model of dissipation whose Q is almost frequency independent-II. Geophys. J. R. Astron. Soc. 13, 529–539 (1967)
Katsikadelis J.T.: Numerical solution of multi-term fractional differential equations. ZAMM, Zeitschrift für Angewandte Mathematik und Mechanik 89(7), 593–608 (2009)
Katsikadelis J.T.: The analog equation method. A boundary-only integral equation method for nonlinear static and dynamic problems in general bodies. Int. J. Theor. Appl. Mech. 27, 13–38 (2002)
Schmidt A., Gaul L.: Finite element formulation of viscoelastic constitutive equations using fractional time derivatives. Nonlinear Dyn. 29(1-4), 37–55 (2002)
Nerantzaki M.S., Babouskos N.G.: Vibrations of inhomogeneous anisotropic viscoelastic bodies described with fractional derivative models. Eng. Anal. Bound. Elem. 6(12), 1894–1907 (2012). doi:10.1016/j.enganabound.2012.07.003
Atanackovic T.M.: A modified zener model of a viscoelastic body. Contin. Mech. Thermodyn. 14, 137–148 (2002)
Atanackovic T.M., Stankovic B.: Dynamics of a viscoelastic rod of fractional derivative type. ZAMM, Zeitschrift für Angewandte Mathematik und Mechanik 82, 377–386 (2002)
Katsikadelis J.T.: The BEM for numerical solution of partial fractional differential equations. Comput. Math. Appl. 62, 891–901 (2011). doi:10.1016/j.camwa.2011.04.001
Katsikadelis J.T.: The Boundary Element Method for Plate Analysis. Academic Press, Elsevier, Oxford (2014)
Babouskos N.G., Katsikadelis J.T.: Nonlinear vibrations of viscoelastic plates of fractional derivative type: an AEM solution. Open Mech. J. 4, 8–20 (2010)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Katsikadelis, J.T. Generalized fractional derivatives and their applications to mechanical systems. Arch Appl Mech 85, 1307–1320 (2015). https://doi.org/10.1007/s00419-014-0969-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00419-014-0969-0