Abstract
From the end of the last decade of the twentieth century and the beginning of the first decade of the twenty-first century there was a great proliferation of derivatives, always with the purpose of generalizing the classics derivative from the calculus of integer order. Many, although containing the fractional name, are only a multiplicative factor ahead of the derivative of order one, as well as others, proposing as kernel a nonsingular function.
Just as the integer order calculus has integer order derivatives (and integrals), in fractional calculus we have fractional derivatives (and integrals), which should at least satisfy the condition of recovering the form of integer order derivatives (and integrals), for a suitable value of the parameter associated with the order of the fractional derivative. As we have already mentioned, there are several ways of approaching the study of fractional derivatives, since there are many distinct formulations of this concept, some of which should not even be denominated fractional. The proliferation of such definitions is due to the demand for an adequate derivative to describe each particular phenomenon. In general, different approaches to differential operators have been created because of the different domains in which they should be used. After presenting the special functions, in particular those associated with fractional calculus, as well as the methodology of the integral transforms, we are now able to introduce and work specifically with fractional calculus.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Capelas de Oliveira, E., Tenreiro Machado, J.A.: A review of definitions for fractional derivatives and integrals. Math. Prob. Ing. 2014, ID 238459 (2014)
Almeida, R.: A Caputo fractional derivative of a function with respect to another function. Commun. Nonlinear Sci. Numer. Simulat. 44, 460–481 (2017)
Sousa, J.V.C., Capelas de Oliveira, E.: On the \(\uppsi \)-Hilfer fractional derivative. Commun. Nonlinear Sci. Numer. Simulat. 60, 72–91 (2018)
Sonin, N.Y.: On differentiation with arbitrary index. Moscou Mat. Sb. 6, 1–38 (1869)
Grünwald, A.K.: Derivationen und deren Anwendung. Z. Angew. Math. Phys. 12, 441–480 (1867)
Letnikov, A.V.: Theory of differentiation with an arbitrary index. Matem. Sbornik. 3, 1–66 (1868)
Prudnikov, A.P., Brychkov, Y.A., Marichev, O.I.: Integral and Series (Elementary Functions). Gordon and Breach Science Publishers, London (1986)
Tricomi, F.G., Erdélyi, A.: The asymptotic expansion of a ratio of gamma functions. Pacific J. Math. 1, 133–142 (1951)
Love, E.R.: Changing the order of integration. J. Austral. Math. Soc. 9, 421–432 (1970)
Miller, K.S., Ross, B.: An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley, New York (1993)
Samko, S.G., Kilbas, A.A., Marichev, O.E.: Fractional Integrals and Derivatives: Theory and Applications. Gordon and Breach Science Publishers, Switzerland (1993)
Caputo, M.: Linear models of dissipation whose \(Q\) is almost frequency independent—II. Geophys. J. Roy. Astron. Soc. 13, 529–539 (1967). Reprinted in Fract. Cal. Appl. Anal. 11, 4–14 (2008)
Caputo, M., Mainardi, F.: Linear models of dissipation in anelastic solids. Riv. Nuovo Cimento 1, 161–198 (1971). Reprinted in Fract. Cal. Appl. Anal. 10, 309-324 (2007)
Dzhrbashian, M. M., Nersesyan, A. B.: Fractional derivatives and the Cauchy problem for fractional differential equations. Izv. Akad. Nauk. Armyan 3, 3–29 (1968) (in Russian)
Davis, H.T.: The Theory of Linear Operators. The Principia Press, Bloomington (1936)
Hilfer, R. (ed.): Applications of Fractional Calculus in Physics. World Scientific, Singapore (2000)
Rubanraj, S., Jernith, S.S.: Fractional calculus of logarithmic functions. Int. J. Math. Comp. Research. 4, 1296–1303 (2016)
Debnath, L., Bhatta, D.: Integral Transform and Their Applications, 2nd edn. Chapman & Hall/CRC, Boca Raton (2007)
Ortigueira, M.D.: Fractional Calculus for Scientists and Engineers. Springer, New York (2011)
Kilbas, A. A., Srivastava, H. M., Trujillo, J. J.: The Theory and Applications of Fractional Differential Equations. North-Holland Mathematics Studies, vol. 204. Elsevier, Amsterdam (2006)
Podlubny, I.: Fractional Differential Equations. Mathematical in Sciences and Engineering, vol. 198. Academic Press, San Diego (1999)
Nigmatullin, R.R.: The realization of the generalized transfer equation in a medium with fractal geometry. Phys. Sta. Sol. B. 133, 425–430 (1986)
Mathai, A.M., Saxena, R.K., Haubold, H.J.: The \(H\)-Function: Theory and Applications. Springer, New York (2010)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2019 Springer Nature Switzerland AG
About this chapter
Cite this chapter
Capelas de Oliveira, E. (2019). Fractional Derivatives. In: Solved Exercises in Fractional Calculus. Studies in Systems, Decision and Control, vol 240. Springer, Cham. https://doi.org/10.1007/978-3-030-20524-9_5
Download citation
DOI: https://doi.org/10.1007/978-3-030-20524-9_5
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-20523-2
Online ISBN: 978-3-030-20524-9
eBook Packages: EngineeringEngineering (R0)