Evaluation of Fractional Integrals and Derivatives of Elementary Functions: Overview and Tutorial

Several fractional-order operators are available and an in-depth knowledge of the selected operator is necessary for the evaluation of fractional integrals and derivatives of even simple functions. In this paper, we reviewed some of the most commonly used operators and illustrated two approaches to generalize integer-order derivatives to fractional order; the aim was to provide a tool for a full understanding of the specific features of each fractional derivative and to better highlight their differences. We hence provided a guide to the evaluation of fractional integrals and derivatives of some elementary functions and studied the action of different derivatives on the same function. In particular, we observed how Riemann–Liouville and Caputo’s derivatives converge, on long times, to the Grünwald–Letnikov derivative which appears as an ideal generalization of standard integer-order derivatives although not always useful for practical applications.

[1]  Richard B. Paris.  Asymptotics of the special functions of fractional calculus , 2019, Basic Theory.

[2]  Yuri Luchko,et al.  Desiderata for Fractional Derivatives and Integrals , 2019, Mathematics.

[3]  F. Mainardi.  Fractional Calculus , 2018, Fractional Calculus.

[4]  M. Ciesielski,et al.  An Exact Solution of the Second-Order Differential Equation with the Fractional/Generalised Boundary Conditions , 2018 .

[5]  V. E. Tarasov.  No nonlocality. No fractional derivative , 2018, Commun. Nonlinear Sci. Numer. Simul..

[6]  F. Ferrari.  Weyl and Marchaud Derivatives: A Forgotten History , 2017, 1711.08070.

[7]  Andrea Giusti,et al.  A comment on some new definitions of fractional derivative , 2017, Nonlinear Dynamics.

[8]  B. West.  Nature's Patterns and the Fractional Calculus , 2017 .

[9]  Piotr Ostalczyk,et al.  Discrete Fractional Calculus: Applications In Control And Image Processing , 2015 .

[10]  José António Tenreiro Machado,et al.  What is a fractional derivative? , 2015, J. Comput. Phys..

[11]  Y. Povstenko.  Linear Fractional Diffusion-Wave Equation for Scientists and Engineers , 2015 .

[12]  Roberto Garrappa,et al.  Numerical Evaluation of Two and Three Parameter Mittag-Leffler Functions , 2015, SIAM J. Numer. Anal..

[13]  T. Kaczorek,et al.  Fractional Differential Equations , 2015 .

[14]  Vasily E. Tarasov,et al.  No violation of the Leibniz rule. No fractional derivative , 2013, Commun. Nonlinear Sci. Numer. Simul..

[15]  Roberto Garrappa,et al.  Evaluation of generalized Mittag–Leffler functions on the real line , 2013, Adv. Comput. Math..

[16]  M. T. Cicero.  FRACTIONAL CALCULUS AND WAVES IN LINEAR VISCOELASTICITY , 2012 .

[17]  R. Hermann.  Fractional calculus an introduction for physicists , 2011 .

[18]  Luigi Fortuna,et al.  Fractional Order Systems: Modeling and Control Applications , 2010 .

[19]  M. Ortigueira.  Comments on ''Modeling fractional stochastic systems as non-random fractional dynamics driven Brownian motions" , 2009 .

[20]  R. Gorenflo,et al.  Fractional Calculus: Integral and Differential Equations of Fractional Order , 2008, 0805.3823.

[21]  R. Gorenflo,et al.  Time-fractional derivatives in relaxation processes: a tutorial survey , 2008, 0801.4914.

[22]  Roberto Garrappa,et al.  Some Formulas for Sums of Binomial Coefficients and Gamma Functions , 2007 .

[23]  H. Srivastava,et al.  THEORY AND APPLICATIONS OF FRACTIONAL DIFFERENTIAL EQUATIONS. NORTH-HOLLAND MATHEMATICS STUDIES , 2006 .

[24]  Manuel Duarte Ortigueira,et al.  From Differences to Derivatives , 2004 .

[25]  K. Miller,et al.  An Introduction to the Fractional Calculus and Fractional Differential Equations , 1993 .

[26]  H. Kober.  ON FRACTIONAL INTEGRALS AND DERIVATIVES , 1940 .