On some analytic properties of tempered fractional calculus

Abstract We consider the integral and derivative operators of tempered fractional calculus, and examine their analytic properties. We discover connections with the classical Riemann–Liouville fractional calculus and demonstrate how the operators may be used to obtain special functions such as hypergeometric and Appell’s functions. We also prove an analogue of Taylor’s theorem and some integral inequalities to enrich the mathematical theory of tempered fractional calculus.

[1]  Kazuya Kobayashi,et al.  On Generalized Gamma Functions Occurring in Diffraction Theory , 1991 .

[2]  O. Marichev,et al.  Fractional Integrals and Derivatives: Theory and Applications , 1993 .

[3]  K. Diethelm,et al.  Fractional Calculus: Models and Numerical Methods , 2012 .

[4]  R. Hilfer.  Applications Of Fractional Calculus In Physics , 2000 .

[5]  W. Deng,et al.  High Accuracy Algorithm for the Differential Equations Governing Anomalous Diffusion: Algorithm and Models for Anomalous Diffusion , 2018 .

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

[7]  Dumitru Baleanu,et al.  On some new properties of fractional derivatives with Mittag-Leffler kernel , 2017, Commun. Nonlinear Sci. Numer. Simul..

[8]  D. Anderson,et al.  Newly Defined Conformable Derivatives , 2015 .

[9]  Dumitru Baleanu,et al.  Fractional calculus: models and numerical methods (2nd edition) , 2016 .

[10]  S. Arabia,et al.  Properties of a New Fractional Derivative without Singular Kernel , 2015 .

[11]  Some Incomplete Hypergeometric Functions and Incomplete Riemann-Liouville Fractional Integral Operators , 2019, Mathematics.

[12]  St'ephane Dugowson,et al.  Les différentielles métaphysiques : histoire et philosophie de la généralisation de l'ordre de la dérivation , 1994 .

[13]  T. Abdeljawad,et al.  Generalized fractional derivatives generated by a class of local proportional derivatives , 2017 .

[14]  Mark M. Meerschaert,et al.  Tempered fractional calculus , 2015, J. Comput. Phys..

[15]  Dumitru Baleanu,et al.  Series representations for fractional-calculus operators involving generalised Mittag-Leffler functions , 2018, Commun. Nonlinear Sci. Numer. Simul..

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

[17]  R. Agarwal,et al.  A comparative study on generating function relations for generalized hypergeometric functions via generalized fractional operators , 2018 .

[18]  M. Caputo,et al.  A new Definition of Fractional Derivative without Singular Kernel , 2015 .

[19]  W. Deng,et al.  Well-posedness and numerical algorithm for the tempered fractional differential equations , 2019, Discrete & Continuous Dynamical Systems - B.

[20]  É. Picard.  Sur une extension aux fonctions de deux variables du problème de Riemann relatif aux fonctions hypergéométriques , 1881 .

[21]  T. R. Prabhakar.  A SINGULAR INTEGRAL EQUATION WITH A GENERALIZED MITTAG LEFFLER FUNCTION IN THE KERNEL , 1971 .

[22]  Dumitru Baleanu,et al.  On fractional calculus with general analytic kernels , 2019, Appl. Math. Comput..

[23]  M. Özarslan,et al.  Incomplete Caputo fractional derivative operators , 2018, Advances in Difference Equations.

[24]  A. Atangana,et al.  New Fractional Derivatives with Nonlocal and Non-Singular Kernel: Theory and Application to Heat Transfer Model , 2016, 1602.03408.

[25]  R. K. Saxena,et al.  Generalized mittag-leffler function and generalized fractional calculus operators , 2004 .

[26]  M. Ali Özarslan,et al.  Some generating relations for extended hypergeometric functions via generalized fractional derivative operator , 2010, Math. Comput. Model..

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

[28]  B. Ross,et al.  A BRIEF HISTORY AND EXPOSITION OF THE FUNDAMENTAL THEORY OF FRACTIONAL CALCULUS , 1975 .

[29]  W. Deng,et al.  Well-posedness and numerical algorithm for the tempered fractional ordinary differential equations , 2015, 1501.00376.

[30]  D. Baleanu,et al.  The mean value theorem and Taylor’s theorem for fractional derivatives with Mittag–Leffler kernel , 2018, Advances in Difference Equations.

[31]  Zaid M. Odibat,et al.  Generalized Taylor's formula , 2007, Appl. Math. Comput..

[32]  P. Agarwal,et al.  An extension of Caputo fractional derivative operator and its applications , 2016 .

[33]  R. G. Buschman.  Decomposition of an Integral Operator by Use of Mikusiński Calculus , 1972 .

[34]  M. Caputo.  Linear Models of Dissipation whose Q is almost Frequency Independent-II , 1967 .