On fractional calculus with general analytic kernels

Abstract

Many possible definitions have been proposed for fractional derivatives and integrals, starting from the classical Riemann-Liouville formula and its generalisations and modifying it by replacing the power function kernel with other kernel functions. We demonstrate, under some assumptions, how all of these modifications can be considered as special cases of a single, unifying, model of fractional calculus. We provide a fundamental connection with classical fractional calculus by writing these general fractional operators in terms of the original Riemann-Liouville fractional integral operator. We also consider inversion properties of the new operators, prove analogues of the Leibniz and chain rules in this model of fractional calculus, and solve some fractional differential equations using the new operators.

On some analytic properties of tempered fractional calculus

“…As discussed in Section 1 above, many types of fractional calculus can be related directly or indirectly to the Riemann-Liouville operators (1)- (2). The recent work of [14] proposes a general framework to cover many different fractional operators all of which can be written as infinite series of Riemann-Liouville differintegrals. It is proved therein that the GPF integral (6) can be considered as a special case of a generalised operator A a I α,β t f (t) which satisfies the series formula…”

On the solution of time‐fractional dynamical model of Brusselator reaction‐diffusion system arising in chemical reactions

“…Some essential investigation on FC has been analyzed in the past few years, and different books have been published by many authors such as Baleanu et al, 9 Miller and Ross, 10 Kilbas et al, 11 and Podlubny. 12 Several approximate and analytical techniques have been established for such type of problems (see other studies [13][14][15][16][17][18][19][20][21][22][23][24][25][26] ).…”

On the conformable fractional logistic models

“…These differential and integral operators are frequently used to construct mathematical models in several scientific areas. In particular, they have been successfully applied to the study of the logistics model (see previous studies [1][2][3][4][5][6] ).…”

“…For a complementary study on the recent developments in the field of the fractional calculus as well as its applications, see the literature. [1][2][3][4][5][6] It is important to note that the global fractional derivatives (eg, Caputo and Riemann-Liouville) are not collecting mere local information. By contrast, fractional operators keep track of the history of the process being studied; this feature allows modeling the nonlocal and distributed responses that commonly appear in natural and physical phenomena.…”