Fractional calculus on time scales with Taylor’s theorem

Abstract

We present a definition of the Riemann-Liouville fractional calculus for arbitrary time scales through the use of time scales power functions, unifying a number of theories including continuum, discrete and fractional calculus. Basic properties of the theory are introduced including integrability conditions and index laws. Special emphasis is given to extending Taylor's theorem to incorporate our theory.MSC 2010 : Primary 26A33, Secondary 39A12

Discussion on the Leibniz rule and Laplace transform of fractional derivatives using series representation

“…which shows the self-contradictory problem of formula (23). For j > n, C a D α−j t is calculated by the way of the Riemann-Liouville integral, although there is no reason for this assumption.…”

“…Proof. To illustrate the violation of (23), two aspects will be deployed. Likewise, if we put α in the interval (n − 1, n) with positive integer n, then a question will emerge instantly, namely, how to interpret C a D α−j t g (t) , j > n. Recalling Leibniz rule for Riemann-Liouville derivative, one directly assumes that C a D α−j t g (t) = R a I j−α t g (t) for j > n.…”

“…Additionally, the Laplace transform of Riemann-Liouville derivative contains fractional derivatives as initial value which is not practical enough [10,21,22]. With the adoption of series representation [23,24], the Caputo Leibniz rule and Riemann-Liouville Laplace transform might be established anew.…”

Fractional Sobolev’s Spaces on Time Scales via Conformable Fractional Calculus and Their Application to a Fractional Differential Equation on Time Scales

“…Nevertheless, these studies are all about integer order dynamic equations on time scales. The existence and multiplicity of solutions for fractional dynamic equations on time scales has received considerably less attention (see [11,12]).…”

Integral Transform Methods for Solving Fractional Dynamic Equations on Time Scales

“…However, this approach suffers by some technical difficulties, connected to the inverse Laplace transform (see [8]). Recently, in [31,32] (see also [33]), the authors independently suggested an axiomatic definition of power functions on arbitrary time scale.…”