Cauchy’s formula on nonempty closed sets and a new notion of Riemann–Liouville fractional integral on time scales

Abstract

We prove Cauchy's formula for repeated integration on time scales. The obtained relation gives rise to new notions of fractional integration and differentiation on arbitrary nonempty closed sets.

Right fractional Sobolev space via Riemann$-$Liouville derivatives on time scales and an application to fractional boundary value problem on time scales

“…Motivated by Definition 4 in [14] and Definition 2.1 in [22], we can present the right Riemann−Liouville fractional integral and derivative on time scales as follows: Definition 2.5. (Fractional integral on time scales) Suppose h is an integrable function on J.…”

“…Motivated by Definition 4 and Equation ( 21) in [14] and Theorem 2.1 in [4], we can present the right Caputo fractional derivative on time scales as follows: Definition 2.7. (Caputo fractional derivative on time scales) Let t ∈ T, 0 < α ≤ 1 and h : T → R. The left Caputo fractional derivative of order α of h is defined by…”

“…So far, the research of time scale theory has attracted extensive attention all over the world. It can be applied to engineering, physics, economics, population dynamics and other fields ( [7][8][9][10][11][12][13][14]).…”

Hybrid Method for Simulation of a Fractional COVID-19 Model with Real Case Application

“…These characteristics allow FDEs to model effectively not only non-Markovian processes but also non-Gaussian phenomena [39]. Different non-classical fractional-order derivatives and different kinds of FDEs were proposed [40][41][42]. Among them, one has the Atangana-Baleanu-Caputo (ABC) derivative, which is a nonlocal fractional derivative with a nonsingular kernel, connected with various applications.…”

Nabla Fractional Derivative and Fractional Integral on Time Scales

“…After the inception of the topic, a number of papers were published see [12][13][14][15][16][17][18][19]. Recently, D. F. M. Torres introduced a generalized definition of Hilger derivative and integrals in a pure sense of Riemann-Liouville (RL) derivative [20,21]. Many research works have been completed in a conformable delta and nabla fractional derivative and integrals [22][23][24].…”