Generalized fractional derivatives generated by a class of local proportional derivatives

On some analytic properties of tempered fractional calculus

“…Definition 2.4 (GPF derivatives [17]). Let [a, b] be a real interval and ρ ∈ R, α ∈ C be parameters with 0 < ρ ≤ 1 and Re(α) ≥ 0.…”

“…Proposition 2.6 (The semigroup property [21,17]). Let [a, b] be a real interval and α 1 , α 2 , β ∈ C be parameters with Re(α i ) > 0, Re(β) > 0.…”

“…As a consequence, we have the following identities for the generalised proportional fractional integral and derivative, which were not observed in [17]:…”

On more general forms of proportional fractional operators

“…Notwithstanding, in [21,22], the authors introduced a newly defined local derivative that tend to the original function as the order tends to zero and hence improved the conformable derivatives. In addition to this, the non-local fractional operators that emerge from iterating the abovementioned derivative were held forth in [23].…”

“…Motivated by the above mentioned background, we extend the work done in [23] introduce a new generalized fractional calculus based on the proportional derivatives of a function with respect to another function in paralel with the definition discussed in [21]. The kernel obtained in the fractional operators which will be proposed contains an exponential function and is function dependent.…”