Discrete generalized fractional operators defined using h‐discrete Mittag‐Leffler kernels and applications to AB fractional difference systems

Abstract

This study investigates the h ‐fractional difference operators with h ‐discrete generalized Mittag‐Leffler kernels ( in the sense of Riemann type (namely, the A B R ) and Caputo type (namely, the A B C ). For which, we will discuss the region of convergent. Then, we study the h ‐discrete Laplace transforms to formulate their corresponding A B ‐fractional sums. Also, it is useful in obtaining the semi‐group properties. We will prove the action of fractional sums on the A B C type h ‐fractional differences and then it can be used to solve the system of A B C h ‐fractional difference. By using the h ‐discrete Laplace transforms and the Picard successive approximation technique, we will solve the nonhomogeneous linear A B C h ‐fractional difference equation with constant coefficient, and also we will remark the h ‐discrete Laplace transform method for the continuous counterpart. Meanwhile, we will obtain a nontrivial solution for the homogeneous linear A B C h ‐fractional difference initial value problem with constant coefficient for the case δ ≠ 1. We will formulate the relation between the A B C and A B R h ‐fractional differences by using the h ‐discrete Laplace transform. By iterating the fractional sums of order −( ϕ , δ , 1), we will generate the h ‐fractional sum‐differences, and in view of this, a semigroup property will be proved. Due to these new powerful techniques, we can calculate the nabla h ‐discrete transforms for the A B h ‐fractional sums and the A B iterated h ‐fractional sum‐differences. Furthermore, we will obtain some particular cases that can be found in examples and remarks. Finally, we will discuss the higher order case of the h ‐discrete fractional differences and sums.

Topics & Concepts