Fractional difference operators with discrete generalized Mittag–Leffler kernels

By Thabet Abdeljawad. Chaos, Solitons & Fractals, 2019

75 citations recorded by the source corpus.

Abstract In this article we define fractional difference operators with discrete generalized Mittag–Leffler kernels of the form ( E θ , μ ¯ γ ( λ , t − ρ ( s ) ) for both Riemann type (ABR) and Caputo type (ABC) cases, where AB stands for Atangana–Baleanu. Then, we employ the discrete Laplace transforms to formulate their corresponding A B − fractional sums, and prove useful and applicable versions of their semi-group properties. The action of fractional sums on the ABC type fractional differences is proved and used to solve the A B C − fractional difference initial value problems. The nonhomogeneous linear ABC fractional difference equation with constant coefficient is solved by both the discrete Laplace transforms and the successive approximation, and the Laplace transform method is remarked for the continuous counterpart. In fact, for the case μ ≠ 1, we obtain a nontrivial solution for the homogeneous linear A B C − type initial value problem with constant coefficient. The relation between the ABC and ABR fractional differences are formulated by using the discrete Laplace transform. We iterate the fractional sums of order − ( θ , μ , 1 ) to generate fractional sum-differences for which a semigroup property is proved. The nabla discrete transforms for the A B − fractional sums and the A B − iterated fractional sum-differences are calculated. Examples and remarks are given to clarify and confirm the obtained results and some of their particular cases are highlighted. Finally, the discrete extension to the higher order case is discussed.

Subfields: Applied Mathematics, Modeling and Simulation

View Paper: https://www.semanticscholar.org/paper/55ac3d08a0a7478e61823f8f0b64271901ba28ce

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In Applied Mathematics

In Modeling and Simulation

Citations in the corpus

European option pricing models described by fractional operators with classical and generalized Mittag‐Leffler kernels (2020, methodology): “Moreover, in this section, we recall the definition of the fractional operators with singular kernel and their generalized versions which have been recently proposed in the literature by Abdeljawad et al. [26, 28].”
Analysis of some generalized ABC – Fractional logistic models (2019, methodology): “The Laplace transforms also were used there in [22] and completed in [23] for both the continuous and discrete cases to obtain an explicit solution as stated above in Lemma 3.”
Integral inequalities for a fractional operator of a function with respect to another function with nonsingular kernel (2020, background): “1 The discrete version of (15) was obtained in [4] and its generalization to the case of generalized ML function case together with its discrete counterpart can be found in [2] and [3], respectively.”