On Riemann and Caputo fractional differences

By Thabet Abdeljawad. Computers and Mathematics with Applications, 2011

675 citations recorded by the source corpus.

In this paper, we define left and right Caputo fractional sums and differences, study some of their properties and then relate them to Riemann-Liouville ones studied before by Miller K. S. and Ross B., Atici F.M. and Eloe P. W., Abdeljawad T. and Baleanu D., and a few others. Also, the discrete version of the Q-operator is used to relate the left and right Caputo fractional differences. A Caputo fractional difference equation is solved. The solution proposes discrete versions of Mittag-Leffler functions.

Subfields: Applied Mathematics, Modeling and Simulation, Numerical Analysis

View Paper: https://www.semanticscholar.org/paper/7c225587e474855f30e6fbec1c444456336801f1

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Citations in the corpus

The -Transform Method and Delta Type Fractional Difference Operators (2015, neutral): “In [33] versions of solutions of scalar fractional-order difference equation are given.”
Fractional Sums and Differences with Binomial Coefficients (2013, background): “Using the change of variable u = a + b − s, in [18] it was shown that”
Discrete chaos in fractional delayed logistic maps (2015, background): “After that, several authors developed the theory of the fractional difference equations on time scales [9], such as the initial value problems [10], the discrete calculus of variations [11], the Laplace transform [12,13], the properties of the Caputo and the Riemann-Liouville difference [14], and so”
On fractional derivatives with exponential kernel and their discrete versions (2016, background): “On the other side the discrete version of any fractional derivative is one of the interesting topics nowadays [8]-[20].”
On Delta and Nabla Caputo Fractional Differences and Dual Identities (2011, methodology): “(ii) the delta α-order Caputo right fractional difference [9] of a function f defined on bN is defined by”
The Poisson distribution, abstract fractional difference equations, and stability (2017, neutral): “To mention a few, see [1, 2, 4, 5, 23, 25, 26, 31, 40, 44].”
Some further results of the laplace transform for variable–order fractional difference equations (2019, methodology): “The Picard’s method was considered for Cauchy problems of exact solutions of fractional difference equations [1].”