On the Definitions of Nabla Fractional Operators

By Thabet Abdeljawad, Ferhan Merdivenci Atici. 2012

185 citations recorded by the source corpus.

We show that two recent definitions of discrete nabla fractional sum operators are related. Obtaining such a relation between two operators allows one to prove basic properties of the one operator by using the known properties of the other. We illustrate this idea with proving power rule and commutative property of discrete fractional sum operators. We also introduce and prove summation by parts formulas for the right and left fractional sum and difference operators, where we employ the Riemann-Liouville definition of the fractional difference. We formalize initial value problems for nonlinear fractional difference equations as an application of our findings. An alternative definition for the nabla right fractional difference operator is also introduced.

Subfields: Applied Mathematics, Modeling and Simulation

View Paper: https://www.semanticscholar.org/paper/9a79003d84cca29e1ff01565ccf11db7f19339e9

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In Applied Mathematics

In Modeling and Simulation

Citations in the corpus

The Poisson distribution, abstract fractional difference equations, and stability (2017, unknown): “To mention a few, see [1, 2, 4, 5, 23, 25, 26, 31, 40, 44].”
Integration by parts and its applications of a new nonlocal fractional derivative with Mittag-Leffler nonsingular kernel (2016, background): “Integration by parts is of great importance in fractional calculus [4] and discrete fractional calculus [9, 16, 17, 18].”
Fractional Sums and Differences with Binomial Coefficients (2013, methodology): “(49) In [30], the authors used a delta Leibniz’s Rule to prove the following formula for nabla right fractional differences:”
lp ‐maximal regularity for fractional difference equations on UMD spaces (2015, unknown): “To mention a few references, see [1], [2], [8], [9], [28], [30], [31], [35], [40] and [43].”
On Delta and Nabla Caputo Fractional Differences and Dual Identities (2011, methodology): “Integration by Parts for Caputo Fractional Differences In this section we state the integration by parts formulas for nabla fractional sums and differences obtained in [19] then use the dual identities to obtain delta integration by part formulas.”
Exponential functions of discrete fractional calculus (2013, background): “Probably more work has been developed for the fractional backward or nabla difference and we refer the reader to [1, 4, 5, 6, 9, 17, 18].”