DISCRETE FRACTIONAL CALCULUS WITH THE NABLA OPERATOR

By Ferhan Merdivenci Atici, Paul W. Eloe. 2009

495 citations recorded by the source corpus.

Properties of discrete fractional calculus in the sense of a backward difference are introduced and developed. Exponential laws and a product rule are developed and relations to the forward fractional calculus are explored. Properties of the Laplace transform for the nabla derivative on the time scale of integers are developed and a fractional finite difference equation is solved with a transform method. As a corollary, two new identities for the gamma function are exhibited.

Subfields: Applied Mathematics, Modeling and Simulation

View Paper: https://www.semanticscholar.org/paper/db00f8fb0815609f7de0fdebd105ee1828959f41

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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, methodology): “The remarkable fact is that we use here a particular choice of the definition introduced by Atici and Eloe in [9] for the nabla operator.”
The Theory of Discrete Fractional Calculus: Development and Application (2011, unknown): “In Chapter 2, we present a full and rigorous theory for composing fractional sum and di¤erence operators, including correcting and broadening the power rule stated incorrectly in [2], [3] and [4].”
Fixed points and asymptotic stability of nonlinear fractional difference equations (2011, background): “Fractional difference equations have also been studied more intensively of late [2-6, 12].”
ANALYSIS OF DISCRETE FRACTIONAL OPERATORS (2015, methodology): “For this purpose we consider a forward fractional difference operator of Riemann-Liouville type as in the papers [2–7], We then prove some monotonicity criteria for a function f which is defined on N0 and has a sign (positive or negative) for ∆f when ν is between 0 and 1.”
A transference principle for nonlocal operators using a convolutional approach: fractional monotonicity and convexity (2020, methodology): “By contrast, a fractional forward difference of order α > 0 may be defined by (see Gray and Zhang [35] or Atici and Eloe [11, 12, 13]) (1.1) (∆νaf)(t) := 1 Γ(−ν) t+ν∑ s=a (t− s− 1)−ν−1f(s), t ∈ Na+N−ν , where N ∈ N1 is the unique integer satisfying N − 1 ν N , and the map t 7→ tν is defined by t”
Comparison theorems and asymptotic behavior of solutions of discrete fractional equations (2015, background): “There has been some work to develop relations between the forward and backward fractional operators, ∆a and ∇a (see [2]) and fractional calculus on time scales (see [4]).”
Two monotonicity results for nabla and delta fractional differences (2015, background): “There has also been some work to develop relations between the forward and backward fractional operators, Δa and ∇a (see [3,9]) and fractional calculus on time scales [1].”
Convexity for nabla and delta fractional differences (2015, background): “There has been some work to develop relations between the forward and backward fractional operators, D and 7 [2,4] and fractional calculus on time scales [5].”
Fractional Sums and Differences with Binomial Coefficients (2013, background): “4 Discrete Dynamics in Nature and Society Lemma 5 (see [15]).”