Fractional q-Calculus on a time scale

By Ferhan Merdivenci Atici, Paul W. Eloe. 2007

203 citations recorded by the source corpus.

Abstract The study of fractional q-calculus in this paper serves as a bridge between the fractional q-calculus in the literature and the fractional q-calculus on a time scale , where to ∈ ℝ and 0 < q < 1. By use of time scale calculus notation, we find the proof of many results more straight forward. We shall develop some properties of fractional q-calculus, we shall develop some properties of a q-Laplace transform, and then we shall employ the q-Laplace transform to solve fractional q-difference equations.

Subfields: Applied Mathematics, Mathematical Physics, Modeling and Simulation

View Paper: https://www.semanticscholar.org/paper/f4ecceab0b335e7b52a8bec7530a552337e6f47d

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Citations in the corpus

SOLUTIONS TO A DISCRETE RIGHT-FOCAL FRACTIONAL BOUNDARY VALUE PROBLEM (2010, background): “These and other related results and their proofs can be found, for example, in [3–6].”
Two-point boundary value problems for finite fractional difference equations (2011, methodology): “In [2], the authors have recently developed and applied a transform method for fractional q-calculus problems; an appropriate bibliography to illustrate the richness of the theory of fractional q-calculus is provided in this reference.”
Nontrivial solutions for fractional q-difference boundary value problems (2010, background): “More recently, perhaps due to the explosion in research within the fractional calculus setting (see the books [13, 14]), new developments in this theory of fractional q-difference calculus were made, specifically, q-analogues of the integral and differential fractional operators properties such as q”
On Riemann‐Liouville fractional q–difference equations and their application to retarded logistic type model (2018, unknown): “The proof is simply completed by applying the q–fractional integral q∇ a to both sides of (13), using relation (8), using Lemma 2 applied to f (t) = q∇ qa x(t) and the semigroup property q∇ a q∇ −μ a f (t) = q∇ −(α+μ) a f (t), we deduce that q∇ a q∇αqax(t) = q∇ a ∇q(q∇ −(1−α) qa x(t)) = ∇qq∇ qa q∇ −”
DISCRETE FRACTIONAL CALCULUS WITH THE NABLA OPERATOR (2009, background): “Miller and Ross [15] initiated the study and the authors [2, 3, 4] have more recently been developing discrete forward fractional calculus.”
Initial value problems in discrete fractional calculus (2008, methodology): “The authors [2] also developed and applied a transform method for fractional q -calculus problems.”
Existence and uniqueness of positive solutions for three-point boundary value problem with fractional q-differences (2012, background): “, q-analogues of the integral and differential fractional operators properties such as the q-Laplace transform, q-Taylor’s formula [19, 20], just to mention some.”