On a New Definition of the Fractional Difference

Gray, H. L.; Zhang, Nien fan

doi:10.2307/2008620

Cited by 61 publications

(77 citation statements)

References 3 publications

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“…Unfortunately, the method in [10] requires calculation of an infinite series which is often time-consuming, or even worse, infeasible. A more useful version of discrete fractional calculus was then proposed by Granger and Joyeux in [11] where the infinite series was replaced by a finite one. So far, a large volume of pioneering works on discrete fractional calculus have been reported, see, e.g., [12][13][14][15][16].…”

Section: Introductionmentioning

confidence: 99%

Lyapunov functions for nabla discrete fractional order systems

et al. 2019

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This paper investigates the time-domain response of nabla discrete fractional order systems by exploring several useful properties of the nabla discrete Laplace transform and the discrete Mittag-Leffler function. In particular, we establish two fundamental properties of a nabla discrete fractional order system with nonzero initial instant: i) the existence and uniqueness of the system time-domain response; and ii) the dynamic behavior of the zero input response. Finally, one numerical example is provided to show the validity of the theoretical results.

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Section: Introductionmentioning

confidence: 99%

Lyapunov functions for nabla discrete fractional order systems

et al. 2019

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“…Discrete fractional calculus remained without serious developing til the beginning of the last decade in the last century. Twenty years after the articles [1,2], many authors started to attack discrete fractional calculus very extensively ( [3]- [19]). Recently, some authors started to study monotonicity and convexity properties of delta and nabla (left Riemann) fractional differences ( [30]- [34]).…”

Section: Introduction and Preliminaries About Fractional Sums And Difmentioning

confidence: 99%

Monotonicity results for delta and nabla caputo and Riemann fractional differences via dual identities

Abdeljawad

Abdalla

2017

Filomat

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Recently, some authors have proved monotonicity results for delta and nabla fractional differences separately. In this article, we use dual identities relating delta and nabla fractional difference operators to prove shortly the monotonicity properties for the (left Riemann) nabla fractional differences using the corresponding delta type properties. Also, we proved some monotonicity properties for the Caputo fractional differences. Finally, we use the Q−operator dual identities to prove monotonicity results for the right fractional difference operators.Keywords: right (left) delta and nabla fractional sums, right (left) delta and nabla Riemann and Caputo fractional differences, Q-operator, dual identity.

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“…The fractional operators are obtained using the forward operators delta [16,8] or backward operators nabla [11,2] or combined delta and nabla fractional operators [3].…”

Section: Introductionmentioning

confidence: 99%

On the Approximation of Riemann–Liouville Integral by Fractional Nabla h-Sum and Applications

Khitri-Kazi-Tani

Dib

2017

Mediterr. J. Math.

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First of all, in this paper, we prove the convergence of the nabla hsum to the Riemann-Liouville integral in the space of continuous functions and in some weighted spaces of continuous functions. The connection with time scales convergence is discussed. Secondly, the efficiency of this approximation is shown through some Cauchy fractional problems with singularity at the initial value. The fractional Brusselator system is solved as a practical case.

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On a New Definition of the Fractional Difference

Abstract: Abstract.A new definition of the fractional difference is introduced. Many properties based on this definition are established including an extensive exponential law and the important Leibniz rule. The results are then applied to solving second-order linear difference equations.

Cited by 61 publications

References 3 publications

Lyapunov functions for nabla discrete fractional order systems

Lyapunov functions for nabla discrete fractional order systems

Monotonicity results for delta and nabla caputo and Riemann fractional differences via dual identities

On the Approximation of Riemann–Liouville Integral by Fractional Nabla h-Sum and Applications

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