A New Insight Into the Grünwald–Letnikov Discrete Fractional Calculus

2019 ◽  
Vol 14 (4) ◽  
Author(s):  
Yiheng Wei ◽  
Weidi Yin ◽  
Yanting Zhao ◽  
Yong Wang
Keyword(s):  
Fractional Calculus ◽  
Discrete Fractional Calculus ◽  
Fractional Difference ◽  
Integer Order ◽  
Convolution Operation ◽  
Order Case ◽  
Insight Into

The primary work of this paper is to investigate some potential properties of Grünwald–Letnikov discrete fractional calculus. By employing a concise and convenient description, this paper not only establishes excellent relationships between fractional difference/sum and the integer order case but also generalizes the Z-transform and convolution operation.

scholarly journals SOME FURTHER EXTENSIONS CONSIDERING DISCRETE PROPORTIONAL FRACTIONAL OPERATORS

Fractals ◽  
2021 ◽  
pp. 2240026
Author(s):  
SAIMA RASHID ◽  
SOBIA SULTANA ◽  
YELIZ KARACA ◽  
AASMA KHALID ◽  
YU-MING CHU
Keyword(s):  
Fractional Calculus ◽  
Existence And Uniqueness ◽  
Complex Dynamics ◽  
Fractional Operator ◽  
Fractional Operators ◽  
Discrete Fractional Calculus ◽  
Dynamic Features ◽  
Fractional Difference ◽  
Fractional Systems ◽  
Special Cases

In this paper, some attempts have been devoted to investigating the dynamic features of discrete fractional calculus (DFC). To date, discrete fractional systems with complex dynamics have attracted the most consideration. By considering discrete [Formula: see text]-proportional fractional operator with nonlocal kernel, this study contributes to the major consequences of the certain novel versions of reverse Minkowski and related Hölder-type inequalities via discrete [Formula: see text]-proportional fractional sums, as presented. The proposed system has an intriguing feature not investigated in the literature so far, it is characterized by the nabla [Formula: see text] fractional sums. Novel special cases are reported with the intention of assessing the dynamics of the system, as well as to highlighting the several existing outcomes. In terms of applications, we can employ the derived consequences to investigate the existence and uniqueness of fractional difference equations underlying worth problems. Finally, the projected method is efficient in analyzing the complexity of the system.

scholarly journals Analysis of discrete fractional operators

2015 ◽  
Vol 9 (1) ◽  
pp. 139-149 ◽  
Author(s):  
Ferhan Atici ◽  
Meltem Uyanik
Keyword(s):  
Fractional Calculus ◽  
Difference Operator ◽  
Mean Value ◽  
Mean Value Theorem ◽  
Fractional Operators ◽  
Discrete Fractional Calculus ◽  
Fractional Difference ◽  
The Mean ◽  
Discrete Domain

In this paper, we introduce two new monotonicity concepts for a nonnegative or nonpositive valued function defined on a discrete domain. We give examples to illustrate connections between these new monotonicity concepts and the traditional ones. We then prove some monotonicity criteria based on the sign of the fractional difference operator of a function f, ??f with 0 < ? < 1. As an application, we state and prove the mean value theorem on discrete fractional calculus.

scholarly journals On Discrete Fractional Integral Inequalities for a Class of Functions

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-13 ◽  
Author(s):  
Saima Rashid ◽  
Hijaz Ahmad ◽  
Aasma Khalid ◽  
Yu-Ming Chu
Keyword(s):  
Fractional Calculus ◽  
Difference Equations ◽  
Logistic Map ◽  
Neural Systems ◽  
The Novel ◽  
Discrete Fractional Calculus ◽  
Fractional Difference ◽  
Fractional Difference Equations ◽  
Systems With Memory ◽  
Positive Functions

Discrete fractional calculus ℱ C is proposed to depict neural systems with memory impacts. This research article aims to investigate the consequences in the frame of the discrete proportional fractional operator. ℏ -discrete exponential functions are assumed in the kernel of the novel generalized fractional sum defined on the time scale ℏ ℤ . The nabla ℏ -fractional sums are accounted in particular. The governing high discretization of problems is an advanced version of the existing forms that can be transformed into linear and nonlinear difference equations using appropriately adjusted transformations invoking property of observing the new chaotic behaviors of the logistic map. Based on the theory of discrete fractional calculus, explicit bounds for a class of positive functions n n ∈ ℕ concerned are established. These variants can be utilized as a convenient apparatus in the qualitative analysis of solutions of discrete fractional difference equations. With respect to applications, we can apply the introduced outcomes to explore boundedness, uniqueness, and continuous reliance on the initial value problem for the solutions of certain underlying worth problems of fractional difference equations.

scholarly journals On Sumudu Transform Method in Discrete Fractional Calculus

2012 ◽  
Vol 2012 ◽  
pp. 1-16 ◽  
Author(s):  
Fahd Jarad ◽  
Kenan Taş
Keyword(s):  
Fractional Calculus ◽  
Time Scale ◽  
Initial Value Problems ◽  
Discrete Fractional Calculus ◽  
Initial Value ◽  
Fractional Difference ◽  
Transform Method ◽  
Sumudu Transform ◽  
Definition Of ◽  
General Time

In this paper, starting from the definition of the Sumudu transform on a general time scale, we define the generalized discrete Sumudu transform and present some of its basic properties. We obtain the discrete Sumudu transform of Taylor monomials, fractional sums, and fractional differences. We apply this transform to solve some fractional difference initial value problems.

Stability analysis of impulsive fractional difference equations

2018 ◽  
Vol 21 (2) ◽  
pp. 354-375 ◽  
Author(s):  
Guo–Cheng Wu ◽  
Dumitru Baleanu
Keyword(s):  
Stability Analysis ◽  
Asymptotic Stability ◽  
Difference Equation ◽  
Fractional Calculus ◽  
Difference Equations ◽  
Discrete Fractional Calculus ◽  
Fractional Difference ◽  
Numerical Result ◽  
Fractional Difference Equations ◽  
Fractional Difference Equation

AbstractWe revisit motivation of the fractional difference equations and some recent applications to image encryption. Then stability of impulsive fractional difference equations is investigated in this paper. The fractional sum equation is considered and impulsive effects are introduced into discrete fractional calculus. A class of impulsive fractional difference equations are proposed. A discrete comparison principle is given and asymptotic stability of nonlinear fractional difference equation are discussed. Finally, an impulsive Mittag–Leffler stability is defined. The numerical result is provided to support the analysis.

scholarly journals On Discrete Delta Caputo–Fabrizio Fractional Operators and Monotonicity Analysis

2021 ◽  
Vol 5 (3) ◽  
pp. 116
Author(s):  
Pshtiwan Othman Mohammed ◽  
Thabet Abdeljawad ◽  
Faraidun Kadir Hamasalh
Keyword(s):  
Fractional Calculus ◽  
Mean Value ◽  
Mean Value Theorem ◽  
Fractional Operators ◽  
Discrete Fractional Calculus ◽  
Laplace Transform Method ◽  
Fractional Difference ◽  
Transform Method ◽  
Monotonicity Analysis ◽  
Monotonicity Results

The discrete delta Caputo-Fabrizio fractional differences and sums are proposed to distinguish their monotonicity analysis from the sense of Riemann and Caputo operators on the time scale Z. Moreover, the action of Q− operator and discrete delta Laplace transform method are also reported. Furthermore, a relationship between the discrete delta Caputo-Fabrizio-Caputo and Caputo-Fabrizio-Riemann fractional differences is also studied in detail. To better understand the dynamic behavior of the obtained monotonicity results, the fractional difference mean value theorem is derived. The idea used in this article is readily applicable to obtain monotonicity analysis of other discrete fractional operators in discrete fractional calculus.

scholarly journals MODELING AFTERSHOCKS BY FRACTIONAL CALCULUS: EXACT DISCRETIZATION VERSUS APPROXIMATE DISCRETIZATION

Fractals ◽  
2021 ◽  
pp. 2140038
Author(s):  
HUA KONG ◽  
GUANG YANG ◽  
CHENG LUO
Keyword(s):  
Fractional Calculus ◽  
Heavy Tails ◽  
Least Square Method ◽  
Least Square ◽  
The Other ◽  
Discrete Fractional Calculus ◽  
Unknown Parameters ◽  
Fractional Difference ◽  
Difference Formula ◽  
The Difference

This paper suggests two fractional differences for aftershock modeling with heavy tails. Discrete fractional calculus is a straightforward tool on isolated time scale. On the other hand, the fractional difference also can be derived by standard finite difference method when the difference formula is convergent. The two methods are both adopted and compared in the results. The unknown parameters are determined by use of the least square method where Ya’an earthquake aftershock data is used. It is reported that the discrete fractional calculus is an exact discretization tool without any loss in the memory effects which leads to better results.

scholarly journals On the nonoscillatory behavior of solutions of three classes of fractional difference equations

2020 ◽  
Vol 40 (5) ◽  
pp. 549-568
Author(s):  
Said Rezk Grace ◽  
Jehad Alzabut ◽  
Sakthivel Punitha ◽  
Velu Muthulakshmi ◽  
Hakan Adıgüzel
Keyword(s):  
Fractional Calculus ◽  
Difference Equations ◽  
Riccati Transformation ◽  
Discrete Fractional Calculus ◽  
Fractional Difference ◽  
Transformation Technique ◽  
Fractional Difference Equations ◽  
Difference Formula

In this paper, we study the nonoscillatory behavior of three classes of fractional difference equations. The investigations are presented in three different folds. Unlike most existing nonoscillation results which have been established by employing Riccati transformation technique, we employ herein an easily verifiable approach based on the fractional Taylor's difference formula, some features of discrete fractional calculus and mathematical inequalities. The theoretical findings are demonstrated by examples. We end the paper by a concluding remark.

Discrete Fractional Diffusion Equation of Chaotic Order

2016 ◽  
Vol 26 (01) ◽  
pp. 1650013 ◽  
Author(s):  
Guo-Cheng Wu ◽  
Dumitru Baleanu ◽  
He-Ping Xie ◽  
Sheng-Da Zeng
Keyword(s):  
Porous Media ◽  
Fractional Calculus ◽  
Diffusion Equation ◽  
Discrete Time ◽  
Chaotic Map ◽  
Fractional Diffusion ◽  
Fractional Diffusion Equation ◽  
Variable Order ◽  
Diffusion Modeling ◽  
Discrete Fractional Calculus

Discrete fractional calculus is suggested in diffusion modeling in porous media. A variable-order fractional diffusion equation is proposed on discrete time scales. A function of the variable order is constructed by a chaotic map. The model shows some new random behaviors in comparison with other variable-order cases.

Sign in / Sign up

Export Citation Format

Share Document