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 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 Riemann—Liouville and Caputo Fractional Forward Difference Monotonicity Analysis

Mathematics ◽  
2021 ◽  
Vol 9 (11) ◽  
pp. 1303
Author(s):  
Pshtiwan Othman Mohammed ◽  
Thabet Abdeljawad ◽  
Faraidun Kadir Hamasalh
Keyword(s):  
Time Scale ◽  
Initial Value Problems ◽  
Mean Value ◽  
Mean Value Theorem ◽  
Discrete Fractional Calculus ◽  
Initial Value ◽  
Forward Difference ◽  
The Mean ◽  
Monotonicity Analysis ◽  
Monotonicity Results

Monotonicity analysis of delta fractional sums and differences of order υ∈(0,1] on the time scale hZ are presented in this study. For this analysis, two models of discrete fractional calculus, Riemann–Liouville and Caputo, are considered. There is a relationship between the delta Riemann–Liouville fractional h-difference and delta Caputo fractional h-differences, which we find in this study. Therefore, after we solve one, we can apply the same method to the other one due to their correlation. We show that y(z) is υ-increasing on Ma+υh,h, where the delta Riemann–Liouville fractional h-difference of order υ of a function y(z) starting at a+υh is greater or equal to zero, and then, we can show that y(z) is υ-increasing on Ma+υh,h, where the delta Caputo fractional h-difference of order υ of a function y(z) starting at a+υh is greater or equal to −1Γ(1−υ)(z−(a+υh))h(−υ)y(a+υh) for each z∈Ma+h,h. Conversely, if y(a+υh) is greater or equal to zero and y(z) is increasing on Ma+υh,h, we show that the delta Riemann–Liouville fractional h-difference of order υ of a function y(z) starting at a+υh is greater or equal to zero, and consequently, we can show that the delta Caputo fractional h-difference of order υ of a function y(z) starting at a+υh is greater or equal to −1Γ(1−υ)(z−(a+υh))h(−υ)y(a+υh) on Ma,h. Furthermore, we consider some related results for strictly increasing, decreasing, and strictly decreasing cases. Finally, the fractional forward difference initial value problems and their solutions are investigated to test the mean value theorem on the time scale hZ utilizing the monotonicity results.

scholarly journals Difference monotonicity analysis on discrete fractional operators with discrete generalized Mittag-Leffler kernels

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Pshtiwan Othman Mohammed ◽  
Faraidun Kadir Hamasalh ◽  
Thabet Abdeljawad
Keyword(s):  
Mean Value ◽  
Mean Value Theorem ◽  
Fractional Operators ◽  
Fractional Difference ◽  
Monotonicity Properties ◽  
The Mean ◽  
Monotonicity Analysis

AbstractIn this paper, we present the monotonicity analysis for the nabla fractional differences with discrete generalized Mittag-Leffler kernels $( {}^{ABR}_{a-1}{\nabla }^{\delta ,\gamma }y )(\eta )$ ( a − 1 A B R ∇ δ , γ y ) ( η ) of order $0<\delta <0.5$ 0 < δ < 0.5 , $\beta =1$ β = 1 , $0<\gamma \leq 1$ 0 < γ ≤ 1 starting at $a-1$ a − 1 . If $({}^{ABR}_{a-1}{\nabla }^{\delta ,\gamma }y ) ( \eta )\geq 0$ ( a − 1 A B R ∇ δ , γ y ) ( η ) ≥ 0 , then we deduce that $y(\eta )$ y ( η ) is $\delta ^{2}\gamma $ δ 2 γ -increasing. That is, $y(\eta +1)\geq \delta ^{2} \gamma y(\eta )$ y ( η + 1 ) ≥ δ 2 γ y ( η ) for each $\eta \in \mathcal{N}_{a}:=\{a,a+1,\ldots\}$ η ∈ N a : = { a , a + 1 , … } . Conversely, if $y(\eta )$ y ( η ) is increasing with $y(a)\geq 0$ y ( a ) ≥ 0 , then we deduce that $({}^{ABR}_{a-1}{\nabla }^{\delta ,\gamma }y )(\eta ) \geq 0$ ( a − 1 A B R ∇ δ , γ y ) ( η ) ≥ 0 . Furthermore, the monotonicity properties of the Caputo and right fractional differences are concluded to. Finally, we find a fractional difference version of the mean value theorem as an application of our results. One can see that our results cover some existing results in the literature.

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 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.

scholarly journals Monotonicity Analysis of Fractional Proportional Differences

2020 ◽  
Vol 2020 ◽  
pp. 1-11
Author(s):  
Iyad Suwan ◽  
Shahd Owies ◽  
Muayad Abussa ◽  
Thabet Abdeljawad
Keyword(s):  
Time Scale ◽  
Mean Value ◽  
Mean Value Theorem ◽  
The Mean ◽  
Monotonicity Analysis ◽  
Monotonicity Results

In this work, the nabla discrete new Riemann–Liouville and Caputo fractional proportional differences of order 0<ε<1 on the time scale ℤ are formulated. The differences and summations of discrete fractional proportional are detected on ℤ, and the fractional proportional sums associated to ∇cRχε,ρz with order 0<ε<1 are defined. The relation between nabla Riemann–Liouville and Caputo fractional proportional differences is derived. The monotonicity results for the nabla Caputo fractional proportional difference are proved; specifically, if ∇c−1Rχε,ρz>0 then χz is ερ −increasing, and if χz is strictly increasing on ℕc and χc>0, then ∇c−1Rχε,ρz>0. As an application of our findings, a new version of the fractional proportional difference of the mean value theorem (MVT) on ℤ is proved.

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.

A Class of Nonlinear Singular Difference Inequality in Engineering

2014 ◽  
Vol 577 ◽  
pp. 824-827
Author(s):  
Wu Sheng Wang ◽  
Zong Yi Hou
Keyword(s):  
Mean Value ◽  
Mean Value Theorem ◽  
Jensen Inequality ◽  
Fractional Difference ◽  
Difference Inequality ◽  
Amplification Method ◽  
Weakly Singular ◽  
Change Of Variable ◽  
Explicit Bounds ◽  
The Mean

In this paper, we discuss a class of new nonlinear weakly singular difference inequality. Using change of variable, discrete Jensen inequality, amplification method, the mean-value theorem for integrals and Gamma function, explicit bounds for the unknown functions in the inequality is given clearly. The derived results can be applied in the study of fractional difference equations in Engineering.

Application of Fractional Calculus to Fluid Mechanics

2002 ◽  
Vol 124 (3) ◽  
pp. 803-806 ◽  
Author(s):  
Vladimir V. Kulish ◽  
Jose´ L. Lage
Keyword(s):  
Shear Stress ◽  
Fractional Calculus ◽  
Fluid Mechanics ◽  
Time Dependent ◽  
Laplace Transform Method ◽  
Transform Method ◽  
Boundary Shear ◽  
Stokes Problems ◽  
The Laplace Transform ◽  
Boundary Shear Stress

In this note we present the application of fractional calculus, or the calculus of arbitrary (noninteger) differentiation, to the solution of time-dependent, viscous-diffusion fluid mechanics problems. Together with the Laplace transform method, the application of fractional calculus to the classical transient viscous-diffusion equation in a semi-infinite space is shown to yield explicit analytical (fractional) solutions for the shear-stress and fluid speed anywhere in the domain. Comparing the fractional results for boundary shear-stress and fluid speed to the existing analytical results for the first and second Stokes problems, the fractional methodology is validated and shown to be much simpler and more powerful than existing techniques.

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