A Correlation Between Solutions of Uncertain Fractional Forward Difference Equations and Their Paths

Srivástava, H. M.; Mohammed, Pshtiwan Othman

doi:10.3389/fphy.2020.00280

Cited by 11 publications

(9 citation statements)

References 41 publications

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“…In [14,15], the authors introduced the discrete fractional sums and differences which produced directly from the Riemann-Liouville (RL) fractional integrals and derivatives, respectively. To review the history of discrete fractional operators, their properties and information related to discrete fractional calculus applications one can refer to [16][17][18][19][20][21][22] and the references therein. Nowadays, being new fractional integral and derivative operators make the researchers attempt to introduce a new definition of discrete fractional sum and difference operators corresponding to them.…”

Section: Introductionmentioning

confidence: 99%

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

2021

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In 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.

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Section: Introductionmentioning

confidence: 99%

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

2021

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“…In the last few years, researchers introduced various models involving fractional derivative and integral operators of arbitrary order (see [10]). Some recent contributions to the theory of fractional differential (or difference) equations and its applications can be found in [11][12][13][14][15][16][17] and references therein.…”

Section: Introductionmentioning

confidence: 99%

Adomian Decomposition and Fractional Power Series Solution of a Class of Nonlinear Fractional Differential Equations

et al. 2021

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Nonlinear fractional differential equations reflect the true nature of physical and biological models with non-locality and memory effects. This paper considers nonlinear fractional differential equations with unknown analytical solutions. The Adomian decomposition and the fractional power series methods are adopted to approximate the solutions. The two approaches are illustrated and compared by means of four numerical examples.

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“…Moreover, DFC has been suitably characterized the term "memory" especially in physics, economics, mathematics, biology, engineering, control etc., and its study is not only interesting from a purely mathematical point of view, but has been found extremely useful for modeling super-diffusion processes, which naturally appear in many applications in biology, probability, physics, economics, medicine and ecology (see [1][2][3][4]). Additionally, there are some recent works on variable-order fractional difference equations such as [5][6][7][8][9][10] in discrete fractional calculus.…”

Section: Introductionmentioning

confidence: 99%

On Discrete Delta Caputo–Fabrizio Fractional Operators and Monotonicity Analysis

2021

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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.

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A Correlation Between Solutions of Uncertain Fractional Forward Difference Equations and Their Paths

Cited by 11 publications

References 41 publications

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

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

Adomian Decomposition and Fractional Power Series Solution of a Class of Nonlinear Fractional Differential Equations

On Discrete Delta Caputo–Fabrizio Fractional Operators and Monotonicity Analysis

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