Fractional h‐differences with exponential kernels and their monotonicity properties

Suwan, Iyad; Owies, Shahd; Abdeljawad, Thabet

doi:10.1002/mma.6213

“…Fernandez and Mohammed 37 have derived results for the Hermite-Hadamard inequality and its related in the context of fractional integrals and derivatives defined by Mittag-Leffler kernels. Suwan et al 38 formulated the nabla fractional differences of order in (0,1) with discrete exponential kernels on the time scale, which generalized some earlier results. They have also applied their results to prove a fractional difference version of the mean value theorem.…”

mentioning

confidence: 52%

Preface to a thematic special issue of methods of mathematics and fractional calculus

Zeidan

¹

,

Herbst

²

,

Lu

³

2021

Math Methods in App Sciences

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Preface to a thematic special issue of methods of mathematics and fractional calculus Mathematics is a common and widespread language that is not spoken very well in different parts of the world. This language also occurs in diverse forms that are required in different methods and applications. Today, mathematics is used everywhere from sciences and engineering to foreign languages research and the frontiers of social sciences. The key objective of this thematic special issue has been to offer typical attention on the latest advances and trends in different methods of mathematics and fractional calculus. Given the increasingly wide branches of mathematics, it is impossible to encapsulate a thorough representation of the selected topics in a single journal issue. However, it is possible to proffer a thematic special issue to bring together diverse fields of mathematics as well as a range of researchers from different parts of the world to represent a valued panel that addresses the different aspects and methods of mathematics, from pure, applied mathematics to fractional calculus.Manuscripts in this thematic special issue were requested through a general call and by invitation. It also includes selected papers that were presented at the International Conference on Fractional Differentiation and its Applications-ICFDA'18 held in Amman, Jordan, on July 16-18, 2018. All submissions were subject to a rigorous peer-review process following the well-known policies and standards of the Mathematical Methods in the Applied Sciences journal. Every submission was reviewed by at least two internationally established experts in the topic of that paper. The total number of papers received in response to the call of this thematic special issue was 90. Forty-nine papers were rejected after the peer-review process, and 41 were accepted upon up to three revisions and after an agreement was reached by the referees and the editors. Therefore, the rejection rate for this issue is 54.45%, which is one of the indicators of the high standards and quality considered when preparing this thematic special issue.In the following, we give a sense of the 41 accepted papers and the diversity of this thematic special issue where all papers are listed in references by their authors, title and DOI. The first paper by Dassios and Baleanu 1 investigates a class of singular linear systems of fractional differential equations with given inconsistent initial conditions. Validation of the proposed theory and analysis were carried out through numerical examples. Boumenir and Tuan 2 show that it is possible to uniquely recover the coefficient of an one-dimensional fractional diffusion equation from a single boundary measurement. An algorithm based on the well-known Gelfand-Levitan inverse spectral theory of Sturm-Liouville operators was developed and validated. The paper by Senol et al 3 deals with numerical methods for a KdV-like model. It was shown that the developed methods for the model equations are user-friendly and can be easily implemented to other...

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“…• Additionally, we have established a new version of the MVT in the frame fractional differences in the setting of generalized AB. • In the case of the case hZ in the setting of discrete ML-kernel (AB) [30] and discrete exponential kernel [34], it was noticed that the monotonicity factor depends on the step h. However, for the discrete power law case [31] the monotonicity factor is independent of the step h. Since our results in this article generalize those in [29], it is of interest to generalize the results in this article for the hZ case so that the monotonicity factor will depend on δ, γ , and h! • We have been able to address the monotonicity analysis for the ML kernels with parameters 0 < δ < 0.5, β = 1, and 0 < γ ≤ 1.…”

Section: Resultsmentioning

confidence: 99%

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

Mohammed

¹

,

Hamasalh

²

,

Abdeljawad

³

2021

Self Cite

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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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“…where A is some nonnegative number and Δ 𝛼 y is some type of fractional difference of y. Studies of this form include papers by Atici et al, 23 Du et al, 24 Goodrich, 25 Goodrich and Jonnalagadda, 26 Goodrich et al, 27 Jia et al, [28][29][30] Liu et al, 31 Mohammed et al, 32,33 Suwan et al, 34 and Suwan et al 35 ; we point out, off hand, that, somewhat curiously, there does not seem to be the same sort of activity in the continuous fractional frame, and, in fact, the only work of which we are aware in this direction is a paper by Diethelm. 36 Usually in these studies, the number A in (1.2) has been taken to be 0 since this is the natural and obvious analogue of the integer-order setting in which one has the result "(Δy)(𝜏) ≥ 0 ⇒ y is increasing."…”

Section: Some Additional Conditions (If Necessary)mentioning

confidence: 99%

“…On the one hand, there are papers that have studied a single fractional difference—i.e., theorems that take the following form:

\{\begin{matrix} ((), {normalΔ}^{α} monospacey) false(n false) \geq A \\ Some Additional Conditions (if necessary) \end{matrix} \Rightarrow monospacey is positive and/or monotone and/or convex,

where A is some nonnegative number and Δ α y is some type of fractional difference of y. Studies of this form include papers by Atici et al, 23 Du et al, 24 Goodrich, 25 Goodrich and Jonnalagadda, 26 Goodrich et al, 27 Jia et al, 28–30 Liu et al, 31 Mohammed et al, 32,33 Suwan et al, 34 and Suwan et al 35 ; we point out, off hand, that, somewhat curiously, there does not seem to be the same sort of activity in the continuous fractional frame, and, in fact, the only work of which we are aware in this direction is a paper by Diethelm 36 . Usually in these studies, the number A in () has been taken to be 0 since this is the natural and obvious analogue of the integer‐order setting in which one has the result “

false(normalΔ monospacey false) false(τ false) \geq 0 \Rightarrow monospacey is increasing .

” Nonetheless, the recent paper by Goodrich and Lizama 37 studied in extensive detail the case in which A > 0, and a subsequent paper by Goodrich and Muellner 38 also considered such cases.…”

Section: Introductionmentioning

confidence: 99%

On positivity and monotonicity analysis for discrete fractional operators with discrete Mittag–Leffler kernel

Mohammed

¹

,

Goodrich

²

,

Hamasalh

³

et al. 2022

Math Methods in App Sciences

11

0

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We consider conditions under which the positivity of a fractional difference implies either positivity, monotonicity, or convexity, and we consider both the non-sequential and sequential cases. The particular difference we study is the discrete nabla ABC-type difference, and it possesses a Mittag-Leffler kernel; this leads to some different results when compared to other types of kernels. We conclude by applying our results to certain classes of initial value problems in discrete fractional calculus.

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Fractional h‐differences with exponential kernels and their monotonicity properties

Cited by 11 publications

References 36 publications

Preface to a thematic special issue of methods of mathematics and fractional calculus

Preface to a thematic special issue of methods of mathematics and fractional calculus

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

On positivity and monotonicity analysis for discrete fractional operators with discrete Mittag–Leffler kernel

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