Matematik Bölümü Yayın Koleksiyonu
Permanent URI for this collectionhttps://hdl.handle.net/20.500.12416/413
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Article Citation - WoS: 38Citation - Scopus: 40A New Application of the Fractional Logistic Map(Editura Acad Romane, 2016) Huang, Lan-Lan; Baleanu, Dumitru; Baleanu, Dumitru; Wu, Guo-Cheng; Zeng, Sheng-Da; MatematikThe fractional chaotic map started to be applied in physics and engineering to properly treat some real-world phenomena. A shuffling method is proposed based on the fractional logistic map. The fractional difference order is used as a key. An image encryption scheme is designed by using the XOR operation and the security analysis is given. The obtained results demonstrate that the fractional difference order makes the encryption scheme highly secure.Letter Citation - WoS: 46Citation - Scopus: 47Discrete Fractional Watermark Technique(Zhejiang Univ, 2020) Shiri, Babak; Baleanu, Dumitru; Wang, Zai-rongThe fractional logistic map holds rich dynamics and is adopted to generate chaotic series. A watermark image is then encrypted and inserted into the original images. Since the encryption image takes the fractional order within (0, 1], it increases the key space and becomes difficult to attack. This study provides a robust watermark method in the protection of the copyright of hardware, images, and other electronic files.Article Citation - WoS: 8Citation - Scopus: 8Positivity Analysis for the Discrete Delta Fractional Differences of the Riemann-Liouville and Liouville-Caputo Types(Amer inst Mathematical Sciences-aims, 2022) Srivastava, Hari Mohan; Baleanu, Dumitru; Elattar, Ehab E.; Hamed, Y. S.; Mohammed, Pshtiwan OthmanIn this article, we investigate some new positivity and negativity results for some families of discrete delta fractional difference operators. A basic result is an identity which will prove to be a useful tool for establishing the main results. Our first main result considers the positivity and negativity of the discrete delta fractional difference operator of the Riemann-Liouville type under two main conditions. Similar results are then obtained for the discrete delta fractional difference operator of the Liouville-Caputo type. Finally, we provide a specific example in which the chosen function becomes nonincreasing on a time set.Article Citation - WoS: 9Citation - Scopus: 9New Classifications of Monotonicity Investigation for Discrete Operators With Mittag-Leffler Kernel(Amer inst Mathematical Sciences-aims, 2022) Goodrich, Christopher S.; Brzo, Aram Bahroz; Baleanu, Dumitru; Hamed, Yasser S.; Mohammed, Pshtiwan OthmanThis paper deals with studying monotonicity analysis for discrete fractional operators with Mittag-Leffler in kernel. The v-monotonicity definitions, namely v-(strictly) increasing and v-(strictly) decreasing, are presented as well. By examining the basic properties of the proposed discrete fractional operators together with v-monotonicity definitions, we find that the investigated discrete fractional operators will be v(2)-(strictly) increasing or v(2)-(strictly) decreasing in certain domains of the time scale Na:= {a, a + 1, ... }. Finally, the correctness of developed theories is verified by deriving mean value theorem in discrete fractional calculus.Article Citation - WoS: 22Citation - Scopus: 21Modified Fractional Difference Operators Defined Using Mittag-Leffler Kernels(Mdpi, 2022) Srivastava, Hari Mohan; Baleanu, Dumitru; Abualnaja, Khadijah M.; Mohammed, Pshtiwan OthmanThe discrete fractional operators of Riemann-Liouville and Liouville-Caputo are omnipresent due to the singularity of the kernels. Therefore, convexity analysis of discrete fractional differences of these types plays a vital role in maintaining the safe operation of kernels and symmetry of discrete delta and nabla distribution. In their discrete version, the generalized or modified forms of various operators of fractional calculus are becoming increasingly important from the viewpoints of both pure and applied mathematical sciences. In this paper, we present the discrete version of the recently modified fractional calculus operator with the Mittag-Leffler-type kernel. Here, in this article, the expressions of both the discrete nabla derivative and its counterpart nabla integral are obtained. Some applications and illustrative examples are given to support the theoretical results.Article Citation - WoS: 464Citation - Scopus: 528Discrete Fractional Logistic Map and Its Chaos(Springer, 2014) Baleanu, Dumitru; Wu, Guo-ChengA discrete fractional logistic map is proposed in the left Caputo discrete delta's sense. The new model holds discrete memory. The bifurcation diagrams are given and the chaotic behaviors are numerically illustrated.Article Citation - WoS: 1Citation - Scopus: 1Analytical Results for Positivity of Discrete Fractional Operators With Approximation of the Domain of Solutions(Amer inst Mathematical Sciences-aims, 2022) O'Regan, Donal; Baleanu, Dumitru; Hamed, Y. S.; Elattar, Ehab E.; Mohammed, Pshtiwan OthmanWe study the monotonicity method to analyse nabla positivity for discrete fractional operators of Riemann-Liouville type based on exponential kernels, where ((CFR)(c0)del F-theta)(t) > -epsilon Lambda(theta - 1) (del F)(c(0) + 1) such that (del F)(c(0) + 1) >= 0 and epsilon > 0. Next, the positivity of the fully discrete fractional operator is analyzed, and the region of the solution is presented. Further, we consider numerical simulations to validate our theory. Finally, the region of the solution and the cardinality of the region are discussed via standard plots and heat map plots. The figures confirm the region of solutions for specific values of epsilon and theta.Article Citation - WoS: 4Citation - Scopus: 4Analysis of Positivity Results for Discrete Fractional Operators by Means of Exponential Kernels(Amer inst Mathematical Sciences-aims, 2022) O'Regan, Donal; Brzo, Aram Bahroz; Abualnaja, Khadijah M.; Baleanu, Dumitru; Mohammed, Pshtiwan Othman; O’regan, DonalIn this study, we consider positivity and other related concepts such as alpha-convexity and alpha-monotonicity for discrete fractional operators with exponential kernel. Namely, we consider discrete Delta fractional operators in the Caputo sense and we apply efficient initial conditions to obtain our conclusions. Note positivity results are an important factor for obtaining the composite of double discrete fractional operators having different orders.Article Citation - WoS: 4Citation - Scopus: 4Monotonicity and Positivity Analyses for Two Discrete Fractional-Order Operator Types With Exponential and Mittag-Leffler Kernels(Elsevier, 2023) Srivastava, Hari Mohan; Baleanu, Dumitru; Al-Sarairah, Eman; Sahoo, Soubhagya Kumar; Chorfi, Nejmeddine; Mohammed, Pshtiwan OthmanThe discrete analysed fractional operator technique was employed to demonstrate positive findings concerning the Atangana-Baleanu and discrete Caputo-Fabrizo fractional operators. Our tests utilized discrete fractional operators with orders between 1 < phi < 2, as well as between 1 < phi < 3/2. We employed the initial values of Mittag-Leffler functions and applied the principle of mathematical induction to ensure the positivity of the discrete fractional operators at each time step. As a result, we observed a significant impact of the positivity of these operators on (del(Q)) (tau) within Np0+1 according to the Riemann- Liouville interpretation. Furthermore, we established a correlation between the discrete fractional operators based on the Liouville-Caputo and Riemann-Liouville definitions. In addition, we emphasized the positivity of (del(Q)) (tau) in the Liouville-Caputo sense by utilizing this relationship. Two examples are presented to validate the results.(c) 2023 The Author(s). Published by Elsevier B.V. on behalf of King Saud University. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).Article Citation - WoS: 7Citation - Scopus: 8Analytical and Numerical Negative Boundedness of Fractional Differences With Mittag-Leffler Kernel(Amer inst Mathematical Sciences-aims, 2023) Dahal, Rajendra; Hamed, Y. S.; Goodrich, Christopher S.; Baleanu, Dumitru; Mohammed, Pshtiwan OthmanWe show that a class of fractional differences with Mittag-Leffler kernel can be negative and yet monotonicity information can still be deduced. Our results are complemented by numerical approximations. This adds to a growing body of literature illustrating that the sign of a fractional difference has a very complicated and subtle relationship to the underlying behavior of the function on which the fractional difference acts, regardless of the particular kernel used.
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