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: 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: 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: 6Citation - Scopus: 7Reprint Of: Chaos Synchronization of the Discrete Fractional Logistic Map(Elsevier, 2015) Baleanu, Dumitru; Wu, Guo-ChengIn this paper, master slave synchronization for the fractional difference equation is studied with a nonlinear coupling method. The numerical simulation shows that the designed synchronization method can effectively synchronize the fractional logistic map. The Caputo-like delta derivative is adopted as the difference operator. (C) 2014 Elsevier B.V. All rights reserved.Article Citation - WoS: 153Citation - Scopus: 162Chaos Synchronization of the Discrete Fractional Logistic Map(Elsevier, 2014) Baleanu, Dumitru; Wu, Guo-ChengIn this paper, master-slave synchronization for the fractional difference equation is studied with a nonlinear coupling method. The numerical simulation shows that the designed synchronization method can effectively synchronize the fractional logistic map. The Caputo-like delta derivative is adopted as the difference operator. (C) 2014 Elsevier B.V. All rights reserved.Letter Discrete fractional watermark technique(Zhejiang Univ, 2020) Wang, Zai-rong; Shiri, Babak; Baleanu, DumitruThe 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: 134Citation - Scopus: 147Discrete Chaos in Fractional Sine and Standard Maps(Elsevier, 2014) Baleanu, Dumitru; Zeng, Sheng-Da; Wu, Guo-ChengFractional standard and sine maps are proposed by using the discrete fractional calculus. The chaos behaviors are then numerically discussed when the difference order is a fractional one. The bifurcation diagrams and the phase portraits are presented, respectively. (C) 2013 Elsevier B.V. All rights reserved.Article Citation - WoS: 64Citation - Scopus: 77Image Encryption Technique Based on Fractional Chaotic Time Series(Sage Publications Ltd, 2016) Baleanu, Dumitru; Lin, Zhen-Xiang; Wu, Guo-ChengChaos in discrete fractional maps has been reported very recently. In this study, the chaotic time series of fractional order is used in the scrambling technique and a novel image encryption scheme is designed. The fractional difference order and the chaotic coefficient play crucial roles in controlling chaotic behaviors. The encrypted results are analyzed, which shows that the encryption scheme is highly secure.