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    Hiperbolik Denklem İçeren Bir Optimal Kontrol Probleminin Nümerik Çözümü Üzerine
    (2020) Araz, Seda İğret
    Bu makalede, hiperbolik denklem içeren optimal kontrol problemlerinin bir sınıfını çözmek için bir nümerikalgoritma sunulmaktadır. Bir regüler uzayda optimal çözümün var ve tek olduğu gösterilmektedir. Eşlenikproblemi elde ettikten ve amaç fonksiyonelinin türevini hesapladıktan sonra, Gradyen metoduyla nümerikyaklaşımlar elde edilmektedir. Hesaplanan sonuçlar, önerilen metodun optimal kontrol problemleri için iyinümerik yaklaşımlar üretebildiğini göstermektedir.
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    New Numerical Scheme With Newton Polynomial: Theory, Methods, and Applications
    (Elsevier, 2021) Atangana, Abdon; Araz, Seda İğret
    New Numerical Scheme with Newton Polynomial: Theory, Methods, and Applications provides a detailed discussion on the underpinnings of the theory, methods and real-world applications of this numerical scheme. The book's authors explore how this efficient and accurate numerical scheme is useful for solving partial and ordinary differential equations, as well as systems of ordinary and partial differential equations with different types of integral operators. Content coverage includes the foundational layers of polynomial interpretation, Lagrange interpolation, and Newton interpolation, followed by new schemes for fractional calculus. Final sections include six chapters on the application of numerical scheme to a range of real-world applications. Over the last several decades, many techniques have been suggested to model real-world problems across science, technology and engineering. New analytical methods have been suggested in order to provide exact solutions to real-world problems. Many real-world problems, however, cannot be solved using analytical methods. To handle these problems, researchers need to rely on numerical methods, hence the release of this important resource on the topic at hand. © 2021 Elsevier Inc. All rights reserved.
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    On Optimal Control of the Initial Status in a Hyperbolic System
    (2018) Araz, Seda İğret
    In this study, optimal control problem governed by a hyperbolic problem with Dirichlet conditions is considered. It isdemonstrated that the optimal solution for the considered optimal control problem is exist and unique and it is obtainedadjoint problem. Derivative of the cost functional is calculated utilizing from adjoint problem. Finally, necessaryoptimality conditions for hyperbolic system are derived.
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    Piecewise derivatives versus short memory concept: analysis and application
    (Amer Inst Mathematical Sciences-Aims, 2022) Atangana, Abdon; Araz, Seda İğret
    We have provided a detailed analysis to show the fundamental difference between the concept of short memory and piecewise differential and integral operators. While the concept of short memory leads to different long tails in different intervals of time or space as a result of a power law with different fractional orders, the concept of piecewise helps to depict crossover behaviors of different patterns. We presented some examples with different numerical simulations. In some cases piecewise models led to transitional behavior from deterministic to stochastic, this is indeed the reason why this concept was introduced.
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    Retraction notice to ‘Corrigendum to “New numerical method for ordinary differential equations: Newton polynomial”’ [J. Comput. Appl. Math. (2019) 112622] (Journal of Computational and Applied Mathematics (2020) 371, (S0377042719306739), (10.1016/j.cam.2019.112668))
    (Elsevier B.V., 2021) Atangana, Abdon; Araz, Seda İğret
    This article has been retracted: please see Elsevier Policy on Article Withdrawal (https://www.elsevier.com/about/our-business/policies/article-withdrawal). This is a corrigendum to a retracted article. © 2021 Elsevier B.V.
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    Retraction notice to “New numerical method for ordinary differential equations: Newton polynomial” [J. Comput. Appl. Math. 371 (2020) 112668] (Journal of Computational and Applied Mathematics (2020) 372, (S0377042719306272), (10.1016/j.cam.2019.112622))
    (Elsevier B.V., 2021) Atangana, Abdon; Araz, Seda İğret
    This article has been retracted: please see Elsevier Policy on Article Withdrawal (https://www.elsevier.com/about/our-business/policies/article-withdrawal). This article has been retracted at the request of the Principal Editors: In the paper, the authors claim that the Adams-Bashforth methods are based on Lagrange interpolation polynomials and that Lagrange polynomials are less accurate than Newton polynomials. The authors state they have devised a new method which is built on Newton interpolation in order to provide better accuracy. After reviewing concerns that were raised by the community, the Principal Editors invited further independent experts to review the claims made by the authors. They find that changing the basis of the polynomial space (Lagrange, Newton, or others) does not change the interpolant polynomial, which is unique when the number of data fits the polynomial degree. Therefore, devising a method based on Newton polynomials, instead of Lagrange polynomials, does not affect the accuracy of the method, but leads to the same method. As such, the Principal Editors have concluded that the findings of the paper are unreliable. This retraction is not related to a breach of ethics. The authors do not agree to this retraction. Apologies are offered to readers of the journal that this was not detected during the submission process. © 2021 Elsevier B.V.
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    Step forward in epidemiological modeling: Introducing the rate indicator function to capture waves
    (Elsevier B.V., 2022) Atangana, Abdon; Araz, Seda İğret
    In the last centuries, mathematical models have been used to depict the dynamic evolution of infectious diseases’ spread. The aim is to determine the total numbers of infected, recovered, and susceptible individuals; however, they only represent accumulated values due to the complexities of collected data. Having the graphical representations of these classes as accumulated values, one may not directly be able to predict waves to determine a day-to-day number of newly infected cases as some additional calculations are required. Collected data from real-world situations are represented as day-to-day new infections; they help determine in addition numbers of waves. However, current existing mathematical models cannot be used for wave prediction. While knowing the predicted numbers of waves, policymakers can take adequate measures to control the situation. To solve this problem, we questioned the fact that the rates of infection and recovery are constant and suggested an indicator rate function obtained using experimental data. To see the effect of this function, we considered a simple SIR model, which was modified by introducing rate indicator functions for infected and recovered classes. To include nonlocal behaviors in the mathematical model, different types of differential operators, including classical and fractional derivatives, were used. The models were solved numerically using well-known numerical schemes. The numerical solutions were plotted for different theoretical parameters; the results depict real-world behaviors. To test the efficiency of this new approach, we collected data from South Africa and compared them with our model. Not only our model could predict waves, but also it fits experimental data very accurately. This new approach will open new doors of investigation toward a revolution in epidemiological modeling. © 2022 The Authors