Yazar "Araz, Seda Igret" seçeneğine göre listele

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    A fractional optimal control problem with final observation governed by wave equation
    (Amer Inst Physics, 2019) Araz, Seda Igret
    In this paper, we deal with the problem of controlling the source function for an optimal control problem involving the fractional wave equation. We show that an optimal solution exists and it is unique for the considered fractional optimal control problem. We calculate the Frechet derivative of the cost functional by means of an adjoint problem and derive necessary optimality conditions. Also, we introduce an efficient numerical approximation for the fractional wave equation with the Atangana-Baleanu derivative.
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    A successive midpoint method for nonlinear differential equations with classical and Caputo-Fabrizio derivatives
    (Amer Inst Mathematical Sciences-Aims, 2023) Atangana, Abdon; Araz, Seda Igret
    In this study, we present a numerical scheme for solving nonlinear ordinary differential equations with classical and Caputo-Fabrizio derivatives using consecutive interval division and the midpoint approach. By doing so, we increased the accuracy of the midpoint approach, which is dependent on the number of interval divisions. In the example of the Caputo-Fabrizio differential operator, we established the existence and uniqueness of the solution using the Caratheodory-Tonelli sequence. We solved numerous nonlinear equations and determined the global error to test the accuracy of the proposed scheme. When the differential equation met the circumstances under which it was generated, the results revealed that the procedure was quite accurate.
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    Advanced analysis in epidemiological modeling: detection of waves
    (Amer Inst Mathematical Sciences-Aims, 2022) Atangana, Abdon; Araz, Seda Igret
    Mathematical concepts have been used in the last decades to predict the behavior of the spread of infectious diseases. Among them, the reproductive number concept has been used in several published papers to study the stability of the mathematical model used to predict the spread patterns. Some conditions were suggested to conclude if there would be either stability or instability. An analysis was also meant to determine conditions under which infectious classes will increase or die out. Some authors pointed out limitations of the reproductive number, as they presented its inability to help predict the spread patterns. The concept of strength number and analysis of second derivatives of the mathematical models were suggested as additional tools to help detect waves. This paper aims to apply these additional analyses in a simple model to predict the future.
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    Analysis of a Covid-19 model: Optimal control, stability and simulations
    (Elsevier, 2021) Araz, Seda Igret
    Mathematical tools called differential and integral operators are used to model real world problems in all fields of science as they are able to replicate some behaviors observed in real world like fading memory, long-range dependency, power law, random walk and many others. Very recently the world has faced a serious challenge since the breakout of corona-virus started in Wuhan, China. The deathly disease has killed about 1720000 and infected more than 2 millions humans around the globe since December 2019 to 21 of April 2020. In this paper, we analyzed a mathematical model for the spread of COVID-19, we first start with stability analysis, present the optimal control for the system. The model was extended to the concept of non-local operators for each case, we presented the positiveness of the system solutions. We presented numerical solutions are presented for different scenarios. (C) 2020 The Authors. Published by Elsevier B.V. on behalf of Faculty of Engineering, Alexandria University.
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    Analysis of a new partial integro-differential equation with mixed fractional operators
    (Pergamon-Elsevier Science Ltd, 2019) Atangana, Abdon; Araz, Seda Igret
    We have introduced a new partial integro-differential equation with mixed fractional operators. The differential operator can be taken as Caputo while the integral is consider to be Caputo-Fabrizio or the Atangana-Baleanu integral. We presented the well poseness of the new class of partial differential equation. We presented the conditions which the existence and uniqueness are obtained. We presented the derivation of exact solution under some conditions. We suggested a numerical scheme that will be used to solve such mathematical equations. We presented some illustratives examples. (C) 2019 Elsevier Ltd. All rights reserved.
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    ATANGANA-SEDA NUMERICAL SCHEME FOR LABYRINTH ATTRACTOR WITH NEW DIFFERENTIAL AND INTEGRAL OPERATORS
    (World Scientific Publ Co Pte Ltd, 2020) Atangana, Abdon; Araz, Seda Igret
    In this paper, we present a new numerical scheme for a model involving new mathematical concepts that are of great importance for interpreting and examining real world problems. Firstly, we handle a Labyrinth chaotic problem with fractional operators which include exponential decay, power-law and Mittag-Leffler kernel. Moreover, this problem is solved via Atangana-Seda numerical scheme which is based on Newton polynomial. The accuracy and efficiency of the method can be easily seen with numerical simulations.
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    Crossover behaviors via piecewise concept: A model of tumor growth and its response to radiotherapy
    (Elsevier, 2022) Arik, Irem Akbulut; Araz, Seda Igret
    This study aims to combine 3 different models, which take into account the pre-treatment, during the treatment and post-treatment processes of the tumor growth, with the piecewise derivative, and to consider these processes as a whole thanks to this new concept. This concept leads us to analyze and predict the process from the beginning to the end of the tumor, as it offers the possibility to observe many behaviors from crossover to stochastic processes. Moreover, the piecewise differential operators, which can be constructed with operators such as classical, Caputo, Caputo-Fabrizio, Atangana-Baleanu and stochastic derivative, have opened new doors to readers in different disciplines and enable them to capture different behaviors in different time intervals. Thus, researchers can achieve successful results in capturing reality by applying these operators to real-world problems.
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    Deterministic-Stochastic modeling: A new direction in modeling real world problems with crossover effect
    (Amer Inst Mathematical Sciences-Aims, 2022) Atangana, Abdon; Araz, Seda Igret
    Many real world problems depict processes following crossover behaviours. Modelling processes following crossover behaviors have been a great challenge to mankind. Indeed real world problems following crossover from Markovian to randomness processes have been observed in many scenarios, for example in epidemiology with spread of infectious diseases and even some chaos. Deterministic and stochastic methods have been developed independently to develop the future state of the system and randomness respectively. Very recently, Atangana and Seda introduced a new concept called piecewise differentiation and integration, this approach helps to capture processes with crossover effects. In this paper, an example of piecewise modelling is presented with illustration to chaos problems. Some important analysis including a piecewise existence and uniqueness and piecewise numerical scheme are presented. Numerical simulations are performed for different cases.
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    Extension of Atangana-Seda numerical method to partial differential equations with integer and non-integer order
    (Elsevier, 2020) Atangana, Abdon; Araz, Seda Igret
    In this study, we extend newly introduced numerical method to partial differential and integral equations with integer and non-integer order. This numerical approximation suggested by Atangana and Seda was constructed with Newton polynomial. Moreover it is accurate and effi-cient for solving partial differential and integral equations. Also, we present numerical simulation for solution of the considered equation. The numerical results show that this numerical approach is useful and accurate for obtaining numerical solution of such equations. (C) 2020 The Authors. Published by Elsevier B.V. on behalf of Faculty of Engineering, Alexandria University. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/ licenses/by-nc-nd/4.0/).
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    Extension of Chaplygin's existence and uniqueness method for fractal-fractional nonlinear differential equations
    (Amer Inst Mathematical Sciences-Aims, 2024) Atangana, Abdon; Araz, Seda Igret
    The existence and uniqueness of solutions to nonlinear ordinary differential equations with fractal-fractional derivatives, with Dirac-delta, exponential decay, power law, and generalized Mittag-Leffler kernels, have been the focus of this work. To do this, we used the Chaplygin approach, which entails creating two lower and upper sequences that converge to the solution of the equations under consideration. We have for each case provided the conditions under which these sequences are obtained and converge.
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    Extension of successive midpoint scheme for nonlinear differential equations with global nonlocal operators
    (Elsevier, 2025) Atangana, Abdon; Araz, Seda Igret
    This research looked at nonlinear ordinary differential equations with global differential operators and the Dirac-delta and exponential decay kernels. A recently developed numerical approach based on the repetitive use of the well-known midpoint quadrature approximation. Although no theoretical analysis was offered, the method was applied to solve several nonlinear equations in chaos and epidemiology. The observed findings demonstrate the effect of the chosen function g (t), for example, a simple SIR model produced chaotic and crossover behaviors.
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    Fractional stochastic modelling illustration with modified Chua attractor
    (Springer Heidelberg, 2019) Atangana, Abdon; Araz, Seda Igret
    Very recently a new concept to capture more complexities in nature was suggested. The concept combines two important concepts of modeling including fractional differentiation and stochastic approach. In this work, we aim to investigate new chaotic attractors using the modified Chuan models and the new approach. We use the log-normal distribution to convert constant parameters into distribution. Then we use 3 different types of differential operators including Caputo, Caputo-Fabrizio and Atangana-Baleanu derivatives. We solve the new equations by using the newly introduced numerical scheme. Our numerical simulations display very new attractors.
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    Mathematical model of COVID-19 spread in Turkey and South Africa: theory, methods, and applications
    (Springer, 2020) Atangana, Abdon; Araz, Seda Igret
    A comprehensive study about the spread of COVID-19 cases in Turkey and South Africa has been presented in this paper. An exhaustive statistical analysis was performed using data collected from Turkey and South Africa within the period of 11 March 2020 to 3 May 2020 and 05 March and 3 of May, respectively. It was observed that in the case of Turkey, a negative Spearman correlation for the number of infected class and a positive Spearman correlation for both the number of deaths and recoveries were obtained. This implied that the daily infections could decrease, while the daily deaths and number of recovered people could increase under current conditions. In the case of South Africa, a negative Spearman correlation for both daily deaths and daily infected people were obtained, indicating that these numbers may decrease if the current conditions are maintained. The utilization of a statistical technique predicted the daily number of infected, recovered, and dead people for each country; and three results were obtained for Turkey, namely an upper boundary, a prediction from current situation and lower boundary. The histograms of the daily number of newly infected, recovered and death showed a sign of lognormal and normal distribution, which is presented using the Bell curving method parameters estimation. A new mathematical model COVID-19 comprised of nine classes was suggested; of which a formula of the reproductive number, well-poseness of the solutions and the stability analysis were presented in detail. The suggested model was further extended to the scope of nonlocal operators for each case; whereby a numerical method was used to provide numerical solutions, and simulations were performed for different non-integer numbers. Additionally, sections devoted to control optimal and others dedicated to compare cases between Turkey and South Africa with the aim to comprehend why there are less numbers of deaths and infected people in South Africa than Turkey were presented in detail.
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    Modeling third waves of Covid-19 spread with piecewise differential and integral operators: Turkey, Spain and Czechia
    (Elsevier, 2021) Atangana, Abdon; Araz, Seda Igret
    Several collected data representing the spread of some infectious diseases have demonstrated that the spread does not really exhibit homogeneous spread. Clear examples can include the spread of Spanish flu and Covid19. Collected data depicting numbers of daily new infections in the case of Covid-19 from countries like Turkey, Spain show three waves with different spread patterns, a clear indication of crossover behaviors. While modelers have suggested many mathematical models to depicting these behaviors, it becomes clear that their mathematical models cannot really capture the crossover behaviors, especially passage from deterministic resetting to stochastics. Very recently Atangana and Seda have suggested a concept of piecewise modeling consisting in defining a differential operator piece-wisely. The idea was first applied in chaos and outstanding patterns were captured. In this paper, we extend this concept to the field of epidemiology with the aim to depict waves with different patterns. Due to the novelty of this concept, a different approach to insure the existence and uniqueness of system solutions are presented. A piecewise numerical approach is presented to derive numerical solutions of such models. An illustrative example is presented and compared with collected data from 3 different countries including Turkey, Spain and Czechia. The obtained results let no doubt for us to conclude that this concept is a new window that will help mankind to better understand nature.
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    NEW CLASS OF VOLTERRA INTEGRO-DIFFERENTIAL EQUATIONS WITH FRACTAL-FRACTIONAL OPERATORS: EXISTENCE, UNIQUENESS AND NUMERICAL SCHEME
    (Amer Inst Mathematical Sciences-Aims, 2021) Araz, Seda Igret
    In this paper, we introduce a new fractional integro-differential equation involving newly introduced differential and integral operators socalled fractal-fractional derivatives and integrals. We present a numerical scheme that is convenient for obtaining solution of such equations. We give the general conditions for the existence and uniqueness of the solution of the considered equation using Banach fixed-point theorem. Both the suggested new equation and new numerical scheme will considerably contribute for our readers in theory and applications.
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    New concept in calculus: Piecewise differential and integral operators
    (Pergamon-Elsevier Science Ltd, 2021) Atangana, Abdon; Araz, Seda Igret
    In the last decades, many methodologies have been suggested to depict behaviors of some complex world's problems arising in many academic fields. One of these problems is the multi-steps behavior displayed by some problems. A concept of piecewise derivative is introduced in this paper with the aim to model real world problems following multiples processes. We have presented some important properties of these definitions. We considered different scenarios and presented numerical schemes that could be used to solve such problems. Illustrative examples, including chaotic and epidemiological models are presented to see the effectiveness of the suggested concept. (c) 2021 Elsevier Ltd. All rights reserved.
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    New numerical approximation for Chua attractor with fractional and fractal-fractional operators
    (Elsevier, 2020) Atangana, Abdon; Araz, Seda Igret
    In this study, we present new numerical scheme for modified Chua attractor model with fractional operators. However we give numerical solution of the considered model with fractal-fractional operators. Also, we offer error analysis for a general Cauchy problem with fractional and fractal-fractional operators. For numerical solution of the considered equation, we use new numerical scheme which is established with an efficient polynomial known as Newton interpolation polynomial. The results are discussed with some examples and simulations. (C) 2020 The Authors. Published by Elsevier B.V. on behalf of Faculty of Engineering, Alexandria University.
  • Küçük Resim
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    New numerical method for ordinary differential equations: Newton polynomial
    (Elsevier, 2020) Atangana, Abdon; Araz, Seda Igret
    An error estimate of optimal order is established for the correspondingnumerical solutions in a scaled residual norm. In addition, a mathematical convergenceis established in a weak L2topology for the new numerical method. Numerical resultsare reported to demonstrate the efficiency of the primal–dual weak Galerkin method aswell as the accuracy of the numerical approximations
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    Nonlinear equations with global differential and integral operators: Existence, uniqueness with application to epidemiology
    (Elsevier, 2021) Atangana, Abdon; Araz, Seda Igret
    Very recently, the concept of instantaneous change was extended with the aim to accommodate prediction of more complex real world problems that could not be predicted or depicted by the existing rate of change. The extension gave birth to a more general differential operator that to be a derivative associate to the well-known Riemann-Stieltjes integral. In addition to this, using specific functions, one is able to recover all existing local differential operators defined as rate of change. This extended concept is still at its genesis and more works need to be done to establish a Riemann-Stieltjes calculus. In this paper, we aim to present a detailed analysis of an important class of differential equations called stochastic equations with the new classes of differential operators with the global derivative with integer and non-integer orders. We considered many classes as nonlinear Cauchy problems, then we presented existence and the uniqueness of their solutions using the linear growth and the Lipchitz conditions. We derived numerical solutions for each class and presented the error analysis. To show the applicability of these operators, we considered three epidemiological problems, including the zombie virus spread model, the zika virus spread model and Ebola model. We solved each model using the suggested numerical scheme and presented the numerical solutions for different values of fractional order and the global function g(t). Our results showed that, more complex real world problems could be depicted using these classes of differential equations.
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    Numerical Regularization of Optimal Control for the Coefficient Function in a Wave Equation
    (Springer International Publishing Ag, 2019) Subasi, Murat; Araz, Seda Igret
    This study deals with optimal control of the coefficient function in a wave equation. After displaying the ill-posedness of the problem, a regularized version is considered instead. The stages of finding the optimal control and approximation processes to this control are investigated, respectively. The results of regularization process are tested with numerical examples.