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A novel Covid-19 model with fractional differential operators with singular and non-singular kernels: Analysis and numerical scheme based on Newton polynomial
(Elsevier, 2021) Atangana, Abdon; Araz, Seda IGret
To capture more complexities associated to the spread of Covid-19 within a given population, we considered a system of nine differential equations that include a class of susceptible, 5 sub-classes of infected population, recovered, death and vaccine. The mathematical model was suggested with a lockdown function such that after the lockdown, the function follows a fading memory rate, a concept that is justified by the effect of social distancing that suggests, susceptible class should stay away from infected objects and humans. We presented a detailed analysis that includes reproductive number and stability analysis. Also, we introduced the concept of fractional Lyapunov function for Caputo, Caputo-Fabrizio and the Atangana-Baleanu fractional derivatives. We established the sign of the fractional Lyapunov function in all cases. Additionally we proved that, if the fractional order is one, we recover the results Lyapunov for the model with classical differential operators. With the nonlinearity of the differential equations depicting the complexities of the Covid-19 spread especially the cases with nonlocal operators, and due to the failure of existing analytical methods to provide exact solutions to the system, we employed a numerical method based on the Newton polynomial to derive numerical solutions for all cases and numerical simulations are provided for different values of fractional orders and fractal dimensions. Collected data from Turkey case for a period of 90 days were compared with the suggested mathematical model with Atangana-Baleanu fractional derivative and an agreement was reached for alpha 1:009. (C) 2021 THE AUTHORS. Published by Elsevier BV 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/).
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.
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.
Analysis of a derivative with two variable orders
(Amer Inst Mathematical Sciences-Aims, 2022) Atangana, Abdon; Akgul, Ali
In this paper, we investigate a derivative with the two variable orders. The first one shows the variable order fractal dimension and the second one presents the fractional order. We consider these derivatives with the power law kernel, exponential decay kernel and Mittag-Leffler kernel. We give the theory of this derivative in details. We also present the numerical approximation. The results we obtained in this work are very useful for researchers to improve many things for fractal fractional derivative with two variable orders.
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.
Analysis of fractal fractional differential equations
(Elsevier, 2020) Atangana, Abdon; Akgul, Ali; Owolabi, Kolade M.
Nonlocal differential and integral operators with fractional order and fractal dimension have been recently introduced and appear to be powerful mathematical tools to model complex real world problems that could not be modeled with classical and nonlocal differential and integral operators with single order. To stress further possible application of such operators, we consider in this work an advection-dispersion model, where the velocity is considered to be 1. We consider three cases of the models, when the kernels are power law, exponential decay law and the generalized Mittag-Leffler kernel. For each case, we present a detailed analysis including, numerical solution, stability analysis and error analysis. We present some numerical simulation. (C) 2020 The Authors. Published by Elsevier B.V. on behalf of Faculty of Engineering, Alexandria University.
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.
Can transfer function and Bode diagram be obtained from Sumudu transform
(Elsevier, 2020) Atangana, Abdon; Akgul, Ali
In the last past year researchers have relied on the ability of Laplace transform to solve partial, ordinary linear equations with great success. Important analysis in signal analysis including the transfer function, Bode diagram, Nyquist plot and Nichols plot are obtained based on the Laplace transform. The output of the analysis depends only on the results obtained from Laplace transform. However, one weakness of Laplace transform is that the Laplace transform of even func-tion is odd while the Laplace transform of an old function is even which is lack of conservation of properties. On the other hand there exist a similar integral transform known as Sumudu transform has the ability to conserve the properties of the function from real space to complex space. The question that arises in the work, is the following: Can we apply the Sumudu transform to construct new transfer functions that will lead to new Bode, Nichols and Nyquist plots? this question is answered in this work. (c) 2019 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/).
Conformable derivative: A derivative associated to the Riemann-Stieltjes integral
(Natural Sciences Publishing, 2022) Atangana, Abdon; Akgül, Ali; Khan, Muhammad Altaf; Ibrahim, Rabha Waell
In mathematics, the Riemann-Stieltjes integral (Formula Presented) is known to be the more general version of the well-known Riemann integral that is used in classical integral calculus. This integral has found application in several fields. However, there is no clear derivative associate to the generalized integral, except the concept of global derivative that was suggested very recently. Nevertheless, a more generalized differential operator called conformable derivative was suggested and many concerns have been raised as what such operator cannot be considered as derivative. In this paper, we proved that not only the operator is a derivative but a derivative associate to the well-known Riemann-Stieltjes integral. We have in addition present some applications of the conformable derivative in image processing and dynamical processes including chaos and epidemiology © 2022. NSP Natural Sciences Publishing Cor
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.
Existence, uniqueness and numerical solution of stochastic fractional differential equations with integer and non-integer orders
(Amer Inst Mathematical Sciences-Aims, 2024) Araz, Seda I. G. R. E. T.; Cetin, Mehmet Akif; Atangana, Abdon
The parametrized approach is extended in this study to find solutions to differential equations with fractal, fractional, fractal-fractional, and piecewise derivatives with the inclusion of a stochastic component. The existence and uniqueness of the solution to the stochastic Atangana-Baleanu fractional differential equation are established using Caratheodory's existence theorem. For the solution of differential equations using piecewise differential operators, which take into account combining deterministic and stochastic processes utilizing certain significant mathematical tools such as fractal and fractal-fractional derivatives, the applicability of the parametrized technique is being examined. We discuss the crossover behaviors of the model obtained by including these operators and we present some illustrative examples for some problems with piecewise differential operators.
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/).
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.
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.
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.
Integral Transforms and Engineering: Theory, Methods, and Applications
(CRC Press, 2023) Atangana, Abdon; Akgül, Ali
With the aim to better understand nature, mathematical tools are being used nowadays in many different fields. The concept of integral transforms, in particular, has been found to be a useful mathematical tool for solving a variety of problems not only in mathematics, but also in various other branches of science, engineering, and technology. Integral Transforms and Engineering: Theory, Methods, and Applications presents a mathematical analysis of integral transforms and their applications. The book illustrates the possibility of obtaining transfer functions using different integral transforms, especially when mapping any function into the frequency domain. Various differential operators, models, and applications are included such as classical derivative, Caputo derivative, Caputo-Fabrizio derivative, and Atangana-Baleanu derivative. This book is a useful reference for practitioners, engineers, researchers, and graduate students in mathematics, applied sciences, engineering, and technology fields. © 2023 Abdon Atangana, Ali Akgül.
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.
Modeling and forecasting the spread of COVID-19 with stochastic and deterministic approaches: Africa and Europe
(Springer, 2021) Atangana, Abdon; Igret Araz, Seda
Using the existing collected data from European and African countries, we present a statistical analysis of forecast of the future number of daily deaths and infections up to 10 September 2020. We presented numerous statistical analyses of collected data from both continents using numerous existing statistical theories. Our predictions show the possibility of the second wave of spread in Europe in the worse scenario and an exponential growth in the number of infections in Africa. The projection of statistical analysis leads us to introducing an extended version of the well-blancmange function to further capture the spread with fractal properties. A mathematical model depicting the spread with nine sub-classes is considered, first converted to a stochastic system, where the existence and uniqueness are presented. Then the model is extended to the concept of nonlocal operators; due to nonlinearity, a modified numerical scheme is suggested and used to present numerical simulations. The suggested mathematical model is able to predict two to three waves of the spread in the near future.
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.
Modelling and analysis of fractal-fractional partial differential equations: Application to reaction-diffusion model
(Elsevier, 2020) Owolabi, Kolade M.; Atangana, Abdon; Akgul, Ali
In this paper, an extension is paid to an idea of fractal and fractional derivatives which has been applied to a number of ordinary differential equations to model a system of partial differential equations. As a case study, the fractal fractional Schnakenberg system is formulated with the Caputo operator (in terms of the power law), the Caputo-Fabrizio operator (with exponential decay law) and the Atangana-Baleanu fractional derivative (based on the Mittag-Liffler law). We design some algorithms for the Schnakenberg model by using the newly proposed numerical methods. In such schemes, it worth mentioning that the classical cases are recovered whenever alpha = 1 and beta = 1. Numerical results obtained for different fractal-order (beta is an element of (0, 1)) and fractional-order (alpha is an element of (0, 1)) are also given to address any point and query that may arise. (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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