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Öğe An Alternative Approach for Nonlinear Optimization Problem with Caputo - Fabrizio Derivative(E D P Sciences, 2018) Evirgen, Firat; Yavuz, MehmetIn this study, a fractional mathematical model with steepest descent direction is proposed to find optimal solutions for a class of nonlinear programming problem. In this sense, Caputo-Fabrizio derivative is adapted to the mathematical model. To demonstrate the solution trajectory of the mathematical model, we use the multistage variational iteration method (MVIM). Numerical simulations and comparisons on some test problems show that the mathematical model generated using Caputo-Fabrizio fractional derivative is both feasible and efficient to find optimal solutions for a certain class of equality constrained optimization problems.Öğe Analysis and numerical computations of the fractional regularized long-wave equation with damping term(Wiley, 2021) Yavuz, Mehmet; Sulaiman, Tukur Abdulkadir; Usta, Fuat; Bulut, HasanThis study explores the fractional damped generalized regularized long-wave equation in the sense of Caputo, Atangana-Baleanu, and Caputo-Fabrizio fractional derivatives. With the aid of fixed-point theorem in the Atangana-Baleanu fractional derivative with Mittag-Leffler-type kernel, we show the existence and uniqueness of the solution to the damped generalized regularized long-wave equation. The modified Laplace decomposition method (MLDM) defined in the sense of Caputo, Atangana-Baleanu, and Caputo-Fabrizio (in the Riemann sense) operators is used in securing the approximate-analytical solutions of the nonlinear model. The numerical simulations of the obtained solutions are performed with different suitable values of rho, which is the order of fractional parameter. We have seen the effect of the various parameters and variables on the displacement in figures.Öğe Analysis of a fractional pollution model in a system of three interconnecting lakes(Amer Inst Mathematical Sciences-Aims, 2023) Anjam, Yasir Nadeem; Yavuz, Mehmet; Rahman, Mati ur; Batool, AmnaWater pollution is a critical global concern that demands ongoing scrutiny and revision of water resource policies at all levels to safeguard a healthy living environment. In this study, we focus on examining the dynamics of a fractional-order model involving three interconnected lakes, utilizing the Caputo differential operator. The aim is to investigate the issue of lake pollution by analyzing a system of linear equations that represent the interconnecting waterways. To numerically solve the model, we employ two methods: The Laplace transform with the Adomian decomposition method (LADM) and the Homotopy perturbation method (HPM). We compare the obtained numerical solutions from both methods and present the results. The study encompasses three variations of the model: the periodic input model, the exponentially decaying input model, and the linear input model. MATLAB is employed to conduct numerical simulations for the proposed scheme, considering various fractional orders. The numerical results are further supported by informative graphical illustrations. Through simulation, we validate the suitability of the proposed model for addressing the issue at hand. The outcomes of this research contribute to the understanding and management of water pollution, aiding policymakers and researchers in formulating effective strategies for maintaining water quality and protecting our environment.Öğe Analysis of the Influences of Parameters in the Fractional Second-Grade Fluid Dynamics(Mdpi, 2022) Yavuz, Mehmet; Sene, Ndolane; Yildiz, MustafaThis work proposes a qualitative study for the fractional second-grade fluid described by a fractional operator. The classical Caputo fractional operator is used in the investigations. The exact analytical solutions of the constructed problems for the proposed model are determined by using the Laplace transform method, which particularly includes the Laplace transform of the Caputo derivative. The impact of the used fractional operator is presented; especially, the acceleration effect is noticed in the paper. The parameters' influences are focused on the dynamics such as the Prandtl number (Pr), the Grashof numbers (Gr), and the parameter eta when the fractional-order derivative is used in modeling the second-grade fluid model. Their impacts are also analyzed from a physical point of view besides mathematical calculations. The impact of the fractional parameter alpha is also provided. Finally, it is concluded that the graphical representations support the theoretical observations of the paper.Öğe Analytic simulation of MHD boundary layer flow of a chemically reacting upper-convected Maxwell fluid past a vertical surface subjected to double stratifications with variable properties(Springer Heidelberg, 2022) Fayz-Al-Asad, Md.; Oreyeni, Tosin; Yavuz, Mehmet; Olanrewaju, Philip O.The study of thermal stratification has a broad scope of applications in solar engineering owing to its ability to predict the cases of achieving superior energy efficiency. This present communication focuses on the flow of a free convective MHD upper-convected Maxwell fluid in concert temperature-dependent viscosity, thermal conductivity across a stratified surface with nth order of chemical reaction. The governing partial differential equations are transformed into nonlinear ordinary differential equations by introducing relevant similarity variables and approximate analytical solution is determined operating the homotopy analysis method. Influence of different relevant parameters such as Deborah number, stratification, chemical reaction and variable thermophysical parameters on temperature, velocity and concentration distributions is shown to highlight the specifics of heat and mass transfer flow characteristics. It is followed that for the cases of n=1 and n=2, the concentration of species reduces for increasing chemical reaction parameter. It is also noticed that, the values of -f ''(0) decrease while -theta'(0) and -phi(0) increase with increasing Deborah number beta.Öğe Analytical and numerical approaches to nerve impulse model of fractional-order(Wiley, 2020) Yavuz, Mehmet; Yokus, AsifWe consider a fractional-order nerve impulse model which is known as FitzHugh-Nagumo (F-N) model in this paper. Knowing the solutions of this model allows the management of the nerve impulses process. Especially, considering this model as fractional-order ensures to be able to analyze in detail because of the memory effect. In this context, first, we use an analytical solution and with the aim of this solution, we obtain numerical solutions by using two numerical schemes. Then, we demonstrate the walking wave-type solutions of the stated problem. These solutions include complex trigonometric functions, complex hyperbolic functions, and algebraic functions. In addition, the linear stability analysis is performed and the absolute error is occurred by comparing the numerical results with the analytical result. All of the results are depicted by tables and figures. This paper not only points out the exact and numerical solutions of the model but also compares the differences and the similarities of the stated solution methods. Therefore, the results of this paper are important and useful for either neuroscientists and physicists or mathematicians and engineers.Öğe Approximate Solutions of the Model Describing Fluid Flow Using Generalized ?-Laplace Transform Method and Heat Balance Integral Method(Mdpi, 2020) Yavuz, Mehmet; Sene, NdolaneThis paper addresses the solution of the incompressible second-grade fluid models. Fundamental qualitative properties of the solution are primarily studied for proving the adequacy of the physical interpretations of the proposed model. We use the Liouville-Caputo fractional derivative with its generalized version that gives more comprehensive physical results in the analysis and investigations. In this work, both the rho-Laplace homotopy transform method (rho-LHTM) and the heat balance integral method (HBIM) are successfully combined to solve the fractional incompressible second-grade fluid differential equations. Numerical simulations and their physical interpretations of the mentioned incompressible second-grade fluid model are ensured to illustrate the main findings. It is also proposed that one can recognize the differences in physical analysis of diffusions such as ballistic diffusion, super diffusion, and subdiffusion cases by considering the impact of the orders rho and phi.Öğe Chaotic dynamics of a fractional order HIV-1 model involving AIDS-related cancer cells(Pergamon-Elsevier Science Ltd, 2020) Naik, Parvaiz Ahmad; Owolabi, Kolade M.; Yavuz, Mehmet; Zu, JianMathematical models in epidemiology have been studied in the literature to understand the mechanism that underlies AIDS-related cancers, providing us with a better insight towards cancer immunity and viral oncogenesis. In this study, we propose a dynamical fractional order HIV-1 model in Caputo sense which involves the interactions between cancer cells, healthy CD4(+)T lymphocytes, and virus infected CD4(+)T lymphocytes leading to chaotic behavior. The model has been investigated for the existence and uniqueness of its solution via fixed point theory, while the unique non-negative solution remains bounded within the biologically feasible region. The stability analysis of the model is performed and the biological relevance of the equilibria is also discussed in the paper. The numerical simulations are obtained under different instances of fractional order alpha. It is observed that, as the fractional power decreases from 'one' the chaotic behavior becomes more and more attractive. The existence of chaotic attractors for various species interaction has been observed in 2D and 3D cases. The time series evolution of the species show ing different distributions under different fractional order alpha. The results show that order of the fractional derivative has a significant effect on the dynamic process. (c) 2020 Elsevier Ltd. All rights reserved.Öğe CHARACTERIZATIONS OF TWO DIFFERENT FRACTIONAL OPERATORS WITHOUT SINGULAR KERNEL(Edp Sciences S A, 2019) Yavuz, MehmetIn this paper, we analyze the behaviours of two different fractional derivative operators defined in the last decade. One of them is defined with the normalized sinc function (NSF) and the other one is defined with the Mittag-Leffler function (MLF). Both of them have a non-singular kernel. The fractional derivative operator defined with the MLF is developed by Atangana and Baleanu (ABO) in 2016 and the other operator defined with the normalized sinc function (NSFDO) is created by Yang et al. in 2017. These mentioned operators have some advantages to model the real life problems and to solve them. On the other hand, since the Laplace transform (LT) of the ABO can be calculated more easily, it can be preferred to solve linear/nonlinear problems. In this study, we use the perturbation method with coupled the LTs of these operators to analyze their performance in solving some fractional differential equations. Furthermore, by constructing the error analysis, we test the practicability and usefulness of the method.Öğe COMPARING THE NEW FRACTIONAL DERIVATIVE OPERATORS INVOLVING EXPONENTIAL AND MITTAG-LEFFLER KERNEL(Amer Inst Mathematical Sciences-Aims, 2020) Yavuz, Mehmet; Ozdemir, NecatiIn this manuscript, we have proposed a comparison based on newly defined fractional derivative operators which are called as Caputo-Fabrizio (CF) and Atangana-Baleanu (AB). In 2015, Caputo and Fabrizio established a new fractional operator by using exponential kernel. After one year, Atangana and Baleanu recommended a different-type fractional operator that uses the generalized Mittag-Leffler function (MLF). Many real-life problems can be modelled and can be solved by numerical-analytical solution methods which are derived with these operators. In this paper, we suggest an approximate solution method for PDEs of fractional order by using the mentioned operators. We consider the Laplace homotopy transformation method (LHTM) which is the combination of standard homotopy technique (SHT) and Laplace transformation method (LTM). In this study, we aim to demonstrate the effectiveness of the aforementioned method by comparing the solutions we have achieved with the exact solutions. Furthermore, by constructing the error analysis, we test the practicability and usefulness of the method.Öğe Complex dynamics of a discrete-time Bazykin-Berezovskaya prey-predator model with a strong Allee effect(Elsevier, 2022) Naik, Parvaiz Ahmad; Eskandari, Zohreh; Yavuz, Mehmet; Zu, JianThe present paper investigates the critical normal form coefficients for the one-parameter and two-parameter bifurcations of a discrete-time Bazykin-Berezovskaya prey-predator model. Based on the critical coefficients, it can be determined which scenario corresponds to each bifurcation. Further, for a better representation of the study, the complex dynamics of the model are investigated theoretically and numerically using MatcotM, which is a Matlab package. Some graphical representations of the model are presented to verify the obtained results. The outcome of the study reveals that the model undergoes multiple bifurcations including period-doubling, Neimark-Sacker, and strong resonance bifurcations. (C) 2022 Elsevier B.V. All rights reserved.Öğe Comprehensive Investigation of Thermal and Flow Features of Alloy Based Nanofluid Considering Shape and Newtonian Heating Effects via New Fractional Approach(Mdpi, 2023) Muhammad, Shah; Anwar, Talha; Asifa, Mehmet; Yavuz, MehmetThe core purpose of this work is the formulation of a mathematical model by dint of a new fractional modeling approach to study the dynamics of flow and heat transfer phenomena. This approach involves the incorporation of the Prabhakar fractional operator in mathematical analysis to transform the governing system from a conventional framework to a generalized one. This generalized model evaluates the improvement in thermal efficacy of vacuum pump oil because of the inclusion of aluminum alloy nanoparticles. The flow of the under-observation nanofluid starts due to the combined effects of natural convection and the ramped velocity function at the boundary. Meanwhile, an analysis of the energy equation is conducted by taking the Newtonian heating mechanism into consideration. The characteristics of platelet-, brick-, cylinder-, and blade-shaped alloy nanoparticles are incorporated into the primary system using shape-dependent relations for thermal conductivity and viscosity. Both the classical and generalized models are solved to derive the exact solutions by first inserting some dimension-independent quantities and then operating the Laplace transform on the succeeding equations. These solutions are utilized for the development of graphical illustrations to serve the purpose of covering all features of the problem under consideration. Furthermore, changes in energy and flow functions due to the dominant influences of the relevant contributing factors are delineated with appropriate physical arguments. In addition, the numerical results of the skin friction coefficient and Nusselt number are displayed via multiple tables to analyze the disturbance in shear stress and discuss the contribution of the fractional parameters, the volume concentration of the considered nanoparticles, and the shape factor in the boost of the thermal potential of the considered nanofluid. The findings imply that aluminum alloy nanoparticles have the ability to produce a 44% enhancement in the thermal effectiveness of vacuum pump oil. Moreover, the flow velocity is reduced as the loading range of the nanoparticles rises.Öğe A computational approach for shallow water forced Korteweg-De Vries equation on critical flow over a hole with three fractional operators(Ramazan Yaman, 2021) Veeresha, Pundikala; Yavuz, Mehmet; Baishya, ChandraliThe Korteweg-De Vries (KdV) equation has always provided a venue to study and generalizes diverse physical phenomena. The pivotal aim of the study is to analyze the behaviors of forced KdV equation describing the free surface criti-cal flow over a hole by finding the solution with the help of q-homotopy analysis transform technique (q-HATT). he projected method is elegant amalgamations of q-homotopy analysis scheme and Laplace transform. Three fractional oper-ators are hired in the present study to show their essence in generalizing the models associated with power-law distribution, kernel singular, non-local and non-singular. The fixed-point theorem employed to present the existence and uniqueness for the hired arbitrary-order model and convergence for the solution is derived with Banach space. The projected scheme springs the series solution rapidly towards convergence and it can guarantee the convergence associated with the homotopy parameter. Moreover, for diverse fractional order the physi-cal nature have been captured in plots. The achieved consequences illuminates, the hired solution procedure is reliable and highly methodical in investigating the behaviours of the nonlinear models of both integer and fractional order.Öğe Conformable Derivative Operator in Modelling Neuronal Dynamics(Prairie View A & M Univ, Dept Mathematics, 2018) Yavuz, Mehmet; Yaskiran, BurcuThis study presents two new numerical techniques for solving time-fractional one-dimensional cable differential equation (FCE) modeling neuronal dynamics. We have introduced new formulations for the approximate-analytical solution of the FCE by using modified homotopy perturbation method defined with conformable operator (MHPMC) and reduced differential transform method defined with conformable operator (RDTMC), which are derived the solutions for linear-nonlinear fractional PDEs. In order to show the efficiencies of these methods, we have compared the numerical and exact solutions of fractional neuronal dynamics problem. Moreover, we have declared that the proposed models are very accurate and illustrative techniques in determining to approximate-analytical solutions for the PDEs of fractional order in conformable sense.Öğe Differential gradient evolution plus algorithm for constraint optimization problems: A hybrid approach(Ramazan Yaman, 2021) Tabassum, Muhammad Farhan; Akram, Sana; Hassan, Saadia; Karim, Rabia; Naik, Parvaiz Ahmad; Farman, Muhammad; Yavuz, MehmetOptimization for all disciplines is very important and applicable. Optimization has played a key role in practical engineering problems. A novel hybrid meta-heuristic optimization algorithm that is based on Differential Evolution (DE), Gradient Evolution (GE) and Jumping Technique named Differential Gradient Evolution Plus (DGE+) are presented in this paper. The proposed algorithm hybridizes the above-mentioned algorithms with the help of an improvised dynamic probability distribution, additionally provides a new shake off method to avoid premature convergence towards local minima. To evaluate the efficiency, robustness, and reliability of DGE+ it has been applied on seven benchmark constraint problems, the results of comparison revealed that the proposed algorithm can provide very compact, competitive and promising performance.Öğe European option pricing models described by fractional operators with classical and generalized Mittag-Leffler kernels(Wiley, 2022) Yavuz, MehmetIn this paper, we investigate novel solutions of fractional-order option pricing models and their fundamental mathematical analyses. The main novelties of the paper are the analysis of the existence and uniqueness of European-type option pricing models providing to give fundamental solutions to them and a discussion of the related analyses by considering both the classical and generalized Mittag-Leffler kernels. In recent years, the generalizations of classical fractional operators have been attracting researchers' interest globally and they also have been needed to describe the dynamics of complex phenomena. In order to carry out the mentioned analyses, we take the Laplace transforms of either classical or generalized fractional operators into account. Moreover, we evaluate the option prices by giving the models' fractional versions and presenting their series solutions. Additionally, we make the error analysis to determine the efficiency and accuracy of the suggested method. As per the results obtained in the paper, it can be seen that the suggested generalized operators and the method constructed with these operators have a high impact on obtaining the numerical solutions to the option pricing problems of fractional order. This paper also points out a good initiative and tool for those who want to take these types of options into account either individually or institutionally.Öğe European Vanilla Option Pricing Model of Fractional Order without Singular Kernel(Mdpi, 2018) Yavuz, Mehmet; Ozdemir, NecatiRecently, fractional differential equations (FDEs) have attracted much more attention in modeling real-life problems. Since most FDEs do not have exact solutions, numerical solution methods are used commonly. Therefore, in this study, we have demonstrated a novel approximate-analytical solution method, which is called the Laplace homotopy analysis method (LHAM) using the Caputo-Fabrizio (CF) fractional derivative operator. The recommended method is obtained by combining Laplace transform (LT) and the homotopy analysis method (HAM). We have used the fractional operator suggested by Caputo and Fabrizio in 2015 based on the exponential kernel. We have considered the LHAM with this derivative in order to obtain the solutions of the fractional Black-Scholes equations (FBSEs) with the initial conditions. In addition to this, the convergence and stability analysis of the model have been constructed. According to the results of this study, it can be concluded that the LHAM in the sense of the CF fractional derivative is an effective and accurate method, which is computable in the series easily in a short time.Öğe Evaluation of wind energy investment with artificial neural networks(2019) Yıldırım, Hasan Hüseyin; Yavuz, MehmetCountries aiming for sustainability in economic growth and development ensurethe reliability of energy supplies. For countries to provide their energy needsuninterruptedly, it is important for domestic and renewable energy sources to beutilised. For this reason, the supply of reliable and sustainable energy has becomean important issue that concerns and occupies mankind. Of the renewable energysources, wind energy is a clean, reliable and inexhaustible source of energy withlow operating costs. Turkey is a rich nation in terms of wind energy potential.Forecasting of investment efficiency is an important issue before and during theinvestment period in wind energy investment process because of high investmentcosts. It is aimed to forecast the wind energy products monthly with multilayerneural network approach in this study. For this aim a feed forward backpropagation neural network model has been established. As a set of data, windspeed values 48 months (January 2012-December 2015) have been used. Thetraining data set occurs from 36 monthly wind speed values (January 2012-December 2014) and the test data set occurs from other values (January-December2015). Analysis findings show that the trained Artificial Neural Networks (ANNs)have the ability of accurate prediction for the samples that are not used at trainingphase. The prediction errors for the wind energy plantation values are rangedbetween 0.00494-0.015035. Also the overall mean prediction error for thisprediction is calculated as 0.004818 (0.48%). In general, we can say that ANNs beable to estimate the aspect of wind energy plant productions.Öğe Extinction and stationary distribution of a stochastic COVID-19 epidemic model with time-delay(Pergamon-Elsevier Science Ltd, 2022) Ikram, Rukhsar; Khan, Amir; Zahri, Mostafa; Saeed, Anwar; Yavuz, Mehmet; Kumam, PoomWe reformulate a stochastic epidemic model consisting of four human classes. We show that there exists a unique positive solution to the proposed model. The stochastic basic reproduction number R-0(s) is established. A stationary distribution (SD) under several conditions is obtained by incorporating stochastic Lyapunov function. The extinction for the proposed disease model is obtained by using the local martingale theorem. The first order stochastic Runge-Kutta method is taken into account to depict the numerical simulations.Öğe Fractional modeling of plankton-oxygen dynamics under climate change by the application of a recent numerical algorithm(Iop Publishing Ltd, 2021) Kumar, Pushpendra; Suat Erturk, Vedat; Banerjee, Ramashis; Yavuz, Mehmet; Govindaraj, VThis research work is dedicated to studying the dynamics of a coupled plankton-oxygen model in the framework of three non-linear differential equations. As we know that the ocean dynamics have a firm impact on the global climate change and on the creation of the environment. Also, it is recorded that about 70% of the environmental oxygen is manufactured in the oceans due to the photosynthetic bustling of phytoplankton. All the same, the rate of oxygen manufacture based on the temperature of the water and hence can be dominance by global warming. To study the proposed Oxygen-Phytoplankton-Zooplankton system, we use a very recent fractional numerical algorithm. We discuss the existence of the solution for the given model problem because in the case of fractional-order models, the proof of solution existence always becomes an important task. After that, we perform many novel 2-D and 3-D graphs by using Mathematica and Python software to fulfill the requirements of the numerical simulations. We use a generalised form of the well known Liouville-Caputo fractional derivative. Given numerical algorithm is very recent, short, easy, and reliable to use or to apply to the non-linear dynamical models. The main motive of this research work is to study the Plankton-Oxygen dynamics under the climate change by using the proposed fractional-order model with capturing the memory effects and discuss some novel results for the literature on such ecological topics.
