Browsing by Author "Erdogan, Fevzi"

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An Almost Second Order Uniformly Convergent Scheme for a Singularly Perturbed Initial Value Problem
(Springer, 2014) Cen, Zhongdi; Erdogan, Fevzi; Xu, Aimin
In this paper a nonlinear singularly perturbed initial problem is considered. The behavior of the exact solution and its derivatives is analyzed, and this leads to the construction of a Shishkin-type mesh. On this mesh a hybrid difference scheme is proposed, which is a combination of the second order difference schemes on the fine mesh and the midpoint upwind scheme on the coarse mesh. It is proved that the scheme is almost second-order convergent, in the discrete maximum norm, independently of singular perturbation parameter. Numerical experiment supports these theoretical results.
Analysis of Factors Effecting Pisa 2015 Mathematics Literacy Via Educational Data Mining
(Turkish Education Assoc, 2020) Gure, Ozlem Bezek; Kayri, Murat; Erdogan, Fevzi
The aim of this study is to determine the factors affecting PISA 2015 Mathematics literacy by using data mining methods such as Multilayer Perceptron Artificial Neural Networks and Random Forest. Cause and effect relation within the context of the study was tried to be discovered by means of data mining methods at the level of deep learning. In terms of Prediction Ability, the findings of the method whose performance was high were accepted as the factors determining the qualifications in Mathematics literacy in Turkey. In this study, the information, which was collected from a total of 4422 students, 215 (49%) of whom were boys and 2257 (51%) of whom were girls participating in PISA 2015 test, was used. The scores, which the students, having gone in for PISA 2015 test, got from mathematics test, and dependent variables and 25 variables, which were thought to have connection with dependent variables institutionally, were included in the analysis as predictors. As a result of analysis, it was witnessed that Random Forest (RF) method made prediction with smaller errors in terms of a number of performance indicators. The factors that random forest method found important after anxiety variable are Turkish success level of students, mother education level, motivation level, the belief in epistemology, interest level of teachers and class disciplinary environment, respectively. The statistical meaning, significance and impact levels of other variables were tackled together with their details in this study. It is expected that this study will set an example for data mining use in the process of educational studies and that the factors whose affects were found out about the students' mathematics literacy will shed light on National Education system.
Design of a Novel Second-Order Prediction Differential Model Solved by Using Adams and Explicit Runge-Kutta Numerical Methods
(Hindawi Ltd, 2020) Sabir, Zulqurnain; Guirao, Juan L. G.; Saeed, Tareq; Erdogan, Fevzi
In this study, a novel second-order prediction differential model is designed, and numerical solutions of this novel model are presented using the integrated strength of the Adams and explicit Runge-Kutta schemes. The idea of the present study comes to the mind to see the importance of delay differential equations. For verification of the novel designed model, four different examples of the designed model are numerically solved by applying the Adams and explicit Runge-Kutta schemes. These obtained numerical results have been compared with the exact solutions of each example that indicate the performance and exactness of the designed model. Moreover, the results of the designed model have been presented numerically and graphically.
Design of Morlet Wavelet Neural Network for Solving a Class of Singular Pantograph Nonlinear Differential Models
(Ieee-inst Electrical Electronics Engineers inc, 2021) Nisar, Kashif; Sabir, Zulqurnain; Zahoor Raja, Muhammad Asif; Ag. Ibrahim, Ag. Asri; Erdogan, Fevzi; Haque, Muhammad Reazul; Rawat, Danda B.
The aim of this study is to design a layer structure of feed-forward artificial neural networks using the Morlet wavelet activation function for solving a class of pantograph differential Lane-Emden models. The Lane-Emden pantograph differential equation is one of the important kind of singular functional differential model. The numerical solutions of the singular pantograph differential model are presented by the approximation capability of the Morlet wavelet neural networks (MWNNs) accomplished with the strength of global and local search terminologies of genetic algorithm (GA) and interior-point algorithm (IPA), i.e., MWNN-GAIPA. Three different problems of the singular pantograph differential models have been numerically solved by using the optimization procedures of MWNN-GAIPA. The correctness of the designed MWNN-GAIPA is observed by comparing the obtained results with the exact solutions. The analysis for 3, 6 and 60 neurons are also presented to check the stability and performance of the designed scheme. Moreover, different statistical analysis using forty number of trials is presented to check the convergence and accuracy of the proposed MWNN-GAIPA scheme.
Existence, Uniqueness and Blow-Up of Solutions for Generalized Auto-Convolution Volterra Integral Equations
(Elsevier Science inc, 2024) Mostafazadeh, Mahdi; Shahmorad, Sedaghat; Erdogan, Fevzi
In this paper, our intention is to investigate the blow-up theory for generalized auto -convolution Volterra integral equations (AVIEs). To accomplish this, we will consider certain conditions on the main equation. This will establish a framework for our analysis, ensuring that the solution of the equation exists uniquely and is positive. Firstly, we analyze the existence and uniqueness of a local solution for a more general class of AVIEs (including the proposed equation in this paper) under certain hypotheses. Subsequently, we demonstrate the conditions under which this local solution blows up at a finite time. In other words, the solution becomes unbounded at that time. Furthermore, we establish that this blow-up solution can be extended to an arbitrary interval on the non -negative real line, thus referred to as a global solution. These results are also discussed for a special case of generalized AVIEs in which the kernel functions are taken as positive constants.
An Exponentially Fitted Method for Singularly Perturbed Delay Differential Equations
(Hindawi Publishing Corporation, 2009) Erdogan, Fevzi
This paper deals with singularly perturbed initial value problem for linear first-order delay differential equation. An exponentially fitted difference scheme is constructed in an equidistant mesh, which gives first-order uniform convergence in the discrete maximum norm. The difference scheme is shown to be uniformly convergent to the continuous solution with respect to the perturbation parameter. A numerical example is solved using the presented method and compared the computed result with exact solution of the problem. Copyright (C) 2009 Fevzi Erdogan.
A Finite Difference Method on Layer-Adapted Mesh for Singularly Perturbed Delay Differential Equations
(Walter de Gruyter Gmbh, 2020) Erdogan, Fevzi; Sakar, Mehmet Giyas; Saldir, Onur
The purpose of this paper is to present a uniform finite difference method for numerical solution of a initial value problem for semilinear second order singularly perturbed delay differential equation. A numerical method is constructed for this problem which involves appropriate piecewise-uniform Shishkin mesh on each time subinterval. The method is shown to uniformly convergent with respect to the perturbation parameter. A numerical experiment illustrate in practice the result of convergence proved theoretically.
A Finite Difference Scheme for a Class of Singularly Perturbed Initial Value Problems for Delay Differential Equations
(Springer, 2009) Amiraliyev, Gabil M.; Erdogan, Fevzi
This study deals with the singularly perturbed initial value problem for a quasilinear first-order delay differential equation. A numerical method is generated on a grid that is constructed adaptively from a knowledge of the exact solution, which involves appropriate piecewise-uniform mesh on each time subinterval. An error analysis shows that the method is first order convergent except for a logarithmic factor, in the discrete maximum norm, independently of the perturbation parameter. The parameter uniform convergence is confirmed by numerical computations.
Fitted Finite Difference Method for Singularly Perturbed Delay Differential Equations
(Springer, 2012) Erdogan, Fevzi; Amiraliyev, Gabil M.
This paper deals with singularly perturbed initial value problem for linear second-order delay differential equation. An exponentially fitted difference scheme is constructed in an equidistant mesh, which gives first order uniform convergence in the discrete maximum norm. The difference scheme is shown to be uniformly convergent to the continuous solution with respect to the perturbation parameter. A numerical example is solved using the presented method and compared the computed result with exact solution of the problem.
Fitting the Itô Stochastic Differential Equation To the Covid-19 Data in Turkey
(2021) Erdogan, Fevzi; Çalıkuşu, Sevda Özdemir
In this study, COVID-19 data in Turkey is investigated by Stochastic Differential Equation Modeling (SDEM). Firstly, parameters of SDE which occur in mentioned epidemic problem are estimated by using the maximum likelihood procedure. Then, we have obtained reasonable Stochastic Differential Equation (SDE) based on the given COVID-19 data. Moreover, by applying Euler-Maruyama Approximation Method trajectories of SDE are achieved. The performances of trajectories are established by Chi-Square criteria. The results are acquired by using statistical software R-Studio.These results are also corroborated by graphical representation.
The Homotopy Analysis Method for Solving the Time-Fractional Fornberg-Whitham Equation and Comparison With Adomian's Decomposition Method
(Elsevier Science inc, 2013) Sakar, Mehmet Giyas; Erdogan, Fevzi
In this paper, we applied relatively new analytical techniques, the homotopy analysis method (HAM) and the Adomian's decomposition method (ADM) for solving time-fractional Fomberg-Whitham equation. The homotopy analysis method contains the auxiliary parameter, which provides us with a simple way to adjust and control the convergence region of solution series. The fractional derivatives are described in the Caputo sense. A comparison is made the between HAM and ADM results. The present methods performs extremely well in terms of efficiency and simplicity. Numerical results for different particular cases of the problem are presented. (C) 2013 Elsevier Inc. All rights reserved.
Intelligent Computing Technique for Solving Singular Multi-Pantograph Delay Differential Equation
(Springer, 2022) Sabir, Zulqurnain; Wahab, Hafiz Abdul; Nguyen, Tri Gia; Altamirano, Gilder Cieza; Erdogan, Fevzi; Ali, Mohamed R.
The purpose of this study is to introduce a stochastic computing solver for the multi-pantograph delay differential equation (MP-DDE). The MP-DDE is not easy to solve due to the singularities and pantograph terms. An advance computational intelligent paradigm is proposed to solve MP-DDE of the second kind by manipulating the procedures of the artificial neural networks (ANNs) through the optimization of genetic algorithm (GA) and sequential quadratic programming (SQP), i.e., ANNs-GA-SQP. A fitness function is constructed based on MP-DDE of second kind and corresponding boundary conditions. The correctness of the ANNs-GA-SQP is observed by performing the comparison of the proposed and exact solutions. The values of the absolute error (AE) in good measures are provided to solve the MP-DDE of the second kind. The efficacy and correctness of the stochastic computing approach to solve three problems of the MP-DDE of second kind to signify the efficiency, worth, and consistency. Moreover, the statistical soundings are applied to validate the accuracy and consistency.
An Iterative Approximation for Time-Fractional Cahn-Allen Equation With Reproducing Kernel Method
(Springer Heidelberg, 2018) Sakar, Mehmet Giyas; Saldir, Onur; Erdogan, Fevzi
In this article, we construct a novel iterative approach that depends on reproducing kernel method for Cahn-Allen equation with Caputo derivative. Representation of solution and convergence analysis are presented theoretically. Numerical results are given as tables and graphics with intent to show efficiency and power of method. The results demonstrate that approximate solution uniformly converges to exact solution for Cahn-Allen equation with fractional derivative.
Novel Behaviors To the Nonlinear Evolution Equation Describing the Dynamics of Ionic Currents Along Microtubules
(E D P Sciences, 2017) Baskonus, Haci Mehmet; Erdogan, Fevzi; Ozkul, Arif; Asmouh, Itham
In this work, we consider the Bernoulli sub-equation function method for obtaining novel behaviors to the nonlinear evolution equation describing the dynamics of ionic currents along Microtubules. We obtain new results by using this technique. We plot two-and three-dimensional surfaces of the results by using Wolfram Mathematica 9. At the end of this manuscript, we submit a conclusion in the comprehensive manner.
A Numerical Scheme for Semilinear Singularly Perturbed Reaction-Diffusion Problems
(Walter de Gruyter Gmbh, 2020) Yamac, Kerem; Erdogan, Fevzi
In this study we investigated the singularly perturbed boundary value problems for semilinear reaction-difussion equations. We have introduced a basic and computational approach scheme based on Numerov's type on uniform mesh. We indicated that the method is uniformly convergence, according to the discrete maximum norm, independently of the parameter of epsilon. The proposed method was supported by numerical example.
Numerical Solution of Fractional Order Burgers' Equation With Dirichlet and Neumann Boundary Conditions by Reproducing Kernel Method
(Mdpi, 2020) Saldir, Onur; Sakar, Mehmet Giyas; Erdogan, Fevzi
In this research, obtaining of approximate solution for fractional-order Burgers' equation will be presented in reproducing kernel Hilbert space (RKHS). Some special reproducing kernel spaces are identified according to inner products and norms. Then an iterative approach is constructed by using kernel functions. The convergence of this approach and its error estimates are given. The numerical algorithm of the method is presented. Furthermore, numerical outcomes are shown with tables and graphics for some examples. These outcomes demonstrate that the proposed method is convenient and effective.
Numerical Solution of Time-Fractional Kawahara Equation Using Reproducing Kernel Method With Error Estimate
(Springer Heidelberg, 2019) Saldir, Onur; Sakar, Mehmet Giyas; Erdogan, Fevzi
We present a new approach depending on reproducing kernel method (RKM) for time-fractional Kawahara equation with variable coefficient. This approach consists of obtaining an orthonormal basis function on specific Hilbert spaces. In this regard, some special Hilbert spaces are defined. Kernel functions of these special spaces are given and basis functions are obtained. The approximate solution is attained as serial form. Convergence analysis, error estimation and stability analysis are presented after obtaining the approximate solution. To show the power and effect of the method, two examples are solved and the results are given as table and graphics. The results demonstrate that the presented method is very efficient and convenient for Kawahara equation with fractional order.
Numerical Solution of Time-Fractional Nonlinear Pdes With Proportional Delays by Homotopy Perturbation Method
(Elsevier Science inc, 2016) Sakar, Mehmet Giyas; Uludag, Fatih; Erdogan, Fevzi
In this paper, homotopy perturbation method (HPM) is applied to solve fractional partial differential equations (PDEs) with proportional delay in t and shrinking in x. The method do not require linearization or small perturbation. The fractional derivatives are taken in the Caputo sense. The present method performs extremely well in terms of efficiency and simplicity. Numerical results for different particular cases of alpha are presented graphically. (C) 2016 Elsevier Inc. All rights reserved.
On the New Hyperbolic and Trigonometric Structures To the Simplified Mch and Srlw Equations
(Springer Heidelberg, 2017) Bulut, Hasan; Sulaiman, Tukur Abdulkadir; Erdogan, Fevzi; Baskonus, Haci Mehmet
With the aid of the Wolfram Mathematica software, the powerful sine-Gordon expansion method is used in constructing some new solutions to the two well-known nonlinear models, namely, the simplified modified Camassa-Holm and symmetric regularized long-wave equations. We obtain some novel complex, trigonometric and hyperbolic function solutions to these two equations. We also plot the three-dimensional and two-dimensional graphics for each solutions obtained using the same programming code in Wolfram Mathematica software. Finally, we submit a comprehensive conclusions.
A Parameter Robust Method for Singularly Perturbed Delay Differential Equations
(Springer, 2010) Erdogan, Fevzi
Uniform finite difference methods are constructed via nonstandard finite difference methods for the numerical solution of singularly perturbed quasilinear initial value problem for delay differential equations. A numerical method is constructed for this problem which involves the appropriate Bakhvalov meshes on each time subinterval. The method is shown to be uniformly convergent with respect to the perturbation parameter. A numerical example is solved using the presented method, and the computed result is compared with exact solution of the problem.