Browsing by Author "Cimen, Erkan"

Now showing 1 - 20 of 23

Convergence Analysis of Approximate Method for a Singularly Perturbed Differential-Difference Problem
(Univ Prishtines, 2019) Cimen, Erkan; Amiraliyev, Gabil M.
In this paper, we analyze a singularly perturbed convection-diffusion delay problem with Robin condition. In order to solve this problem numerically, we construct a fitted difference scheme on a uniform mesh. The scheme is based on the method of integral identities with the use of exponential basis functions and interpolating quadrature rules with the weight and remainder terms in integral form. We prove that the method is first order convergence in the discrete maximum norm. Also, we present numerical results that support the theoretical results.
Convergence Analysis of Finite Difference Method for Singularly Perturbed Nonlocal Differential-Difference Problem
(Univ Miskolc inst Math, 2018) Cimen, Erkan; Cakir, Musa
This study is concerned with a singularly perturbed three-point boundary-value problem with delay. Firstly, bounds on the solution and its derivative of the solution to be used later in the paper are obtained. To solve it numerically, we use an exponentially fitted difference scheme on an equidistant mesh which is established by the method of integral identities with the use of exponential basis functions and interpolating quadrature rules with weight and remainder term in integral form. Then, the stability and convergence analysis of difference scheme is given and it is uniformly convergent in perturbation parameter. Furthermore, numerical results which show the effectiveness of the method are presented.
The Difference Schemes for Solving Singularly Perturbed Three-Point Boundary Value Problem
(Springer, 2020) Cakir, Musa; Cimen, Erkan; Amiraliyev, Gabil M.
In this paper, we propose and analyze numerical treatment for a singularly perturbed convection-diffusion boundary value problem with nonlocal condition. First, the boundary layer behavior of the exact solution and its first derivative have been estimated. Then we construct a finite difference scheme on a uniform mesh. We prove the uniform convergence of the proposed difference scheme and give an error estimate. We also present numerical examples, which demonstrate computational efficiency of the proposed method.
An Efficient Method for Solving Second-Order Delay Differential Equation
(Univ Miskolc inst Math, 2021) Cimen, Erkan; Uncu, Sevket
In this paper, the initial-value problem for a linear second order delay differential equation is considered. To solve this problem numerically, an appropriate difference scheme is constructed by using the method of integral identities which contains basis functions and interpolating quadrature rules with weight and remainder term in integral form. Besides, the method is proved to be first-order convergent in discrete maximum norm. The numerical illustration provided support the theoretical results. Finally, the proposed method is compared with the im
A First Order Convergent Numerical Method for Solving the Delay Differential Problem
(Lebanese Univ, 2019) Cimen, Erkan
In this paper, the boundary-value problem for a parameter dependent linear first order delay differential equation is analyzed. A finite difference method for approximate solution of this problem is presented. The method is based on fitted difference scheme on a uniform mesh which is achieved by using the method of integral identities which includes the exponential basis functions and applying interpolating quadrature formulas which contain the remainder term in integral form. Also, the method is proved first-order convergent in the discrete maximum norm. Moreover, a numerical example is solved using both the presented method and the Euler method and compared the computed results.
High-Order Finite Difference Technique for Delay Pseudo-Parabolic Equations
(Elsevier Science Bv, 2017) Amiraliyev, Gabil M.; Cimen, Erkan; Amirali, Ilhame; Cakir, Musa
One dimensional initial boundary delay pseudo-parabolic problem is being considered. To solve this problem numerically, we construct higher order difference method for approximation to the considered problem and obtain the error estimate for its solution. Based on the method of energy estimate the fully discrete scheme is shown to be convergent of order four in space and of order two in time. Numerical example is presented. (C) 2017 Elsevier B.V. All rights reserved.
A Novel Approximation on the Solution of Systems of Ordinary Differential Equations
(Rgn Publ, 2024) Uncu, Sevket; Cimen, Erkan
In this paper, the initial-value problem for the system of first-order differential equations is considered. To solve this problem, we construct a fitted difference scheme using the finite difference method, which is based on integral identities for the quadrature formula with integral term remainder terms. Next, we prove first-order convergence for the method in the discrete maximum norm. Although this scheme has the same rate of convergence, it has more efficiency and accuracy compared to the classical Euler scheme. Two test problems are solved by using the proposed method and the classical Euler method, which confirm the theoretical findings. The numerical results obtained from here show that the proposed method is reliable, efficient, and accurate.
A Novel Numerical Approach for Solving Delay Differential Equations Arising in Population Dynamics
(Amer inst Mathematical Sciences-aims, 2023) Obut, Tugba; Cimen, Erkan; Cakir, Musa
In this paper, the initial-value problem for a class of first order delay differential equations, which emerges as a model for population dynamics, is considered. To solve this problem numerically, using the finite difference method including interpolating quadrature rules with the basis functions, we construct a fitted difference scheme on a uniform mesh. Although this scheme has the same rate of convergence, it has more efficiency and accuracy compared to the classical Euler scheme. The different models, Nicolson's blowfly and Mackey-Glass models, in population dynamics are solved by using the proposed method and the classical Euler method. The numerical results obtained from here show that the proposed method is reliable, efficient, and accurate.
A Numerical Approach for Fredholm Delay Integro Differential Equation
(Rgn Publ, 2021) Cimen, Erkan; Enterili, Kubra
This paper deal with the initial-value problem for a linear first order Fredholm delay integro differential equation. To solve this problem numerically, a finite difference scheme is presented, which based on the method of integral identities with the use of exponential form basis function. As a result of the error analysis, it is proved that the method is first-order convergent in the discrete maximum norm. Finally, an example is provided that supports the theoretical results.
A Numerical Approach for Solving Nonlinear Fredholm Integro-Differential Equation With Boundary Layer
(Springer Heidelberg, 2022) Cakir, Musa; Ekinci, Yilmaz; Cimen, Erkan
The study deals with an initial-value problem for a singularly perturbed nonlinear Fredholm integro-differential equation. Parameter explicit theoretical bounds on the continuous solution and its derivative are derived. To solve the approximate solution to this problem, a new difference scheme is constructed with the finite difference method by using the interpolated quadrature rules with the remaining terms in integral form. Parameter uniform error estimates for the approximate solution are established. It is proved that the method converges in the discrete maximum norm, uniformly with respect to the perturbation parameter. Numerical results are given to illustrate the parameter-uniform convergence of the numerical approximations.
Numerical Method for a Neutral Delay Differential Problem
(Lebanese Univ, 2017) Cimen, Erkan; Ekinci, Yilmaz
In this paper, the initial-value problem for a linear first order neutral delay differential equation is considered. To solve this problem numerically, we construct a fitted difference scheme on a uniform mesh which is succeeded by the method of integral identities with the use of exponential basis functions and interpolating quadrature rules with weight and remainder term in integral form. Also, the method is first-order convergent in the discrete maximum norm. Furthermore, numerical illustrations provide support of the theoretical results.
Numerical Method for a Singularly Perturbed Convection-Diffusion Problem With Delay
(Elsevier Science inc, 2010) Amiraliyev, Gabil M.; Cimen, Erkan
This paper deals with the singularly perturbed boundary value problem for a linear second-order delay differential equation. For the numerical solution of this problem, we use an exponentially fitted difference scheme on a uniform mesh which is accomplished by the method of integral identities with the use of exponential basis functions and interpolating quadrature rules with weight and remainder term in integral form. It is shown that one gets first order convergence in the discrete maximum norm, independently of the perturbation parameter. Numerical results are presented which illustrate the theoretical results. (C) 2010 Elsevier Inc. All rights reserved.
Numerical Solution of a Boundary Value Problem Including Both Delay and Boundary Layer
(Vilnius Gediminas Tech Univ, 2018) Cimen, Erkan
Difference method on a piecewise uniform mesh of Shishkin type, for a singularly perturbed boundary-value problem for a nonlinear second order delay differential equation is analyzed. Also, the method is proved that it gives essentially first order parameter-uniform convergence in the discrete maximum norm. Furthermore, numerical results are presented in support of the theory.
Numerical Solution of Volterra Integro-Differential Equation With Delay
(Journal Mathematics & Computer Science-jmcs, 2020) Cimen, Erkan; Yatar, Sabahattin
We consider an initial value problem for a linear first-order Volterra delay integro-differential equation. We develop a novel difference scheme for the approximate solution of this problem via a finite difference method. The method is based on the fitted difference scheme on a uniform mesh which is achieved by using the method of integral identities which includes the exponential basis functions and applying to interpolate quadrature formulas that contain the remainder term in integral form. Also, the method is proved to be first-order convergent in the discrete maximum norm. Furthermore, a numerical experiment is performed to verify the theoretical results. Finally, the proposed scheme is compared with the implicit Euler scheme.
Numerical Treatment of a Quasilinear Initial Value Problem With Boundary Layer
(Taylor & Francis Ltd, 2016) Cakir, Musa; Cimen, Erkan; Amirali, Ilhame; Amiraliyev, Gabil M.
The paper deals with the singularly perturbed quasilinear initial value problem exhibiting initial layer. First the nature of solution of differential problem before presenting method for its numerical solution is discussed. The numerical solution of the problem is performed with the use of a finite-fitted difference scheme on an appropriate piecewise uniform mesh (Shishkin-type mesh). 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. Finally, numerical results supporting the theory are presented.
Numerical Treatment of Nonlocal Boundary Value Problem With Layer Behaviour
(Belgian Mathematical Soc Triomphe, 2017) Cimen, Erkan; Cakir, Musa
This paper deals with the singularly perturbed nonlocal boundary value problem for a linear first order differential equation. For the numerical solution of this problem, we use a fitted difference scheme on a piecewise uniform Shishkin mesh. An error analysis shows that the method is almost first order convergent, in the discrete maximum norm, independently of the perturbation parameter. Numerical results are presented which illustrate the theoretical results.
On the Solution of the Delay Differential Equation Via Laplace Transform
(Rgn Publ, 2020) Cimen, Erkan; Uncu, Sevket
In this paper, we consider the initial-value problem for a linear second order delay differential equation. We use Laplace transform method for solving this problem. Furthermore, we present examples provided support the theoretical results.
A Priori Estimates for Solution of Singularly Perturbed Boundary Value Problem With Delay in Convection Term
(Univ Prishtines, 2017) Cimen, Erkan
In this paper, the boundary value problems for singularly perturbed linear second order delay differential equation is considered. The boundary layer behavior of the solution and its first and second derivatives have been established. An example which is in agreement with the theoretical analysis is presented.
Uniform Convergence Method for a Delay Differential Problem With Layer Behaviour
(Springer Basel Ag, 2019) Cimen, Erkan; Amiraliyev, Gabil M.
Difference method on a piecewise uniform mesh of Shishkin type, for a singularly perturbed boundary-value problem for a linear second-order delay differential equation is examined. It is proved that it gives essentially a first-order parameter-uniform convergence in the discrete maximum norm. Furthermore, numerical results are presented in support of the theory.
A Uniform Convergent Method for Singularly Perturbed Nonlinear Differential-Difference Equation
(Rgn Publ, 2017) Cimen, Erkan; Amiraliyev, Gabil M.
In this paper, the singularly perturbed boundary-value problem for a nonlinear second order delay differential equation is considered. For the numerical solution of this problem, we use an exponentially fitted difference scheme on a uniform mesh which is succeeded by the method of integral identities with the use of exponential basis functions and interpolating quadrature rules with weight and remainder term in integral form. Also, the method is proved to be first-order convergent in the discrete maximum norm uniformly in the perturbation parameter. Furthermore, numerical illustration provide support of the theoretical results.