Browsing by Author "Cakir, Musa"

Now showing 1 - 20 of 35

Cakir, M.; Gunes, B. a Fitted Operator Finite Difference Approximation for Singularly Perturbed Volterra-Fredholm Integro-Differential Equations (Vol 10, 3560, 2022)
(Mdpi, 2022) Cakir, Musa; Gunes, Baransel
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.
Convergence Analysis of the Numerical Method for a Singularly Perturbed Periodical Boundary Value Problem
(int Scientific Research Publications, 2016) Cakir, Musa; Amirali, Ilhame; Kudu, Mustafa; Amiraliyev, Gabli M.
This work deals with the singularly perturbed periodical boundary value problem for a quasilinear second-order differential equation. The numerical method is constructed on piecewise uniform Shishkin type mesh, which gives first-order uniform convergence in the discrete maximum norm. Numerical results supporting the theory are presented. (C) 2016 All rights reserved.
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.
Exponentially Fitted Difference Scheme for Singularly Perturbed Mixed Integro-Differential Equations
(Walter de Gruyter Gmbh, 2022) Cakir, Musa; Gunes, Baransel
In this study, singularly perturbed mixed integro-differential equations (SPMIDEs) are taken into account. First, the asymptotic behavior of the solution is investigated. Then, by using interpolating quadrature rules and an exponential basis function, the finite difference scheme is constructed on a uniform mesh. The stability and convergence of the proposed scheme are analyzed in the discrete maximum norm. Some numerical examples are solved, and numerical outcomes are obtained.
Exponentially Fitted Difference Scheme for Singularly Perturbed Mixed Integro-Differential Equations(Vol 29,pg 193, 2022)
(Walter de Gruyter Gmbh, 2023) Cakir, Musa; Gunes, Baransel
A Fitted Approximate Method for Solving Singularly Perturbed Volterra-Fredholm Integrodifferential Equations With Integral Boundary Condition
(Springer, 2024) Gunes, Baransel; Cakir, Musa
We consider a novel numerical approach for solving boundary-value problems for the second-order Volterra-Fredholm integrodifferential equation with layer behavior and an integral boundary condition. A finite-difference scheme is proposed on suitable Shishkin-type mesh to obtain an approximate solution of the presented problem. It is proved that the method is first-order convergent in the discrete maximum norm. Two numerical examples are included to show the efficiency of the method.
A Fitted Operator Finite Difference Approximation for Singularly Perturbed Volterra-Fredholm Integro-Differential Equations
(Mdpi, 2022) Cakir, Musa; Gunes, Baransel
This paper presents a epsilon-uniform and reliable numerical scheme to solve second-order singularly perturbed Volterra-Fredholm integro-differential equations. Some properties of the analytical solution are given, and the finite difference scheme is established on a non-uniform mesh by using interpolating quadrature rules and the linear basis functions. An error analysis is successfully carried out on the Boglaev-Bakhvalov-type mesh. Some numerical experiments are included to authenticate the theoretical findings. In this regard, the main advantage of the suggested method is to yield stable results on layer-adapted meshes.
A Fully Discrete Scheme on Piecewise-Equidistant Mesh for Singularly Perturbed Delay Integro-Differential Equations
(Taylor & Francis Ltd, 2024) Cakir, Musa; Gunes, Baransel
This paper chiefly takes into account the singularly perturbed delay Volterra-Fredholm integro-differential equations by numerically. In this context, firstly, priori estimates are given and a new discretization is constructed on piecewise-equidistant mesh by using interpolating quadrature rules [2] and composite integration formulas. Then, the convergence analysis and stability bounds of the presented method are discussed. Finally, numerical results are demonstrated with two test problems.
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 New Difference Method for the Singularly Perturbed Volterra-Fredholm Integro-Differential Equations on a Shishkin Mesh
(Hacettepe Univ, Fac Sci, 2022) Cakir, Musa; Gunes, Baransel
In this research, the finite difference method is used to solve the initial value problem of linear first order Volterra-Fredholm integro-differential equations with singularity. By using implicit difference rules and composite numerical quadrature rules, the difference scheme is established on a Shishkin mesh. The stability and convergence of the proposed scheme are analyzed and two examples are solved to display the advantages of the presented technique.
A New Numerical Approach for a Singularly Perturbed Problem With Two Integral Boundary Conditions
(Springer Heidelberg, 2021) Cakir, Musa; Arslan, Derya
In this study, finite difference method on a Shishkin mesh is applied to solve the singularly perturbed problem with integral boundary conditions. Some properties of the exact solution are obtained. Finite difference scheme on this mesh is constructed. The stability and convergence analysis of the method are shown as first-order convergent at the discrete maximum norm, regardless of the perturbation parameter e. Numerical results are shown by solving an example on the table and figure.
A New Numerical Scheme for Singularly Perturbed Reaction-Diffusion Problems
(Gazi Univ, 2023) Temel, Zelal; Cakir, Musa
This study is related to a novel numerical technique for solving the singularly perturbed reaction -diffusion boundary value problems. First, explicit boundaries for the solution of the problem are established. Then, a finite difference scheme is established on a uniform mesh supported by the method of integral identities using the remainder term in integral form and the exponential rules with weight. The uniform convergence and stability of these schemes are investigated concerning the perturbation parameter in the discrete maximum norm. At last, the numerical results that provide theoretical results are presented.
A Novel Computational Method for Solving Nonlinear Volterra Integro-Differential Equation
(Academic Publication Council, 2021) Cakir, Musa; Gunes, Baransel; Duru, Hakki
In this paper, we study quasilinear Volterra integro-differential equations (VIDEs). Asymptotic estimates are made for the solution of VIDE. Finite difference scheme, which is accomplished by the method of integral identities using interpolating quadrature rules with weight functions and remainder term in integral form, is presented for the VIDE. Error estimates are carried out according to the discrete maximum norm. It is given an effective quasilinearization technique for solving nonlinear VIDE. The theoretical results are performed on numerical examples.
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 Novel Numerical Technique for Solving Singularly Perturbed Differential Equations With Mixed Boundary Conditions
(Univ Prishtines, 2025) Temel, Zelal; Cakir, Musa
We investigate an innovative numerical technique for a singularly perturbed problem that has an integral boundary condition. To solve the problem, first the boundary values and their derivatives are determined. And then, the difference scheme is constructed on the Shishkin mesh. And also, we analyze the uniform convergence and stability of the scheme with a perturbation parameter. Next, the numerical technique's convergence and stability are discussed and tested. Numerical results verify the theoretical conclusions as well.
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.
A Numerical Comparative Study for the Singularly Perturbed Nonlinear Volterra-Fredholm Integro-Differential Equations on Layer-Adapted Meshes
(Univ Miskolc inst Math, 2024) Gunes, Baransel; Cakir, Musa
This article deals with the singularly perturbed nonlinear Volterra-Fredholm integrodifferential equations. Firstly, some priori bounds are presented. Then, the finite difference scheme is constructed on non-uniform mesh by using interpolating quadrature rules [5] and composite numerical integration formulas. The error estimates are derived in the discrete maximum norm. Finally, theoretical results are performed on two examples and they are compared for both Bakhvalov (B-type) and Shishkin (S-type) meshes.
A Numerical Method for a Singularly Perturbed Three-Point Boundary Value Problem
(Hindawi Ltd, 2010) Cakir, Musa; Amiraliyev, Gabil M.
The purpose of this paper is to present a uniform finite difference method for numerical solution of nonlinear singularly perturbed convection-diffusion problem with nonlocal and third type boundary conditions. The numerical method is constructed on piecewise uniform Shishkin type mesh. The method is shown to be convergent, uniformly in the diffusion parameter epsilon, of first order in the discrete maximum norm. Some numerical experiments illustrate in practice the result of convergence proved theoretically.
A Numerical Method for Solving Linear First-Order Volterra Integro-Differential Equations With Integral Boundary Condition
(Prairie View A & M Univ, dept Mathematics, 2024) Temel, Zelal; Cakir, Musa
We investigate an efficient numerical method for the linear first-order Volterra integro-differential equations with integral boundary condition. To solve this problem, boundaries are determined its derivative and the solution. The numerical solutions of the problem are modeled over a uniform mesh using the composite right-side rectangle concept for the integral component and the implicit difference rules for the differential component. Next, the stability and convergence of the numerical approach are discussed. The numerical experiments are presented confirming the accuracy of proposed scheme.