Browsing by Author "Gunes, Baransel"

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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
An Efficient Numerical Scheme on Adaptive Mesh for Solving Singularly Perturbed Quasilinear Boundary Value Problems
(Univ Simon Bolivar, 2024) Duru, Hakki; Demirbas, Mutlu; Gunes, Baransel
This paper investigates the singularly perturbed quasilinear boundary value problem by numerically. Initially, some features of the analytical solution of the presented problem are given. Then, by using quasilinearization technique and interpolating quadrature formulas, the finite difference scheme is constructed on Bakhvalov-type mesh. The convergence estimations of the numerical scheme are provided and three examples are solved to demonstrate the efficiency of the suggested method. The main contribution of this paper is to ensure a uniform finite difference scheme for quasilinear problems with layer behavior.
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
The Finite Difference Method on Adaptive Mesh for Singularly Perturbed Nonlinear 1d Reaction Diffusion Boundary Value Problems
(Czestochowa Univ Technology, inst Mathematics, 2020) Duru, Hakla; Gunes, Baransel
In this paper, we study singularly perturbed nonlinear reaction-diffusion equations. The asymptotic behavior of the solution is examined. The difference scheme which is accomplished by the method of integral identities with using of interpolation quadrature rules with weight functions and remainder term integral form is established on adaptive mesh. Uniform convergence and stability of the difference method are discussed in the discrete maximum norm. The discrete scheme shows that orders of convergent rates are close to 2. An algorithm is presented, and some problems are solved to validate the theoretical results.Y
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.
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 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 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.
Parçalı Düzgün Şebekede Singüler Pertürbe Özellikli Lineer Olmayan Reaksiyon Difüzyon Problemleri İçin Nümerik Çözümler
(2023) Duru, Hakkı; Gunes, Baransel
Bu çalışmada singüler pertürbe özellikli lineer olmayan reaksiyon-difüzyon sınır değer problemi ele alınmıştır. Kalan terimi integral biçiminde olan ve baz fonksiyonu içeren interpolasyon kuadratür kuralları kullanılarak parçalı düzgün şebeke üzerinde fark şeması kurulmuştur. Sunulan metodun kararlı olduğu gösterilmiş ve yakınsaklık analizi yapılmıştır. Kurulan metodun yaklaşık çözüme düzgün yakınsadığı gösterilmiştir. Nümerik sonuçların teorik sonuçları desteklediği örnek üzerinde gösterilmiştir.
A Reliable Numerical Method for the Singularly Perturbed Nonlinear Differential Equation With an Integral Boundary Condition
(Ivane Javakhishvili Tbilisi State Univ, 2024) Cakir, Musa; Gurbuz, Bahar; Gunes, Baransel
This study purposes to present an efficient numerical method for the singularly perturbed nonlinear problems involving an integral boundary condition. Initially, some properties are given for the continuous problem. Then, using interpolating quadrature formulas [3], the finite difference scheme is established on the Bakhvalov-Shishkin mesh (B-S mesh). The error approximations of the suggested scheme are examined in the discrete maximum norm. Finally, some numerical examples are included to confirm the theory.
A Second-Order Numerical Method for Pseudo-Parabolic Equations Having Both Layer Behavior and Delay Parameter
(Ankara Univ, Fac Sci, 2024) Gunes, Baransel; Duru, Hakki
In this paper, singularly perturbed pseudo-parabolic initial-boundary value problems with time-delay parameter are considered by numerically. Initially, the asymptotic properties of the analytical solution are investigated. Then, a discretization with exponential coefficient is suggested on a uniform mesh. The error approximations and uniform convergence of the presented method are estimated in the discrete energy norm. Finally, some numerical experiments are given to clarify the theory.
Singüler Pertürbe Özellikli Konveksiyon Difüzyon Problemleri İçin Çoklu Ölçekler Metodu ve Sonlu Fark Metodunun Karşılaştırılması
(2020) Gunes, Baransel; Chianeh, Afshin Barati; Demirbaş, Mutlu
Bu çalışmada singüler pertürbe özellikli konveksiyon difüzyon problemi için çoklu ölçekler metodu tanıtılmıştır. Bu bağlamda, söz konusu problem kısmi diferansiyel denklemlere dönüştürülmüştür. Ayrıca ağırlık fonksiyonu içeren ve kalan terimi integral biçiminde olan interpolasyon kuadratür kuralları ve lineer baz fonksiyonlarının kullanımı ile üstel katsayılı fark şeması kurulmuştur. Teorik sonuçları doğrulamak için bazı nümerik çalışmalara yer verilmiştir. Bu makalenin temel amacı, singüler pertürbe özellikli konveksiyon-difüzyon problemleri için çoklu ölçekler metodu ile sonlu fark metodunu karşılaştırmaktır.
The Stability and Convergence Analysis for Singularly Perturbed Sobolev Problems With Robin Type Boundary Condition
(Walter de Gruyter Gmbh, 2023) Duru, Hakki; Gunes, Baransel
This paper presents the robust and stable difference scheme to estimate singularly perturbed Sobolev boundary value problems with Robin type boundary condition. Firstly, the asymptotic behavior of the solution is analyzed. By using interpolating quadrature rules and basis functions, a completely exponentially fitted tree-level difference scheme is constructed on the uniform mesh. Then an error estimation is investigated in a discrete energy norm. Two numerical examples are solved and the computational results are tabulated.
A Uniform Discretization for Solving Singularly Perturbed Convection-Diffusion Boundary Value Problems
(Georgian Natl Acad Sciences, 2022) Gunes, Baransel; Demirbas, Mutlu
In this paper, a discrete scheme is presented for solving singularly perturbed convection-diffusion equations. The stability and convergence of the proposed scheme are analyzed in the discrete maximum norm. Error estimates are carried out for both Bakhvalov (B-mesh) and Shishkin-type (S-mesh) meshes. Three numerical examples are solved to authenticate the theoretical findings.
A Uniformly Convergent Numerical Method for Singularly Perturbed Semilinear Integro-Differential Equations With Two Integral Boundary Conditions
(Pleiades Publishing inc, 2023) Gunes, Baransel; Cakir, Musa
This paper purposes to present a new discrete scheme for the singularly perturbed semilinear Volterra-Fredholm integro-differential equation including two integral boundary conditions. Initially, some analytical properties of the solution are given. Then, using the composite numerical integration formulas and implicit difference rules, the finite difference scheme is established on a uniform mesh. Error approximations for the approximate solution and stability bounds are investigated in the discrete maximum norm. Finally, a numerical example is solved to show epsilon-uniform convergence of the suggested difference scheme.