Browsing by Author "Duru, Hakki"

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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.
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.
On Adaptive Mesh for the Initial Boundary Value Singularly Perturbed Delay Sobolev Problems
(Wiley, 2020) Chiyaneh, Akbar Barati; Duru, Hakki
A uniform finite difference method on a B-mesh is applied to solve the initial-boundary value problem for singularly perturbed delay Sobolev equations. To solve the foresold problem, finite difference scheme on a special nonuniform mesh, whose solution converges point-wise independently of the singular perturbation parameter is constructed and analyzed. The present paper also aims at discussing the stability and convergence analysis of the method. An error analysis shows that the method is of second order convergent in the discrete maximum norm independent of the perturbation parameter. A numerical example and the simulation results show the effectiveness of our theoretical results.
A Parameter-Uniform Numerical Method for a Sobolev Problem With Initial Layer
(Springer, 2007) Amiraliyev, G. M.; Duru, Hakki; Amiraliyeva, I. G.
The present study is concerned with the numerical solution, using finite difference method of a one-dimensional initial-boundary value problem for a linear Sobolev or pseudo-parabolic equation with initial jump. In order to obtain an efficient method, to provide good approximations with independence of the perturbation parameter, we have developed a numerical method which combines a finite difference spatial discretization on uniform mesh and the implicit rule on Shishkin mesh(S-mesh) for the time variable. The fully discrete scheme is shown to be convergent of order two in space and of order one expect for a logarithmic factor in time, uniformly in the singular perturbation parameter. Some numerical results confirming the expected behavior of the method are shown.
A Second-Order Difference Scheme for the Singularly Perturbed Sobolev Problems With Third Type Boundary Conditions on Bakhvalov Mesh
(Taylor & Francis Ltd, 2022) Gunes, B.; Duru, Hakki
We consider a singularly perturbed initial-third boundary value Sobolev problems. Firstly, the asymptotic behaviour of the exact solution is analysed. Then, a second-order finite difference scheme is constructed on the special non-uniform mesh. By using energy estimate, the stability and convergence of the proposed scheme are investigated in the discrete maximum norm. Finally, three numerical examples are solved to validate 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.
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.
Uniform Difference Method for a Parameterized Singular Perturbation Problem
(Elsevier Science inc, 2006) Amiraliyev, G. M.; Kudu, Mustafa; Duru, Hakki
We consider a uniform finite difference method on a B-mesh is applied to solve a singularly perturbed quasilinear boundary value problem (BVP) depending on a parameter. We give a uniform first-order error estimates in a discrete maximum norm. Numerical results are presented that demonstrate the sharpness of our theoretical analysis. (c) 2005 Elsevier Inc. All rights reserved.