Browsing by Author "Amiraliyev, Gabil M."

Now showing 1 - 12 of 12

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.
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.
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.
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.
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.
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.
Numerical Solution of Singularly Perturbed Fredholm Integro-Differential Equations by Homogeneous Second Order Difference Method
(Elsevier, 2022) Durmaz, Muhammet Enes; Cakir, Musa; Amirali, Ilhame; Amiraliyev, Gabil M.
This work presents a computational approximate to solve singularly perturbed Fredholm integro-differential equation with the reduced second type Fredholm equation. This problem is discretized by a finite difference approximate, which generates second-order uniformly convergent numerical approximations to the solution. Parameter-uniform approximations are generated using Shishkin type meshes. The performance of the numerical scheme is tested which supports the effectiveness of the technique. (c) 2022 Elsevier B.V. All rights reserved.
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.
A Second Order Numerical Method for Singularly Perturbed Problem With Non-Local Boundary Condition
(Springer Heidelberg, 2021) Cakir, Musa; Amiraliyev, Gabil M.
The aim of this paper is to present a monotone numerical method on uniform mesh for solving singularly perturbed three-point reaction-diffusion boundary value problems. Firstly, properties of the exact solution are analyzed. Difference schemes are established by the method of the integral identities with the appropriate quadrature rules with remainder terms in integral form. It is then proved that the method is second-order uniformly convergent with respect to singular perturbation parameter, in discrete maximum norm. Finally, one numerical example is presented to demonstrate the efficiency of the proposed method.
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.