Browsing by Author "Amiraliyev, G. M."

Now showing 1 - 9 of 9

Difference Schemes for the Singularly Perturbed Sobolev Equations
(World Scientific Publ Co Pte Ltd, 2007) Amiraliyev, G. M.; 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 quasilinear Sobolev or pseudo-parabolic equation with initial jump. We have derived a method based on using finite elements with piecewise linear functions in space and with exponential functions in time and appropriate quadrature formulae with remainder term in integral form. In the initial layer, we introduce a special non-uniform mesh which is constructed by using estimates of derivatives of the exact solution and the analysis of the local truncation error. For the time integration we use the implicit rule. The fully discrete scheme is shown to be convergent of order 2 in space and of order one in time, uniformly in the singular perturbation parameter. Numerical results supporting the theory are presented.
Explicit Finite Difference Methods for the Delay Pseudoparabolic Equations
(Hindawi Publishing Corporation, 2014) Amirali, I.; Amiraliyev, G. M.; Cakir, M.; Cimen, E.
Finite difference technique is applied to numerical solution of the initial-boundary value problem for the semilinear delay Sobolev or pseudoparabolic equation. By the method of integral identities two-level difference scheme is constructed. For the time integration the implicit rule is being used. Based on the method of energy estimates the fully discrete scheme is shown to be absolutely stable and convergent of order two in space and of order one in time. The error estimates are obtained in the discrete norm. Some numerical results confirming the expected behavior of the method are shown.
Numerical Solution of a Singularly Perturbed Three-Point Boundary Value Problem
(Taylor & Francis Ltd, 2007) Cakir, M.; Amiraliyev, G. M.
We consider a uniform finite difference method on an S-mesh (Shishkin type mesh) for a singularly perturbed semilinear one-dimensional convection-diffusion three-point boundary value problem with zeroth-order reduced equation. We show that the method is first-order convergent in the discrete maximum norm, independently of the perturbation parameter except for a logarithmic factor. An effective iterative algorithm for solving the non-linear difference problem and some numerical results are presented.
A Numerical Treatment for Singularly Perturbed Differential Equations With Integral Boundary Condition
(Elsevier Science inc, 2007) Amiraliyev, G. M.; Amiraliyeva, I. G.; Kudu, Mustafa
We consider a uniform finite difference method on Shishkin mesh for a quasilinear first order singularly perturbed boundary value problem (BVP) with integral boundary condition. We prove 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. (c) 2006 Elsevier Inc. All rights reserved.
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.
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.
Uniform Difference Method for Singularly Perturbed Volterra Integro-Differential Equations
(Elsevier Science inc, 2006) Amiraliyev, G. M.; Sevgin, Sebaheddin
Singularly perturbed Volterra integro-differential equations is considered. An exponentially fitted difference scheme is constructed in a uniform mesh which gives first order uniform convergence in the discrete maximum norm. Numerical experiments support the theoretical results. (c) 2006 Elsevier Inc. All rights reserved.
A Uniform Numerical Method for Dealing With a Singularly Perturbed Delay Initial Value Problem
(Pergamon-elsevier Science Ltd, 2010) Amiraliyeva, I. G.; Erdogan, F.; Amiraliyev, G. M.
This work deals with a singularly perturbed initial value problem fora quasi-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. Numerical results are also presented. (C) 2010 Elsevier Ltd. All rights reserved.
Uniform Numerical Method for Singularly Perturbed Delay Differential Equations
(Pergamon-elsevier Science Ltd, 2007) Amiraliyev, G. M.; Erdogan, F.
This paper deals with the singularly perturbed initial value problem for a linear first-order delay differential equation. A numerical method is constructed for this problem which involves an appropriate piecewise-uniform mesh on each time subinterval. The difference scheme is shown to converge to the continuous solution uniformly with respect to the perturbation parameter. Numerical results are presented. (c) 2007 Elsevier Ltd. All rights reserved.