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In this paper, a linear singularly perturbed Fredholm integro-differential initial value problem with integral condition is being considered. On a Shishkin-type mesh, a fitted finite difference approach is applied using a composite trapezoidal rule in both; in the integral part of equation and in the initial condition. The proposed technique acquires a uniform second-order convergence in respect to perturbation parameter. Further provided the numerical results to support the theoretical estimates.

This paper deals with the study a finite-difference approximation of the one dimensional initial-boundary value problem for a pseudo-parabolic equation containing time delay in second derivative. We propose three layer difference scheme and obtain the error estimates for its solution. Based on the method of energy estimates the fully discrete scheme is shown to be convergent of order four in space and order two in time. Numerical results are presented to illustrate the theoretical findings. (C) 2021 Elsevier B.V. All rights reserved. WOS:000697029800006 2-s2.0-85114402983

In this paper stability inequalities for the linear non homogeneous Volterra delay integro-differential equation (VDIDE) is being established. Test example is encountered to show the applicability of the method and to confirm the predicted theoretical analysis.

Yapman, Omer/0000-0003-3117-2932 WOS: 000463302400022 In this paper, the initial value problem for a quasilinear singularly perturbed delay Volterra integro-differential equation was considered. By the method of integral identities with the use of exponential basis functions and interpolating quadrature rules with the weight and remainder term in integral form, a fitted difference scheme is constructed and analysed. It is shown that the method displays first order uniform convergence in perturbation parameter. Some numerical results are given to confirm the theoretical analysis. (C) 2019 Elsevier B.V. All rights reserved.

In this paper, the homogeneous type fitted difference scheme for solving singularly perturbed problem depending on a parameter with integral boundary condition is proposed. We prove that the method is $$O(N^{-2}\ln N)$$ uniform convergent on Shishkin meshes. Numerical results are also presented.

In this paper, a singularly perturbed Volterra integro- differential equation is being surveyed. On a piecewise-uniform Shishkin mesh, a fitted mesh finite difference approach is applied using a composite trapezoidal rule in the case of integral component and a finite difference operator for the derivative component. The proposed technique acquires a uniform convergence in accordance with the perturbation parameter. To improve the accuracy of the computed solution, an extrapolation, specifically Richardson extrapolation, is used measured in the discrete maximum norm and almost second-order convergence is attained. Further numerical results are provided to assist the theoretical estimates. WOS:000697079500001 2-s2.0-85115147823

In this paper, we consider a class of parameterized singularly perturbed problems with integral boundary condition. A finite difference scheme of hybrid type with an appropriate Shishkin mesh is suggested to solve the problem. We prove that the method is of almost second order convergent in the discrete maximum norm. Numerical results are presented, which illustrate the theoretical results. © 2021 Elsevier B.V. 2-s2.0-85119251175

Cimen, Erkan/0000-0002-7258-192X WOS: 000400878000001 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.

KUDU, Mustafa/0000-0002-6610-0587 WOS: 000381546600025 We consider the singularly perturbed initial value problem for a linear first order Volterra integro-differential equation with delay. Our purpose is to construct and analyse a numerical method with uniform convergence in the perturbation parameter. The numerical solution of this problem is discretized using implicit difference rules for differential part and the composite numerical quadrature rules for integral part. On a layer adapted mesh error estimations for the approximate solution are established. Numerical examples supporting the theory are presented. (C) 2016 Elsevier B.V. All rights reserved.

This paper deals with the initial-boundary value problem for linear pseudo-parabolic equation. Using the method of energy estimates the stability bounds obtained for the considered problem. Illustrative example is also presented. © 2019 Academic Publications.

WOS: 000504461100008 In this paper, the author considers the one dimensional initial-boundary problem for a pseudo-parabolic equation with time delay in second spatial derivative. To solve this problem numerically, the author constructs higher order difference method and obtain the error estimate for its solution. Based on the method of energy estimates the fully discrete scheme is shown to be convergent of order four in space and of order two in time. Some numerical examples illustrate the convergence and effectiveness of the numerical method. Scientific and Technological Research Council of Turkey (TUBITAK)Turkiye Bilimsel ve Teknolojik Arastirma Kurumu (TUBITAK) [1059B191700821] This research was fully supported by Scientific and Technological Research Council of Turkey (TUBITAK) [grant number 1059B191700821]. The author would like to thank the Department of Mathematics of the University of Oklahoma for the hospitality during her work on the project, Nikola Petrov and Murad Ozaydin who were instrumental in arranging her stay in Oklahoma. Ilhame Amirali thanks Nikola Petrov for stimulating discussions.

In this paper, we consider a parameterized singularly perturbed second order quasilinear boundary value problem. Asymptotic estimates for the solution and its first and second derivatives have been established. The theoretical estimates have been justified by concrete example.

The method of lines is applied to the boundary-value problem for third order partial differential equation. Explicit expression and order of convergence for the approximate solution are obtained.

This work deals with the numerical solution of singular perturbation system of ordinary differential equations with boundary layer. For the numerical solution of this problem fitted finite difference scheme on a uniform mesh is constructed and analyzed. The uniform error estimates for the approximate solution are obtained.

WOS: 000391119500013 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 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. 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.

International Conference on Differential and Difference Equations and Applications (ICDDEA) -- MAY 18-22, 2015 -- Mil Acad, Amadora, PORTUGAL WOS: 000391876600016 This work deals with the one-dimensional initial-boundary Sobolev or pseudo-parabolic problem with delay. For solving this problem numerically, we construct fourth-order difference-differential scheme and obtain the error estimate for its solution. Further, for the time variable, we use the appropriate Runge-Kutta method for the realization of our differential-difference problem. Numerical results supporting the theory are presented.

WOS: 000441460300026 In this paper, a parameter-uniform numerical method for a parameterized singularly perturbed ordinary differential equation containing integral boundary condition is studied. Asymptotic estimates on the solution and its derivatives are derived. A numerical algorithm based on upwind finite difference operator and an appropriate piecewise uniform mesh is constructed. Parameter-uniform error estimate for the numerical solution is established. Numerical results are presented, which illustrate the theoretical results.