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Öğe An efficient numerical method for a singularly perturbed Fredholm integro-differential equation with integral boundary condition(Springer Heidelberg, 2023) Durmaz, Muhammet Enes; Amirali, Ilhame; Amiraliyev, Gabil M.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.Öğe A Fitted Second-Order Difference Method for a Parameterized Problem with Integral Boundary Condition Exhibiting Initial Layer(Springer Basel Ag, 2021) Kudu, Mustafa; Amirali, Ilhame; Amiraliyev, Gabil M.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)lnN) uniform convergent on Shishkin meshes. Numerical results are also presented.Öğe A new survey to the nonlinear electrical transmission line model(KeAi Communications Co., 2021) Özer, Özen; Başkonuş, Hacı Mehmet; Bulut, Hasan; Amirali, Ilhame; Yel, GülnurIn this paper, we find new travelling wave solutions to the nonlinear electrical transmission line model. Exponential function method is applied. This model is used to the voltage behaviors in electrical transmission lines. Under a suitable choice of control parameters, we plot the figures along with contour surfaces of new solutions by using computational programs such as Mathematica, Maple and Matlab. Some information about the behavior of voltage on the electrical lines is extracted. The hyperbolic, periodic or singular properties are reported. © 2021Öğe A novel approach for the stability inequalities for high-order Volterra delay integro-differential equation(Springer Heidelberg, 2023) Amirali, Ilhame; Acar, HülyaHigh-order linear Volterra delay integro-differential equations are examined in the present paper. Proposed approach, which will be provided for solving high-order linear Volterra delay integro-differential equations, expresses certain key elements of determining the equations' stability bounds and exact solutions. Furthermore, stability inequalities can be generated for each order of derivative using the proposed method.Öğe Numerical solution of linear pseudo-parabolic equation with time delay using three layer difference method(Elsevier, 2024) Amirali, Ilhame; Amiraliyev, Gabil M.In this paper, we consider the one-dimensional initial-boundary problem for a pseudoparabolic equation with a time delay in the second spatial derivative. To solve this problem numerically, we construct a 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. Given numerical results illustrate the convergence and effectiveness of the numerical method. & COPY; 2023 Elsevier B.V. All rights reserved.Öğe Numerical solution of singularly perturbed Fredholm integro-differential equations by homogeneous second order difference method(Elsevier, 2022) Durmaz, Muhammet Enes; Çakır, 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.Öğe A numerical technique for solving nonlinear singularly perturbed Fredholm integro-differential equations(Elsevier, 2024) Panda, Abhilipsa; Mohapatra, Jugal; Amirali, Ilhame; Durmaz, Muhammet Enes; Amiraliyev, Gabil M.This study deals with two numerical schemes for solving a class of singularly perturbed nonlinear Fredholm integro-differential equations. The nonlinear terms are linearized using the quasi -linearization technique. On the layer adapted Shishkin mesh, the numerical solution is initially calculated using the finite difference scheme for the differential part and quadrature rule for the integral part. The method proves to be first -order convergent in the discrete maximum norm. Then, using a post -processing technique we significantly enhance the accuracy from first order to second order. Further, a hybrid scheme on the nonuniform mesh is also constructed and analyzed whose solution converges uniformly, independent of the perturbation parameter and directly gives second order accuracy. Parameter uniform error estimates are demonstrated and the theoretical results are validated through some numerical tests.Öğe On the second-order neutral Volterra integro-differential equation and its numerical solution(Elsevier Science Inc, 2024) Amirali, Ilhame; Fedakar, Burcu; Amiraliyev, Gabil M.In this paper, we consider an initial-value problem for a second-order neutral Volterra integrodifferential equation. First, we give the stability inequality indicating the stability of the problem with respect to the right -side and initial conditions. Further, we develop a finite difference method that uses for differential part second difference derivative, for the integral part appropriate composite trapezoidal and midpoint rectangle rules followed by second-order accurate difference quantities at intermediate points. Error estimate for the approximate solution is established, which shows the second-order accuracy. Finally, the numerical experiments are presented confirming the accuracy of the proposed scheme.Öğe A second-order numerical approximation of a singularly perturbed nonlinear Fredholm integro-differential equation(Elsevier, 2023) Durmaz, Muhammet Enes; Amirali, Ilhame; Mohapatra, Jugal; Amiraliyev, Gabil M.We consider a singularly perturbed problem for a nonlinear first-order Fredholm integrodifferential equation. Our aim is to build and analyze a numerical approach with uniform convergence in the epsilon-parameter. The numerical solution of problem is discretized utilizing interpolation quadrature formulas for the differential part and the composite rectangular rule for the integral part. Error estimations for the approximate solution are established. In support of the idea, numerical examples are provided.(c) 2023 IMACS. Published by Elsevier B.V. All rights reserved.Öğe Second-order numerical method for a neutral Volterra integro-differential equation(Elsevier, 2025) Amirali, Ilhame; Fedakar, Burcu; Amiraliyev, Gabil M.This paper is dedicated to obtaining an approximate solution for a neutral second-order Volterra integro-differential equation. Our method is the second-order accurate finite difference scheme on a uniform mesh. The error analysis is carried out and numerical results are given to support the proposed approach.Öğe A Second-Order Post-processing Technique for Singularly Perturbed Volterra Integro-differential Equations(Springer Basel Ag, 2021) Panda, Abhilipsa; Mohapatra, Jugal; Amirali, IlhameIn 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.Öğe Stability inequalities and numerical solution for neutral Volterra delay integro-differential equation(Elsevier, 2024) Amirali, Ilhame; Acar, HulyaThe aim of the present paper is to introduce a new numerical method for solving Volterra integro-differential equation involving delay and neutral types. Using the numerical quadrature formula we form a finite difference scheme on a uniform mesh. The presented numerical method obtains a second-order convergence in discrete maximum norm. Furthermore, we illustrate the efficacy of the proposed method by constructing examples. & COPY; 2023 Elsevier B.V. All rights reserved.Öğe Stability Properties for the Delay Integro-Differential Equation(Gazi Univ, 2023) Amirali, IlhameIn this paper stability inequalities for the linear nonhomogeneous Volterra delay integro-differential equation (VDIDE) is being established. The particular problems are encountered to show the applicability of the method and to confirm the predicted theoretical analysis.Öğe Three layer difference method for linear pseudo-parabolic equation with delay(Elsevier, 2022) Amirali, Ilhame; Amiraliyev, Gabil M.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.
