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    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.
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    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.
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    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.
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    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.