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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.
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    A Second-Order Post-processing Technique for Singularly Perturbed Volterra Integro-differential Equations
    (Springer Basel Ag, 2021) Panda, Abhilipsa; Mohapatra, Jugal; Amirali, Ilhame
    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.