Browsing by Author "Cakir, Firat"

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    Article
    A Numerical Method on Bakhvalov Shishkin Mesh for Volterra Integro-Differential Equations With A Boundary Layer
    (Ankara Univ, Fac Sci, 2022) Guckir Cakir, Hayriye; Cakir, Firat; Cakir, Musa
    We construct a finite difference scheme for a first-order linear singularly perturbed Volterra integro-differential equation (SPVIDE) on Bakhvalov-Shishkin mesh. For the discretization of the problem, we use the integral identities and deal with the emerging integrals terms with interpolating quadrature rules which also yields remaining terms. The stability bound and the error estimates of the approximate solution are established. Further, we demonstrate that the scheme on Bakhvalov-Shishkin mesh is O(N-1) uniformly convergent, where N is the mesh parameter. The numerical results are also provided for a couple of examples.
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    Article
    A Robust Numerical Technique for Solving Non-Linear Volterra Integro-Differential Equations With Boundary Layer
    (Korean Mathematical Soc, 2022) Cakir, Firat; Cakir, Musa; Cakir, Hayriye Guckir
    In this paper, we study a first-order non-linear singularly per-turbed Volterra integro-differential equation (SPVIDE). We discretize the problem by a uniform difference scheme on a Bakhvalov-Shishkin mesh. The scheme is constructed by the method of integral identities with expo-nential basis functions and integral terms are handled with interpolating quadrature rules with remainder terms. An effective quasi-linearization technique is employed for the algorithm. We establish the error estimates and demonstrate that the scheme on Bakhvalov-Shishkin mesh is O(N-1) uniformly convergent, where N is the mesh parameter. The numerical re-sults on a couple of examples are also provided to confirm the theoretical analysis.