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A Second-Order Post-processing Technique for Singularly Perturbed Volterra Integro-differential Equations

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Abstract

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.

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Abbreviations

v(t):

Solution of model problem

\(\upsilon _0\) :

Initial value of the problem

\(\varepsilon \) :

Small parameter

\(\tau \) :

Transition parameter used in Shishkin mesh

\(\Omega _N\) :

Domain with N points

\(\Omega _{2N}\) :

Domain with 2N points

\(R_i,\;R_{1,i},\; R_{2,i},\; R_{3,i}\) :

Remainder terms from the discrete scheme

\(y(t), y^{N}(t)\) :

Solution of discrete problem on \(\Omega _N\)

\(y^{2N}(t)\) :

Solution of discrete problem on \(\Omega _{2N}\)

\(y_{extp}(t)\) :

Extrapolation formula on \(\Omega ^N\)

\(E_{\varepsilon }^N\) :

Pointwise error formula

\(p_\varepsilon ^N\) :

Rate of convergence

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Authors and Affiliations

  1. Department of Mathematics, National Institute of Technology Rourkela, Rourkela, 769008, India

    Abhilipsa Panda & Jugal Mohapatra

  2. Department of Mathematics, Faculty of Arts and Sciences, Düzce University, 81620, Düzce, Turkey

    Ilhame Amirali

Authors
  1. Abhilipsa Panda
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  2. Jugal Mohapatra
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  3. Ilhame Amirali
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Correspondence to Ilhame Amirali.

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Panda, A., Mohapatra, J. & Amirali, I. A Second-Order Post-processing Technique for Singularly Perturbed Volterra Integro-differential Equations. Mediterr. J. Math. 18, 231 (2021). https://doi.org/10.1007/s00009-021-01873-8

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  • DOI: https://doi.org/10.1007/s00009-021-01873-8

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