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Numerical Treatment on Parabolic Singularly Perturbed Differential Difference Equation via Fitted Operator Scheme

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  • Dagnachew Mengstie Tefera
  • Awoke Andargie Tiruneh
  • Getachew Adamu Derese

Abstract

This paper proposes a new fitted operator strategy for solving singularly perturbed parabolic partial differential equation with delay on the spatial variable. We decomposed the problem into three piecewise equations. The delay term in the equation is expanded by Taylor series, the time variable is discretized by implicit Euler method, and the space variable is discretized by central difference methods. After developing the fitting operator method, we accelerate the order of convergence of the time direction using Richardson extrapolation scheme and obtained O(h2 + k2) uniform order of convergence. Finally, three examples are given to illustrate the effectiveness of the method. The result shows the proposed method is more accurate than some of the methods that exist in the literature.

Suggested Citation

  • Dagnachew Mengstie Tefera & Awoke Andargie Tiruneh & Getachew Adamu Derese, 2021. "Numerical Treatment on Parabolic Singularly Perturbed Differential Difference Equation via Fitted Operator Scheme," Abstract and Applied Analysis, John Wiley & Sons, vol. 2021(1).
    • Handle: RePEc:wly:jnlaaa:v:2021:y:2021:i:1:n:1661661
    DOI: 10.1155/2021/1661661
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    References listed on IDEAS

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    1. Mesfin Mekuria Woldaregay & Gemechis File Duressa, 2021. "Uniformly Convergent Hybrid Numerical Method for Singularly Perturbed Delay Convection-Diffusion Problems," International Journal of Differential Equations, Hindawi, vol. 2021, pages 1-20, September.
    2. Singh, Maneesh Kumar & Natesan, Srinivasan, 2018. "Richardson extrapolation technique for singularly perturbed system of parabolic partial differential equations with exponential boundary layers," Applied Mathematics and Computation, Elsevier, vol. 333(C), pages 254-275.
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    Cited by:

    1. Amare Worku Demsie & Awoke Andargie Tiruneh & Endalew Getnet Tsega, 2025. "A Fitted Linear Multistep Approach for Singularly Perturbed Parabolic‐Type Reaction–Diffusion Problems Using Shishkin Meshes," International Journal of Mathematics and Mathematical Sciences, John Wiley & Sons, vol. 2025(1).
    2. Amare Worku Demsie & Awoke Andargie Tiruneh & Endalew Getnet Tsega, 2025. "A Uniformly Convergent Scheme for Singularly Perturbed Unsteady Reaction–Diffusion Problems," Journal of Applied Mathematics, John Wiley & Sons, vol. 2025(1).

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