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Uniformly Convergent Numerical Scheme for Singularly Perturbed Parabolic PDEs with Shift Parameters

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  • Mesfin Mekuria Woldaregay
  • Gemechis File Duressa

Abstract

In this article, singularly perturbed parabolic differential difference equations are considered. The solution of the equations exhibits a boundary layer on the right side of the spatial domain. The terms containing the advance and delay parameters are approximated using Taylor series approximation. The resulting singularly perturbed parabolic PDEs are solved using the Crank–Nicolson method in the temporal discretization and nonstandard finite difference method in the spatial discretization. The existence of a unique discrete solution is guaranteed using the discrete maximum principle. The uniform stability of the scheme is investigated using solution bound. The uniform convergence of the scheme is discussed and proved. The scheme converges uniformly with the order of convergence , where is number of subintervals in spatial discretization and is mesh length in temporal discretization. Two test numerical examples are considered to validate the theoretical findings of the scheme.

Suggested Citation

  • Mesfin Mekuria Woldaregay & Gemechis File Duressa, 2021. "Uniformly Convergent Numerical Scheme for Singularly Perturbed Parabolic PDEs with Shift Parameters," Mathematical Problems in Engineering, Hindawi, vol. 2021, pages 1-15, November.
    • Handle: RePEc:hin:jnlmpe:6637661
    DOI: 10.1155/2021/6637661
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    Cited by:

    1. Gemechis File Duressa & Imiru Takele Daba & Chernet Tuge Deressa, 2023. "A Systematic Review on the Solution Methodology of Singularly Perturbed Differential Difference Equations," Mathematics, MDPI, vol. 11(5), pages 1-16, February.

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