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An Initial Value Technique for Singularly Perturbed Convection–Diffusion Problems with a Negative Shift

Author

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  • V. Subburayan

    (Bharathidasan University)

  • N. Ramanujam

    (Bharathidasan University)

Abstract

In this paper, a numerical method named as Initial Value Technique (IVT) is suggested to solve the singularly perturbed boundary value problem for the second order ordinary differential equations of convection–diffusion type with a delay (negative shift). In this technique, the original problem of solving the second order equation is reduced to solving two first order differential equations, one of which is singularly perturbed without delay and other one is regular with a delay term. The singularly perturbed problem is solved by the second order hybrid finite difference scheme, whereas the delay problem is solved by the fourth order Runge–Kutta method with Hermite interpolation. An error estimate is derived by using the supremum norm. Numerical results are provided to illustrate the theoretical results.

Suggested Citation

  • V. Subburayan & N. Ramanujam, 2013. "An Initial Value Technique for Singularly Perturbed Convection–Diffusion Problems with a Negative Shift," Journal of Optimization Theory and Applications, Springer, vol. 158(1), pages 234-250, July.
    • Handle: RePEc:spr:joptap:v:158:y:2013:i:1:d:10.1007_s10957-012-0200-9
    DOI: 10.1007/s10957-012-0200-9
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    References listed on IDEAS

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    1. T. Valanarasu & N. Ramanujam, 2007. "Asymptotic Initial-Value Method for Second-Order Singular Perturbation Problems of Reaction-Diffusion Type with Discontinuous Source Term," Journal of Optimization Theory and Applications, Springer, vol. 133(3), pages 371-383, June.
    2. Bellen, Alfredo & Zennaro, Marino, 2003. "Numerical Methods for Delay Differential Equations," OUP Catalogue, Oxford University Press, number 9780198506546.
    3. V.Y. Glizer, 2003. "Asymptotic Analysis and Solution of a Finite-Horizon H ∞ Control Problem for Singularly-Perturbed Linear Systems with Small State Delay," Journal of Optimization Theory and Applications, Springer, vol. 117(2), pages 295-325, May.
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    Cited by:

    1. V. Subburayan & S. Natesan, 2022. "Parameter Uniform Numerical Method for Singularly Perturbed 2D Parabolic PDE with Shift in Space," Mathematics, MDPI, vol. 10(18), pages 1-19, September.
    2. Chen, Shu-Bo & Soradi-Zeid, Samaneh & Dutta, Hemen & Mesrizadeh, Mehdi & Jahanshahi, Hadi & Chu, Yu-Ming, 2021. "Reproducing kernel Hilbert space method for nonlinear second order singularly perturbed boundary value problems with time-delay," Chaos, Solitons & Fractals, Elsevier, vol. 144(C).
    3. 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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