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A Systematic Review on the Solution Methodology of Singularly Perturbed Differential Difference Equations

Author

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  • Gemechis File Duressa

    (Department of Mathematics, Jimma University, Jimma P.O. Box 378, Ethiopia
    These authors contributed equally to this work.)

  • Imiru Takele Daba

    (Department of Mathematics, Dilla University, Dilla P.O. Box 419, Ethiopia
    These authors contributed equally to this work.)

  • Chernet Tuge Deressa

    (Department of Mathematics, Jimma University, Jimma P.O. Box 378, Ethiopia
    These authors contributed equally to this work.)

Abstract

This review paper contains computational methods or solution methodologies for singularly perturbed differential difference equations with negative and/or positive shifts in a spatial variable. This survey limits its coverage to singular perturbation equations arising in the modeling of neuronal activity and the methods developed by numerous researchers between 2012 and 2022. The review covered singularly perturbed ordinary delay differential equations with small or large negative shift(s), singularly perturbed ordinary differential–differential equations with mixed shift(s), singularly perturbed delay partial differential equations with small or large negative shift(s) and singularly perturbed partial differential–difference equations of the mixed type. The main aim of this review is to find out what numerical and asymptotic methods were developed in the last ten years to solve such problems. Further, it aims to stimulate researchers to develop new robust methods for solving families of the problems under consideration.

Suggested Citation

  • 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.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:5:p:1108-:d:1077377
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    References listed on IDEAS

    as
    1. Ababi Hailu Ejere & Gemechis File Duressa & Mesfin Mekuria Woldaregay & Tekle Gemechu Dinka & Stanislaw Migorski, 2022. "An Exponentially Fitted Numerical Scheme via Domain Decomposition for Solving Singularly Perturbed Differential Equations with Large Negative Shift," Journal of Mathematics, Hindawi, vol. 2022, pages 1-13, June.
    2. Wondwosen Gebeyaw Melesse & Awoke Andargie Tiruneh & Getachew Adamu Derese, 2019. "Solving Linear Second-Order Singularly Perturbed Differential Difference Equations via Initial Value Method," International Journal of Differential Equations, Hindawi, vol. 2019, pages 1-10, November.
    3. 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.
    4. Daba, Imiru Takele & Duressa, Gemechis File, 2022. "Collocation method using artificial viscosity for time dependent singularly perturbed differential–difference equations," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 192(C), pages 201-220.
    5. M.K. Kadalbajoo & K.K. Sharma, 2002. "Numerical Analysis of Boundary-Value Problems for Singularly-Perturbed Differential-Difference Equations with Small Shifts of Mixed Type," Journal of Optimization Theory and Applications, Springer, vol. 115(1), pages 145-163, October.
    6. Habtamu Garoma Debela & Solomon Bati Kejela & Ayana Deressa Negassa, 2020. "Exponentially Fitted Numerical Method for Singularly Perturbed Differential-Difference Equations," International Journal of Differential Equations, Hindawi, vol. 2020, pages 1-13, June.
    7. V. Y. Glizer, 2000. "Asymptotic Solution of a Boundary-Value Problem for Linear Singularly-Perturbed Functional Differential Equations Arising in Optimal Control Theory," Journal of Optimization Theory and Applications, Springer, vol. 106(2), pages 309-335, August.
    8. 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.
    9. Mesfin Mekuria Woldaregay & Gemechis File Duressa, 2020. "Higher-Order Uniformly Convergent Numerical Scheme for Singularly Perturbed Differential Difference Equations with Mixed Small Shifts," International Journal of Differential Equations, Hindawi, vol. 2020, pages 1-15, December.
    10. Manoj Kumar & Parul, 2011. "A Recent Development of Computer Methods for Solving Singularly Perturbed Boundary Value Problems," International Journal of Differential Equations, Hindawi, vol. 2011, pages 1-32, November.
    11. 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.
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