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Computational method based on reproducing kernel for solving singularly perturbed differential-difference equations with a delay

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  • Sahihi, Hussein
  • Abbasbandy, Saeid
  • Allahviranloo, Tofigh

Abstract

In this paper, Reproducing Kernel Hilbert Space Method (RKHSM) based on collocation scheme is used to solve singularly perturbed second order differential-difference equations. We implement RKHSM without Gram–Schmidt orthogonalization process for singularly perturbed differential-difference equation with boundary layer behavior and also oscillatory behavior with small delay. RKHSM in this study, is based on the division of the problem domain into two subintervals, one with boundary layer and the other one without such a boundary layer. Several numerical examples are studied to demonstrate the accuracy of the present method. Results of the present scheme indicate that new algorithm has the following advantages: small computational work, fast convergence, and high precision.

Suggested Citation

  • Sahihi, Hussein & Abbasbandy, Saeid & Allahviranloo, Tofigh, 2019. "Computational method based on reproducing kernel for solving singularly perturbed differential-difference equations with a delay," Applied Mathematics and Computation, Elsevier, vol. 361(C), pages 583-598.
    • Handle: RePEc:eee:apmaco:v:361:y:2019:i:c:p:583-598
    DOI: 10.1016/j.amc.2019.06.010
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    References listed on IDEAS

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    1. Geng, F.Z. & Qian, S.P. & Cui, M.G., 2015. "Improved reproducing kernel method for singularly perturbed differential-difference equations with boundary layer behavior," Applied Mathematics and Computation, Elsevier, vol. 252(C), pages 58-63.
    2. Li, Zhi-Yuan & Wang, Yu-Lan & Tan, Fu-Gui & Wan, Xiao-Hui & Yu, Hao & Duan, Jun-Sheng, 2015. "Solving a class of linear nonlocal boundary value problems using the reproducing kernel," Applied Mathematics and Computation, Elsevier, vol. 265(C), pages 1098-1105.
    3. 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.
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    Cited by:

    1. Li, X.Y. & Wu, B.Y., 2020. "A new kernel functions based approach for solving 1-D interface problems," Applied Mathematics and Computation, Elsevier, vol. 380(C).

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