IDEAS home Printed from https://ideas.repec.org/a/spr/joptap/v158y2013i1d10.1007_s10957-012-0200-9.html
   My bibliography  Save this article

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

Author

Listed:
  • 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
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10957-012-0200-9
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.
    File URL: https://libkey.io/10.1007/s10957-012-0200-9?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    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.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    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.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Tan, Zengqiang & Zhang, Chengjian, 2022. "Numerical approximation to semi-linear stiff neutral equations via implicit–explicit general linear methods," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 196(C), pages 68-87.
    2. Eriqat, Tareq & El-Ajou, Ahmad & Oqielat, Moa'ath N. & Al-Zhour, Zeyad & Momani, Shaher, 2020. "A New Attractive Analytic Approach for Solutions of Linear and Nonlinear Neutral Fractional Pantograph Equations," Chaos, Solitons & Fractals, Elsevier, vol. 138(C).
    3. Posch, Olaf & Trimborn, Timo, 2013. "Numerical solution of dynamic equilibrium models under Poisson uncertainty," Journal of Economic Dynamics and Control, Elsevier, vol. 37(12), pages 2602-2622.
    4. Amat, Sergio & José Legaz, M. & Pedregal, Pablo, 2015. "A variable step-size implementation of a variational method for stiff differential equations," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 118(C), pages 49-57.
    5. García, M.A. & Castro, M.A. & Martín, J.A. & Rodríguez, F., 2018. "Exact and nonstandard numerical schemes for linear delay differential models," Applied Mathematics and Computation, Elsevier, vol. 338(C), pages 337-345.
    6. M. Motawi Khashan & Rohul Amin & Muhammed I. Syam, 2019. "A New Algorithm for Fractional Riccati Type Differential Equations by Using Haar Wavelet," Mathematics, MDPI, vol. 7(6), pages 1-12, June.
    7. Olaf Posch & Timo Trimborn, 2010. "Numerical solution of continuous-time DSGE models under Poisson uncertainty," Economics Working Papers 2010-08, Department of Economics and Business Economics, Aarhus University.
    8. Tan, Zengqiang & Zhang, Chengjian, 2018. "Implicit-explicit one-leg methods for nonlinear stiff neutral equations," Applied Mathematics and Computation, Elsevier, vol. 335(C), pages 196-210.
    9. Zhang, G.L. & Song, Minghui & Liu, M.Z., 2015. "Asymptotical stability of the exact solutions and the numerical solutions for a class of impulsive differential equations," Applied Mathematics and Computation, Elsevier, vol. 258(C), pages 12-21.
    10. Kürkçü, Ömür Kıvanç & Aslan, Ersin & Sezer, Mehmet, 2016. "A numerical approach with error estimation to solve general integro-differential–difference equations using Dickson polynomials," Applied Mathematics and Computation, Elsevier, vol. 276(C), pages 324-339.
    11. Xu, Y. & Zhao, J.J. & Sui, Z.N., 2010. "Exponential Runge–Kutta methods for delay differential equations," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 80(12), pages 2350-2361.
    12. Berezansky, Leonid & Braverman, Elena, 2019. "On stability of linear neutral differential equations in the Hale form," Applied Mathematics and Computation, Elsevier, vol. 340(C), pages 63-71.
    13. Ashwin Aravindakshan & Prasad Naik, 2011. "How does awareness evolve when advertising stops? The role of memory," Marketing Letters, Springer, vol. 22(3), pages 315-326, September.
    14. Xiaohua Ding & Huan Su, 2007. "Dynamics of a Discretization Physiological Control System," Discrete Dynamics in Nature and Society, Hindawi, vol. 2007, pages 1-16, February.
    15. Yan, Xiaoqiang & Zhang, Chengjian, 2019. "Solving nonlinear functional–differential and functional equations with constant delay via block boundary value methods," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 166(C), pages 21-32.
    16. Qin, Hongyu & Zhang, Qifeng & Wan, Shaohua, 2019. "The continuous Galerkin finite element methods for linear neutral delay differential equations," Applied Mathematics and Computation, Elsevier, vol. 346(C), pages 76-85.
    17. Qin, Tingting & Zhang, Chengjian, 2015. "Stable solutions of one-leg methods for a class of nonlinear functional-integro-differential equations," Applied Mathematics and Computation, Elsevier, vol. 250(C), pages 47-57.
    18. Wang, Qi, 2015. "Numerical oscillation of neutral logistic delay differential equation," Applied Mathematics and Computation, Elsevier, vol. 258(C), pages 49-59.
    19. Xu, Y. & Zhao, J.J., 2008. "Stability of Runge–Kutta methods for neutral delay-integro-differential-algebraic system," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 79(3), pages 571-583.
    20. Cheng, Xue & Chen, Zhong & Zhang, Qingpu, 2015. "An approximate solution for a neutral functional–differential equation with proportional delays," Applied Mathematics and Computation, Elsevier, vol. 260(C), pages 27-34.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:spr:joptap:v:158:y:2013:i:1:d:10.1007_s10957-012-0200-9. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.