IDEAS home Printed from https://ideas.repec.org/a/spr/joptap/v133y2007i3d10.1007_s10957-007-9167-3.html
   My bibliography  Save this article

Asymptotic Initial-Value Method for Second-Order Singular Perturbation Problems of Reaction-Diffusion Type with Discontinuous Source Term

Author

Listed:
  • T. Valanarasu

    (Bharathidasan University)

  • N. Ramanujam

    (Bharathidasan University)

Abstract

In this paper, a numerical method is presented to solve singularly-perturbed two-point boundary-value problems for second-order ordinary differential equations with a discontinuous source term. First, an asymptotic expansion approximation of the solution of the boundary-value problem is constructed using the basic ideas of the well-known WKB perturbation method. Then, some initial-value problems and terminal-value problems are constructed such that their solutions are the terms of this asymptotic expansion. These initial-value problems and terminal-value problems are singularly-perturbed problems and therefore fitted mesh method (Shishkin mesh) are used to solve these problems. Necessary error estimates are derived and examples are provided to illustrate the method.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:joptap:v:133:y:2007:i:3:d:10.1007_s10957-007-9167-3
    DOI: 10.1007/s10957-007-9167-3
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10957-007-9167-3
    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/s10957-007-9167-3?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. S. Natesan & M. Ramanujam, 1998. "Initial-Value Technique for Singularly-Perturbed Turning-Point Problems Exhibiting Twin Boundary Layers," Journal of Optimization Theory and Applications, Springer, vol. 99(1), pages 37-52, October.
    2. S. Natesan & N. Ramanujam, 1998. "Initial-Value Technique for Singularly Perturbed Boundary-Value Problems for Second-Order Ordinary Differential Equations Arising in Chemical Reactor Theory," Journal of Optimization Theory and Applications, Springer, vol. 97(2), pages 455-470, May.
    3. T. Valanarasu & N. Ramanujan, 2003. "Asymptotic Initial-Value Method for Singularly-Perturbed Boundary-Value Problems for Second-Order Ordinary Differential Equations," Journal of Optimization Theory and Applications, Springer, vol. 116(1), pages 167-182, January.
    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 & 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.

    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. T. Valanarasu & N. Ramanujan, 2003. "Asymptotic Initial-Value Method for Singularly-Perturbed Boundary-Value Problems for Second-Order Ordinary Differential Equations," Journal of Optimization Theory and Applications, Springer, vol. 116(1), pages 167-182, January.
    2. Singh, Satpal & Kumar, Devendra & Ramos, Higinio, 2022. "A uniformly convergent quadratic B-spline based scheme for singularly perturbed degenerate parabolic problems," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 195(C), pages 88-106.
    3. Sahlan, M. Nosrati, 2019. "Four computational approaches for solving a class of boundary value problems arising in chemical reactor industry," Applied Mathematics and Computation, Elsevier, vol. 355(C), pages 253-268.
    4. S. Natesan & N. Ramanujam, 1998. "Booster Method for Singularly-Perturbed One-Dimensional Convection-Diffusion Neumann Problems," Journal of Optimization Theory and Applications, Springer, vol. 99(1), pages 53-72, October.
    5. Chein-Shan Liu & Essam R. El-Zahar & Chih-Wen Chang, 2022. "Higher-Order Asymptotic Numerical Solutions for Singularly Perturbed Problems with Variable Coefficients," Mathematics, MDPI, vol. 10(15), pages 1-20, August.
    6. Ianni, A., 2002. "Reinforcement learning and the power law of practice: some analytical results," Discussion Paper Series In Economics And Econometrics 0203, Economics Division, School of Social Sciences, University of Southampton.
    7. Majumdar, Anirban & Natesan, Srinivasan, 2017. "Alternating direction numerical scheme for singularly perturbed 2D degenerate parabolic convection-diffusion problems," Applied Mathematics and Computation, Elsevier, vol. 313(C), pages 453-473.
    8. Toprakseven, Şuayip & Zhu, Peng, 2023. "Error analysis of a weak Galerkin finite element method for two-parameter singularly perturbed differential equations in the energy and balanced norms," Applied Mathematics and Computation, Elsevier, vol. 441(C).

    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:133:y:2007:i:3:d:10.1007_s10957-007-9167-3. 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.