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A Parameter-Uniform B-Spline Collocation Method for Singularly Perturbed Semilinear Reaction-Diffusion Problems

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  • S. C. S. Rao

    (Indian Institute of Technology Delhi)

  • S. Kumar

    (Indian Institute of Technology Delhi)

  • M. Kumar

    (Indian Institute of Technology Delhi)

Abstract

We consider a Dirichlet boundary value problem for a class of singularly perturbed semilinear reaction-diffusion equations. A B-spline collocation method on a piecewise-uniform Shishkin mesh is developed to solve such problems numerically. The convergence analysis is given and the method is shown to be almost second-order convergent, uniformly with respect to the perturbation parameter ε in the maximum norm. Numerical results are presented to validate the theoretical results.

Suggested Citation

  • S. C. S. Rao & S. Kumar & M. Kumar, 2010. "A Parameter-Uniform B-Spline Collocation Method for Singularly Perturbed Semilinear Reaction-Diffusion Problems," Journal of Optimization Theory and Applications, Springer, vol. 146(3), pages 795-809, September.
    • Handle: RePEc:spr:joptap:v:146:y:2010:i:3:d:10.1007_s10957-010-9683-4
    DOI: 10.1007/s10957-010-9683-4
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    References listed on IDEAS

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    1. M.K. Kadalbajoo & K.C. Patidar, 2002. "Spline Techniques for Solving Singularly-Perturbed Nonlinear Problems on Nonuniform Grids," Journal of Optimization Theory and Applications, Springer, vol. 114(3), pages 573-591, September.
    2. S. C. S. Rao & M. Kumar, 2007. "B-Spline Collocation Method for Nonlinear Singularly-Perturbed Two-Point Boundary-Value Problems," Journal of Optimization Theory and Applications, Springer, vol. 134(1), pages 91-105, July.
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    Cited by:

    1. Chandra Sekhara Rao, S. & Chaturvedi, Abhay Kumar, 2022. "Analysis of an almost fourth-order parameter-uniformly convergent numerical method for singularly perturbed semilinear reaction-diffusion system with non-smooth source term," Applied Mathematics and Computation, Elsevier, vol. 421(C).
    2. Anshima Singh & Sunil Kumar, 2024. "An Efficient Numerical Method Based on Exponential B-splines for a Time-Fractional Black–Scholes Equation Governing European Options," Computational Economics, Springer;Society for Computational Economics, vol. 64(4), pages 1965-2002, October.
    3. S. C. S. Rao & S. Kumar & M. Kumar, 2011. "Uniform Global Convergence of a Hybrid Scheme for Singularly Perturbed Reaction–Diffusion Systems," Journal of Optimization Theory and Applications, Springer, vol. 151(2), pages 338-352, November.

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