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B-Spline Collocation Method for Nonlinear Singularly-Perturbed Two-Point Boundary-Value Problems

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  • S. C. S. Rao

    (Indian Institute of Technology Delhi)

  • M. Kumar

    (Indian Institute of Technology Delhi)

Abstract

A B-spline collocation method is presented for nonlinear singularly-perturbed boundary-value problems with mixed boundary conditions. The quasilinearization technique is used to linearize the original nonlinear singular perturbation problem into a sequence of linear singular perturbation problems. The B-spline collocation method on piecewise uniform mesh is derived for the linear case and is used to solve each linear singular perturbation problem obtained through quasilinearization. The fitted mesh technique is employed to generate a piecewise uniform mesh, condensed in the neighborhood of the boundary layers. The convergence analysis is given and the method is shown to have second-order uniform convergence. The stability of the B-spline collocation system is discussed. Numerical experiments are conducted to demonstrate the efficiency of the method.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:joptap:v:134:y:2007:i:1:d:10.1007_s10957-007-9200-6
    DOI: 10.1007/s10957-007-9200-6
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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.
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    Cited by:

    1. 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.
    2. 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).
    3. Roul, Pradip & Prasad Goura, V.M.K., 2022. "A superconvergent B-spline technique for second order nonlinear boundary value problems," Applied Mathematics and Computation, Elsevier, vol. 414(C).
    4. Roul, Pradip & Madduri, Harshita & Kassner, Klaus, 2019. "A sixth-order numerical method for a strongly nonlinear singular boundary value problem governing electrohydrodynamic flow in a circular cylindrical conduit," Applied Mathematics and Computation, Elsevier, vol. 350(C), pages 416-433.
    5. 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.
    6. Roul, Pradip & Prasad Goura, V.M.K., 2020. "A high order numerical method and its convergence for time-fractional fourth order partial differential equations," Applied Mathematics and Computation, Elsevier, vol. 366(C).

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    1. 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.
    2. 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.

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