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Zbl 1119.65064
Chandra Sekhara Rao, S.; Kumar, Mukesh
Optimal B-spline collocation method for self-adjoint singularly perturbed boundary value problems.
(English)
[J] Appl. Math. Comput. 188, No. 1, 749-761 (2007). ISSN 0096-3003

The authors propose a numerical method based on a non-overlapping domain decomposition by using B-spline for singularly perturbed two-point boundary-value problems of the form $$-\varepsilon^2 u''(x) + b(x)u(x) = f(x), \quad (0,1), \quad u(0) = \rho_1, \quad u(1) = \rho_2.$$ Recently, {\it R. K. Bawa} and {\it S. Natesan} in [Comput. Math. Appl. 50, No. 8--9, 1371--1382 (2005; Zbl 1084.65070)] devised a non-overlapping domain decomposition method for the singular perturbation problem stated above by using quintic spline. In this article, the authors follow the same domain decomposition idea proposed by Bawa and Natesan [loc. cit.], and use B-splines instead of quintic splines. Error estimates are derived for the B-spline scheme following the same steps for the quintic spline scheme as given in Bawa and Natesan [loc. cit.]. Also, the numerical examples given here do not reflect any improvement of the proposed minor change suggested by the authors as the error estimates are of same order as obtained by Bawa and Natesan [loc. cit.].
[Srinivasan Natesan (Assam)]
MSC 2000:
*65L10 Boundary value problems for ODE (numerical methods)
34B05 Linear boundary value problems of ODE
34E15 Asymptotic singular perturbations, general theory (ODE)
65L60 Finite numerical methods for ODE
65L70 Error bounds (numerical methods for ODE)

Keywords: singular perturbation; non-overlapping domain decomposition; asymptotic approximation; B-spline collocation method; two-point boundary-value problems; error estimates; numerical examples

Citations: Zbl 1084.65070

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