A hybrid initial-value technique for singularly perturbed boundary value problems
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A hybrid initial-value technique for singularly perturbed boundary value problems Vinod Kumar · Rajesh K. Bawa · A. K. Lal
Received: 30 August 2012 / Accepted: 10 January 2013 / Published online: 26 March 2013 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2013
Abstract In this paper, we propose a hybrid initial-value technique for singularly perturbed boundary value problems. First, we develop a hybrid scheme to solve the singularly perturbed initial-value problems, and then, the hybrid scheme is used to solve the singularly perturbed boundary value problems. The scheme is shown to converge to the continuous solution uniformly with respect to the perturbation parameter. Necessary error estimates are derived for the scheme. To verify computational efficiency and accuracy, some numerical examples are provided. Keywords Asymptotic expansion approximation · Backward difference operator · Piecewise uniform Shishkin mesh · Singularly perturbed boundary value problem · Trapezoidal method Mathematics Subject Classification (2010)
Primary 65L11; Secondary 65L10
1 Introduction Singularly perturbed problems (SPPs) have received a significant amount of attention in the past and recent years. The presence of small parameter(s) in these problems prevent us from obtaining satisfactory numerical solutions. It is a well-known fact that the solutions of these problems have a multi-scale character. That is, there are thin layer(s) where the solution varies very rapidly, while away from the layer(s) the solution behaves regularly and varies
Communicated by Marcos Raydan. V. Kumar (B) · A. K. Lal SMCA, Thapar University, Patiala 147004, India e-mail: [email protected] R. K. Bawa Department of Computer Science, Punjabi University, Patiala 147002, India
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slowly. To solve these types of problems, mainly there are two approaches, namely fitted operators and fitted mesh methods. The first one has an advantage that it does not require the knowledge of location and width of the boundary layer, however, it is difficult to extend it for higher dimensional problems. Whereas, the disadvantage of the second approach is the requirement of the knowledge of location and width of the boundary layer. Nevertheless, it is gaining popularity because of simple piecewise uniform meshes like Shishkin mesh. For solving various types of SPPs, many techniques are available in the literature, more details can be found in the books by Farrell et al. Farrell et al. (2000) and Roos et al. Roos et al. (1996). In this article, we consider the following class of singularly perturbed boundary value problems (SPBVPs): L u(x) ≡ u (x) + a(x)u (x) + b(x)u(x) = f (x), x ∈ = (0, 1), u(0) = p, u(1) = q,
(1) (2)
where 0 < 1 is a small positive parameter, a(x), b(x) and f (x) are sufficiently smooth functions, such that a(x) ≥ β > 0 and b(x) ≥ 0 on = [0, 1]. Under these assumptions, (1–2) possesses a unique solution u(x) ∈ C 2 () with a boundary layer of width O() at x = 0. Recently, some resear
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