Variable mesh spline approximation method for solving singularly perturbed turning point problems having interior layer

  • Authors:

  • Mohan K. Kadalbajoo;Kailash C. Patidar

  • Affiliations:

  • Department of Mathematics, Indian Institute of Technology, Kanpur, India and Department of Mathematics and Applied Mathematics, University of the Western, Bellville, South Africa;Department of Mathematics, Indian Institute of Technology, Kanpur, India and Department of Mathematics and Applied Mathematics, University of the Western, Bellville, South Africa

  • Venue:
  • Neural, Parallel & Scientific Computations
  • Year:
  • 2010

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Abstract

We present a second order numerical method based on cubic spline on a non-uniform mesh for the singularly perturbed two-point boundary value problems having interior layer in the turning point region. As opposed to our previous work dealing with the turning point problems having boundary layers [Kadalbajoo, M. K. & Patidar, K. C., (2001). Variable mesh spline approximation method for solving singularly perturbed turning point problems having boundary layer(s), Comp. Math. Appl., v. 42(10-11), 1439-1453], the distinctive feature of the problem considered in this paper lies in the fact that the layer appears in the interior of the region around the turning point. Rather than using a piecewise uniform mesh of Shishkin type [Miller, J. J. H., O'Riordan, E. & Shishkin, G. I., (1996). Fitted numerical methods for singular perturbation problems, Singapore: Word Scientific] which can not resolve the interior layer problems efficiently, we design a fully nonuniform mesh in the interior layer region. We then extend the classical approach of Berger et al. [Berger, A. E., Solomon, J. M. & Ciment, M., (1981). An analysis of a uniformly accurate difference method for a singular perturbation problem, Math. Comp., v. 37, 79-94] and Kellogg and Tsan [Kellogg, R. B. & Tsan, A., (1978). Analysis of some difference approximations for a singular perturbation problem without turning points, Math. Comp., v. 32, 1025-1039] to analyze our numerical method. Some numerical results confirming the theoretical estimates are also provided.