Volume 16, Issue 4
Orthogonal Spline Collocation for Singularly Perturbed Reaction Diffusion Problems in One Dimension

Pankaj Mishra, Kapil K. Sharma, Amiya K. Pani & Graeme Fairweather


Int. J. Numer. Anal. Mod., 16 (2019), pp. 647-667.

Published online: 2019-02

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  • Abstract

An orthogonal spline collocation method (OSCM) with C1splines of degree r ≥ 3 is analyzed for the numerical solution of singularly perturbed reaction diffusion problems in one dimension. The method is applied on a Shishkin mesh and quasi-optimal error estimates in weighted Hm norms for m = 1, 2 and in a discrete L2-norm are derived. These estimates are valid uniformly with respect to the perturbation parameter. The results of numerical experiments are presented for C1 cubic splines (r = 3) and C1 quintic splines (r = 5) to demonstrate the efficacy of the OSCM and confirm our theoretical findings. Further, quasi-optimal a priori estimates in L2, L and W1,∞-norms are observed in numerical computations. Finally, superconvergence of order 2r − 2 at the mesh points is observed in the approximate solution and also in its first derivative when r = 5.

  • Keywords

Singularly perturbed reaction diffusion problems, orthogonal spline collocation, Shishkin mesh, quasi-optimal global error estimates, superconvergence.

  • AMS Subject Headings

65L11, 65L60, 65L70

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

pmparasar@students.sau.ac.in (Pankaj Mishra)

kapil.sharma@sau.ac.in (Kapil K. Sharma)

akp@math.iitb.ac.in (Amiya K. Pani)

graeme.fairweather@gmail.com (Graeme Fairweather)

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