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- Title
ORTHOGONAL SPLINE COLLOCATION FOR SINGULARLY PERTURBED REACTION DIFFUSION PROBLEMS IN ONE DIMENSION.
- Authors
MISHRA, PANKAJ; SHARMA, KAPIL K.; PANI, AMIYA K.; FAIRWEATHER, GRAEME
- Abstract
An orthogonal spline collocation method (OSCM) with C¹ splines 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.
- Subjects
SPLINES; COLLOCATION methods; DIFFUSION; DIMENSIONS
- Publication
International Journal of Numerical Analysis & Modeling, 2019, Vol 16, Issue 4, p647
- ISSN
1705-5105
- Publication type
Article