@Article{IJNAM-16-4,
author = {Pankaj, Mishra and Sharma, Kapil, K. and Pani, K., Amiya and Graeme, Fairweather},
title = {Orthogonal Spline Collocation for Singularly Perturbed Reaction Diffusion Problems in One Dimension},
journal = {International Journal of Numerical Analysis and Modeling},
year = {2019},
volume = {16},
number = {4},
pages = {647--667},
abstract = {<p style="text-align: justify;">An orthogonal spline collocation method (OSCM) with $C^1$<span style="font-size: 13.3333px;"> </span>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 $H$<sup>$m$</sup> norms for $m$ = 1, 2 and in a discrete $L$<sup>2</sup>-norm are derived. These estimates are valid
uniformly with respect to the perturbation parameter. The results of numerical experiments are
presented for $C$<sup>1</sup> cubic splines ($r$ = 3) and $C$<sup>1</sup> quintic splines ($r$ = 5) to demonstrate the efficacy
of the OSCM and confirm our theoretical findings. Further, quasi-optimal a $priori$ estimates in $L$<sup>2</sup>, $L$<sup>∞</sup> and $W$<sup>1</sup><sup>,∞</sup>-norms are observed in numerical computations. Finally, superconvergence
of order 2$r$ − 2 at the mesh points is observed in the approximate solution and also in its first
derivative when $r$ = 5.</p>},
issn = {2617-8710},
doi = {https://doi.org/2019-IJNAM-13019},
url = {https://global-sci.com/article/83081/orthogonal-spline-collocation-for-singularly-perturbed-reaction-diffusion-problems-in-one-dimension}
}