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

DOI:

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 $C$1splines 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$$m$ norms for $m$ = 1, 2 and in a discrete $L$2-norm are derived. These estimates are valid uniformly with respect to the perturbation parameter. The results of numerical experiments are presented for $C$1 cubic splines ($r$ = 3) and $C$1 quintic splines ($r$ = 5) to demonstrate the efficacy of the OSCM and confirm our theoretical findings. Further, quasi-optimal a $priori$ estimates in $L$2, $L$ and $W$1,∞-norms are observed in numerical computations. Finally, superconvergence of order 2$r$ − 2 at the mesh points is observed in the approximate solution and also in its first derivative when $r$ = 5.

  • 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)

  • BibTex
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  • TXT
@Article{IJNAM-16-647, author = {Mishra , Pankaj and Sharma , Kapil K. and Pani , Amiya K. and Fairweather , Graeme }, 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 = {

An orthogonal spline collocation method (OSCM) with $C$1splines 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$$m$ norms for $m$ = 1, 2 and in a discrete $L$2-norm are derived. These estimates are valid uniformly with respect to the perturbation parameter. The results of numerical experiments are presented for $C$1 cubic splines ($r$ = 3) and $C$1 quintic splines ($r$ = 5) to demonstrate the efficacy of the OSCM and confirm our theoretical findings. Further, quasi-optimal a $priori$ estimates in $L$2, $L$ and $W$1,∞-norms are observed in numerical computations. Finally, superconvergence of order 2$r$ − 2 at the mesh points is observed in the approximate solution and also in its first derivative when $r$ = 5.

}, issn = {2617-8710}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/ijnam/13019.html} }
TY - JOUR T1 - Orthogonal Spline Collocation for Singularly Perturbed Reaction Diffusion Problems in One Dimension AU - Mishra , Pankaj AU - Sharma , Kapil K. AU - Pani , Amiya K. AU - Fairweather , Graeme JO - International Journal of Numerical Analysis and Modeling VL - 4 SP - 647 EP - 667 PY - 2019 DA - 2019/02 SN - 16 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/ijnam/13019.html KW - Singularly perturbed reaction diffusion problems, orthogonal spline collocation, Shishkin mesh, quasi-optimal global error estimates, superconvergence. AB -

An orthogonal spline collocation method (OSCM) with $C$1splines 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$$m$ norms for $m$ = 1, 2 and in a discrete $L$2-norm are derived. These estimates are valid uniformly with respect to the perturbation parameter. The results of numerical experiments are presented for $C$1 cubic splines ($r$ = 3) and $C$1 quintic splines ($r$ = 5) to demonstrate the efficacy of the OSCM and confirm our theoretical findings. Further, quasi-optimal a $priori$ estimates in $L$2, $L$ and $W$1,∞-norms are observed in numerical computations. Finally, superconvergence of order 2$r$ − 2 at the mesh points is observed in the approximate solution and also in its first derivative when $r$ = 5.

Pankaj Mishra, Kapil K. Sharma, Amiya K. Pani & Graeme Fairweather. (2019). Orthogonal Spline Collocation for Singularly Perturbed Reaction Diffusion Problems in One Dimension. International Journal of Numerical Analysis and Modeling. 16 (4). 647-667. doi:
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