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Volume 13, Issue 1
On the Singularly Perturbed Semilinear Reaction-Diffusion Problem and its Numerical Solution

R. Vulanović & L. Teofanov

Int. J. Numer. Anal. Mod., 13 (2016), pp. 41-57.

Published online: 2016-01

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  • Abstract

We obtain improved derivative estimates for the solution of the semilinear singularly perturbed reaction-diffusion problem in one dimension. This enables us to modify the transition points between the fine and coarse parts of the Shishkin discretization mesh. We prove that the numerical solution, obtained by using the central finite-difference scheme on the modified mesh, retains the same order of convergence uniform in the perturbation parameter as on the standard Shishkin mesh. However, the modified mesh may be denser in the layers than the standard one, and, when this is the case, numerical results show an improvement in the accuracy of the computed solution.

  • Keywords

singularly perturbed boundary-value problem, reaction-diffusion, Shishkin mesh, finite differences, and uniform convergence.

  • AMS Subject Headings

65L10, 65L12, 65L20, 65L70

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{IJNAM-13-41, author = {R. Vulanović and L. Teofanov}, title = {On the Singularly Perturbed Semilinear Reaction-Diffusion Problem and its Numerical Solution}, journal = {International Journal of Numerical Analysis and Modeling}, year = {2016}, volume = {13}, number = {1}, pages = {41--57}, abstract = {

We obtain improved derivative estimates for the solution of the semilinear singularly perturbed reaction-diffusion problem in one dimension. This enables us to modify the transition points between the fine and coarse parts of the Shishkin discretization mesh. We prove that the numerical solution, obtained by using the central finite-difference scheme on the modified mesh, retains the same order of convergence uniform in the perturbation parameter as on the standard Shishkin mesh. However, the modified mesh may be denser in the layers than the standard one, and, when this is the case, numerical results show an improvement in the accuracy of the computed solution.

}, issn = {2617-8710}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/ijnam/425.html} }
TY - JOUR T1 - On the Singularly Perturbed Semilinear Reaction-Diffusion Problem and its Numerical Solution AU - R. Vulanović & L. Teofanov JO - International Journal of Numerical Analysis and Modeling VL - 1 SP - 41 EP - 57 PY - 2016 DA - 2016/01 SN - 13 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/ijnam/425.html KW - singularly perturbed boundary-value problem, reaction-diffusion, Shishkin mesh, finite differences, and uniform convergence. AB -

We obtain improved derivative estimates for the solution of the semilinear singularly perturbed reaction-diffusion problem in one dimension. This enables us to modify the transition points between the fine and coarse parts of the Shishkin discretization mesh. We prove that the numerical solution, obtained by using the central finite-difference scheme on the modified mesh, retains the same order of convergence uniform in the perturbation parameter as on the standard Shishkin mesh. However, the modified mesh may be denser in the layers than the standard one, and, when this is the case, numerical results show an improvement in the accuracy of the computed solution.

R. Vulanović and L. Teofanov. (2016). On the Singularly Perturbed Semilinear Reaction-Diffusion Problem and its Numerical Solution. International Journal of Numerical Analysis and Modeling. 13 (1). 41-57. doi:
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