A uniformly convergent difference scheme on a modified Shishkin mesh for the singularly perturbed reaction-diffusion boundary value problem

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Vedad Pasic http://orcid.org/0000-0003-2115-0422 Enes Duvnjakovic Samir Karasuljic Helena Zarin

Abstract

We are considering a semilinear singular perturbation reaction-diffusion boundary value problem which contains a small perturbation parameter that acts on the highest order derivative. We construct a difference scheme on an arbitrary nonequidistant mesh using a collocation method and Green's function. We show that the constructed difference scheme has a unique solution and that the scheme is stable. The central result of the paper is \(\epsilon\)-uniform convergence of almost second order for the discrete approximate solution on a modified Shishkin mesh. We finally provide two numerical examples which illustrate the theoretical results on the uniform accuracy of the discrete problem, as well as the robustness of the method.

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How to Cite
PASIC, Vedad et al. A uniformly convergent difference scheme on a modified Shishkin mesh for the singularly perturbed reaction-diffusion boundary value problem. Journal of Modern Methods in Numerical Mathematics, [S.l.], v. 6, n. 1, p. 28-43, july 2015. ISSN 2090-4770. Available at: <http://www.m-sciences.com/index.php?journal=jmmnm&page=article&op=view&path%5B%5D=971>. Date accessed: 13 dec. 2017. doi: https://doi.org/10.20454/jmmnm.2015.971.
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