Samir Karasuljić

Samir Karasuljić, Helena Zarin, Enes Duvnjaković

In this paper we consider the numerical solution of a singularly perturbed one-dimensional semilinear reaction-diffusion problem. A class of differential schemes is constructed. There is a proof of the existence and uniqueness of the numerical solution for this constructed class of differential schemes. The central result of the paper is an $\varepsilon$--uniform convergence of the second order $\mathcal{O}\left(1/N^2 \right),$ for the discrete approximate solution on the modified Bakhvalov mesh. At the end of the paper there are numerical experiments, two representatives of the class of differential schemes are tested and it is shown the robustness of the method and concurrence of theoretical and experimental results.

Samir Karasuljić, Enes Duvnjaković, Elvir Memić

In this paper we consider two difference schemes for numerical solving of a one--dimensional singularly perturbed boundary value problem. We proved an $\varepsilon$--uniform convergence for both difference schemes on a Shiskin mesh. Finally, we present four numerical experiments to confirm the theoretical results.

Samir Karasuljić, Enes Duvnjaković, Vedad Pasic et al.

We consider an approximate solution for the one-dimensional semilinear singularly-perturbed boundary value problem, using the previously obtained numerical values of the boundary value problem in the mesh points and the representation of the exact solution using Green's function. We present an $\varepsilon$-uniform convergence of such gained the approximate solutions, in the maximum norm of the order $\mathcal{O}\left(N^{-1}\right)$ on the observed domain. After that, the constructed approximate solution is repaired and we obtain a solution, which also has $\varepsilon$--uniform convergence, but now of order $\mathcal{O}\left(\ln^2N/N^2\right)$ on $[0,1].$ In the end a numerical experiment is presented to confirm previously shown theoretical results.

Enes Duvnjaković, Samir Karasuljić, Vedad Pasic et al.

In this paper 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 $ε$-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.