A fitted second--order difference scheme on a modified Shishkin mesh for a semilinear singularly-perturbed boundary-value problem

In the present paper we consider the numerical solving of a semilinear singular–perturbation reaction–diffusion boundary–value problem having boundary layers. A new difference scheme is constructed, the second order of convergence on a modified Shishkin mesh is shown. The numerical experiments are included in the paper, which confirm the theoretical results.