In the solution of singularly perturbed problems by difference methods some difficulties arise due to the availability of singularities of boundary layer type. In a small neighborhood of such a layer the derivatives of the solutions are infinitely growing when the perturbation parameter \(\varepsilon\) tends to zero. If one don’t take into account this singularity and apply the traditional difference schemes then the errors in the solutions in comparison with the exact solutions are becoming large for \(\varepsilon \to 0\).
In the article the first boundary value problem for a linear differential equation of the second order with a turning point and with a small parameter \(\varepsilon\) to the highest derivative is considered. For the approximate solution of the equation a difference scheme with exponential matching on the grid with a constant step is applied. A uniform in relation to the perturbed parameter convergence of the solutions found by the mentioned schemes to the solutions of the initial differential problem with a first order of correctness is proved