An almost fourth order uniformly convergent scheme for reaction-diffusion problems on a piecewise uniform grid

For the problem: $-\varepsilon y^{\prime\prime}(x) + p(x) y(x) = f(x),\;\: x \in D, \;\: y(0)=\alpha_{0},\, y(1)=\alpha_{1}$ the spline difference schemes on a piecewise mesh having the second order of uniform convergence are given. Also, in this paper is presented a construction of an uniformly convergent scheme with an almost fourth order of uniform convergence on a Shishkin mesh.