Numerical Derivative Using the Piecewise Uniform Mesh

In this paper, a technique of a piecewise-uniform meshes formed on an improvement finite difference algorithm for finding derivatives of functions. The purpose was to overcome difficulties which face numerical derivatives of functions with stiff formula, the main idea is that the formula includes some terms that can lead to rapid variation in the graph of the functions, which have recently been named singular layers in numerical analysis. The fundamental numerical difficulty is related to non-physical oscillations of the solution (instability) when the formula of the function dominates over the formula of its derivatives, this is a characteristic of many fluid flow problems. The use of Shishkin mesh to find derivatives of arbitrary degree and order is the novelty of this paper. The method was applied to find derivatives of some examples until third order and the results were compared with a previous study, mentioned in the paper, to derive the functions numerically.