B-spline collocation and self-adapting differential evolution (jDE) algorithm for a singularly perturbed convection–diffusion problem

Abstract

Many numerical methods applied on a Shishkin mesh are very popular in solving the singularly perturbed problems. However, few approaches are used to obtain the Shishkin mesh transition parameter. Thus, in this paper, we first use the cubic B-spline collocation method on a Shishkin mesh to solve the singularly perturbed convection–diffusion problem with two small parameters. Then, we transform the Shishkin mesh transition parameter selection problem into a nonlinear unconstrained optimization problem which is solved by using the self-adapting differential evolution (jDE) algorithm. To verify the performance of our presented method, a numerical example is employed. It is shown from the experiment results that our approach is efficient. Compared with other evolutionary algorithms, the jDE algorithm performs better and with more stability.