Rational spectral collocation and differential evolution algorithms for singularly perturbed problems with an interior layer

We are concerned with using high accuracy numerical methods for solving singularly perturbed problems whose solutions exhibit an interior layer. A rational spectral collocation method in barycentric form with sinh transformation is firstly applied to find the approximate solution to the given problem. The sinh transformation consists of two parameters: the location and the width of the interior layer. With the tool of the asymptotic analysis, the interior layer can be easily located. Then, in determining the width of the interior layer, we design a nonlinear unconstrained optimization problem which is solved by using the basic differential evolution (DE) algorithm. The feasibility and effectiveness of the proposed method is verified by numerical results. The proposed method can also be applied to other singularly perturbed problems.