Abstract:

In this paper, a numerical method is proposed for solving singularly perturbed turning point problems with boundary layers. This method is based on the asymptotic expansion technique and the reproducing kernel method, and it decomposes the original problem into a boundary layer problem and a regular region problem. The regular region problem is solved by the asymptotic expansion method, while the boundary layer problem is solved by the variable stretching method and the reproducing kernel method based on the collocation method. For singularly perturbed delay differential-difference equations with a single boundary layer, we construct basis functions based on the reproducing kernel function in the $W_2^4$ space. Inside each sub - division cell, the Gaussian-Legendre nodes with 4 points are selected as collocation points. Compared with the original fitted mesh B-spline collocation method, the accuracy and convergence order obtained by our method are higher. Four numerical examples are provided to illustrate the effectiveness of this method. The results of the numerical examples show that this method can provide very accurate approximate solutions and achieve the optimal convergence order.

Key words: singularly perturbed equations, delay problems, optimal convergence rate, reproducing kernel method.