*(English)*Zbl 1069.65086

The problem under consideration is the singularly perturbed boundary value problem (BVP) for the delay differential equation

under the boundary conditions

where $\epsilon $ and $\delta $ are small positive parameters.

The stated BVP for the delay differential equation is approximated by one for the ordinary differential equation (ODE), created by replacing the retarded term ${y}^{\text{'}}(x-\delta )$ by its first order Taylor approximation ${y}^{\text{'}}\left(x\right)-\delta {y}^{\text{'}\text{'}}\left(x\right)$. The approximate BVP for the ODE is approximated by a standard three points difference scheme. The stability and convergence of the method is discussed for two cases corresponding to the location the boundary layer, on the left side (when $a\left(x\right)>0$) and on the right (when $a\left(x\right)<0$). Numerical examples are presented.

##### MSC:

65L10 | Boundary value problems for ODE (numerical methods) |

34K28 | Numerical approximation of solutions of functional-differential equations |

65L20 | Stability and convergence of numerical methods for ODE |

34K26 | Singular perturbations of functional-differential equations |

34K10 | Boundary value problems for functional-differential equations |