*(English)*Zbl 1157.65047

The authors study numerical method for solving the self-adjoint boundary value problem of a singularly perturbed second order linear differential equation

with homogeneous boundary conditions $u\left(0\right)=u\left(1\right)=0$, where $\epsilon $ is a small parameter and $f\left(x\right),\phantom{\rule{4pt}{0ex}}q\left(x\right)$ are smooth functions and satisfy $q\left(x\right)\ge {q}_{*},\phantom{\rule{0.166667em}{0ex}}\forall x\in [0,1]$ for some positive constant ${q}_{*}$. For small $\epsilon $, the solution $u\left(x\right)$ may exhibit exponential boundary layers at both ends of the interval $[0,1]$, which gives rise to difficulties in numerical solutions.

Using a B-splines basis for the space of exponential spline, the authors propose a spline collocation method which leads to an easily solvable tridiagonal algebraic system. It is shown that the method can be implemented on uniform meshes, i.e. there is no need of introducing more nodal points in the boundary layers. Further, the second order uniform convergence of the numerical solution is proven.

The efficiency of the method is demonstrated by several numerical experiments where the present method is shown to be superior to the scheme using cubic B-splines on fitted meshes that was suggested previously by *M. K. Kadalbajoo* and *V. K. Aggarwal* [Appl. Math. Comput. 161, No. 3, 973–987 (2005; Zbl 1073.65062)].

##### MSC:

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

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

65L60 | Finite elements, Rayleigh-Ritz, Galerkin and collocation methods for ODE |

34B05 | Linear boundary value problems for ODE |

34E15 | Asymptotic singular perturbations, general theory (ODE) |