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A computational method for self-adjoint singular perturbation problems using quintic spline. (English) Zbl 1084.65070

The authors consider a two-point boundary value problem, which consists of a singularly perturbed ordinary differential equation (ODE) of second order. Thereby, the scalar ODE is linear and involves a self-adjoint operator. Due to the singularly perturbed problem, the corresponding solution exhibits two boundary layers and one regular region.

Consequently, the authors split the domain of dependence in three subdomains with additional boundary conditions resulting from an asymptotic approximation. Thereby, a potential for parallelism is created. In the two subdomains with the boundary layers, a transformation yields an uncritical ODE problem. A straightforward technique based on quintic splines is used to obtain a numerical solution in each subdomain.

The crucial part of the paper consists in a convergenge analysis. The authors prove that the error in each grid point is uniformly bounded by a term $𝒪\left(\epsilon +{h}^{4}\right)$, where $\epsilon$ is the parameter of the perturbation and $h$ is the step size of the discretisation. Numerical results employing simple examples of ODEs are presented in form of large tables with error data.

##### MSC:
 65L10 Boundary value problems for ODE (numerical methods) 65L20 Stability and convergence of numerical methods for ODE 34E15 Asymptotic singular perturbations, general theory (ODE) 34B05 Linear boundary value problems for ODE 65L70 Error bounds (numerical methods for ODE) 65L50 Mesh generation and refinement (ODE) 65Y05 Parallel computation (numerical methods)