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A numerical method based on finite difference for boundary value problems for singularly perturbed delay differential equations. (English) Zbl 1141.65062

A numerical method is proposed for linear second-order singularly perturbed delay differential equations of the form

$\epsilon {y}^{\text{'}\text{'}}\left(x\right)+a\left(x\right){y}^{\text{'}}\left(x-\delta \right)+b\left(x\right)y\left(x\right)=f\left(x\right),\phantom{\rule{0.166667em}{0ex}}\text{on}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\left(0,1\right),$

subject to Dirichlet boundary conditions. Here, $0<\epsilon \ll 1$ is the perturbation parameter and $\delta$ is the small shift parameter. The authors are mainly focused on the case $\delta =O\left(\epsilon \right)$.

In order to solve this problem classical finite difference schemes are used with the mesh parameter $h=\delta /m$, where $m=pq$, $p$ is a positive integer and $q$ is the mantissa of $\delta$. The truncation error contains the higher-order derivatives of the solution of the continuous problem which involve negative powers of the small (perturbation and delay) parameters. Therefore, the convergence result provided here may not be independent of the parameters, that is, they are not uniformly-convergent. Some numerical examples are presented.

MSC:
 65L10 Boundary value problems for ODE (numerical methods) 65L12 Finite difference methods for ODE (numerical methods) 34K28 Numerical approximation of solutions of functional-differential equations 34K26 Singular perturbations of functional-differential equations 65L70 Error bounds (numerical methods for ODE) 65L20 Stability and convergence of numerical methods for ODE