Convergence analysis of the solution of retarded and neutral delay differential equations by continuous numerical methods.

*(English)*Zbl 0914.65084The authors recently developed a generic approach to solve retarded and neutral delay differential equations. The approach is based on the use of an explicit continuous Runge-Kutta formula and employs defect control. The approach can be applied to problems with state-dependent delays and vanishing delays.

This paper contains convergence properties for the numerical solution associated with methods that implement this approach. They first analyze such properties for retarded delay differential equations and then the analysis is extended to the case of neutral differential equations. Such an extension is particularly important since neutral differential equations have received little attention in the literature. The main result they establish is that the global error of the numerical solution can be efficiently and reliability controlled by directly monitoring the magnitude of the associated defect. This analysis can also be applied directly to other numerical methods (which use defect error control) based on Runge-Kutta formulas for delay differential equations.

This paper contains convergence properties for the numerical solution associated with methods that implement this approach. They first analyze such properties for retarded delay differential equations and then the analysis is extended to the case of neutral differential equations. Such an extension is particularly important since neutral differential equations have received little attention in the literature. The main result they establish is that the global error of the numerical solution can be efficiently and reliability controlled by directly monitoring the magnitude of the associated defect. This analysis can also be applied directly to other numerical methods (which use defect error control) based on Runge-Kutta formulas for delay differential equations.

Reviewer: R.S.Dahiya (Ames)

##### MSC:

65L06 | Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations |

34K40 | Neutral functional-differential equations |

34K05 | General theory of functional-differential equations |

65L05 | Numerical methods for initial value problems |

65L70 | Error bounds for numerical methods for ordinary differential equations |