The numerical integration of neutral functional-differential equations by fully implicit one-step methods.

*(English)*Zbl 0830.65079This paper deals with fully implicit one-step methods for the numerical solution of neutral functional-differential equations (NFDEs) and develops a divided difference representation of these formulas. The authors also discuss how to estimate the local discretization error in an efficient way by comparing two approximations of successive orders.

The predictor-corrector implementation of fully implicit one-step methods is described in a general context of NFDEs, with additional details of implementation for systems of neutral delay differential equations, for Volterra integro-differential equations, and for stiff delay differential equations. If the delay equation is stiff the system of the resulting nonlinear equations is solved by a variant of Newton’s method. Some numerical experiments are presented and discussed.

The predictor-corrector implementation of fully implicit one-step methods is described in a general context of NFDEs, with additional details of implementation for systems of neutral delay differential equations, for Volterra integro-differential equations, and for stiff delay differential equations. If the delay equation is stiff the system of the resulting nonlinear equations is solved by a variant of Newton’s method. Some numerical experiments are presented and discussed.

Reviewer: P.Chocholatý (Bratislava)

##### MSC:

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

65L05 | Numerical methods for initial value problems |

65L12 | Finite difference and finite volume methods for ordinary differential equations |

65R20 | Numerical methods for integral equations |

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

34K05 | General theory of functional-differential equations |

34E13 | Multiple scale methods for ordinary differential equations |

45J05 | Integro-ordinary differential equations |

##### Keywords:

numerical examples; error bound; predictor-corrector method; fully implicit one-step methods; neutral functional-differential equations; divided difference; systems of neutral delay differential equations; Volterra integro-differential equations; stiff delay differential equations; Newton’s method##### Software:

DELSOL
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\textit{Z. Jackiewicz} and \textit{E. Lo}, Z. Angew. Math. Mech. 75, No. 3, 207--221 (1995; Zbl 0830.65079)

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##### References:

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