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**The numerical solution of boundary-value problems for differential equations with state dependent deviating arguments.**
*(English)*
Zbl 0669.65065

This paper deals with a numerical solution of a second order boundary value problem with state dependent deviating argument. The presented method is obtained by approximating the second derivative by a second order difference operator and approximating the solutions at non-grid points by piecewise cubic interpolation. Second order convergence is established and a theorem about the asymptotic expansion of the global discretization error is given. The results are illustrated by some numerical examples.

Reviewer: P.Chocholatý

### MSC:

65L10 | Numerical solution of boundary value problems involving ordinary differential equations |

34K10 | Boundary value problems for functional-differential equations |

### Keywords:

second order; difference operator; Second order convergence; asymptotic expansion; global discretization error; numerical examples
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\textit{V. L. Bakke} and \textit{Z. Jackiewicz}, Apl. Mat. 34, No. 1, 1--17 (1989; Zbl 0669.65065)

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### References:

[1] | V. L. Bakke Z. Jackiewicz: A note on the numerical computation of solutions to second order boundary value problems with state dependent deviating arguments. University of Arkansas Numerical Analysis Technical Report 65110-1, June, 1985. |

[2] | B. Chartres R. Stepleman: Convergence of difference methods for initial and boundary value problems with discontinuous data. Math. Comp., v. 25, 1971, pp. 724-732. · Zbl 0244.65051 |

[3] | P. Chocholaty L. Slahor: A numerical method to boundary value problems for second order delay-differential equations. Numer. Math., v. 33, 1979, pp. 69-75. · Zbl 0432.65046 |

[4] | K. De Nevers K. Schmitt: An application of the shooting method to boundary value problems for second order delay equations. J. Math. Anal. Appl., v. 36, 1971, pp. 588-597. · Zbl 0219.34050 |

[5] | L. J. Grimm K. Schmitt: Boundary value problems for delay-differential equations. Bull. Amer. Math. Soc., v. 74, 1968, pp. 997-1000. · Zbl 0167.38504 |

[6] | L. J. Grimm K. Schmitt: Boundary value problems for differential equations with deviating arguments. Aequationes Math., v. 4, 1970, p. 176-190. · Zbl 0198.13201 |

[7] | G. A. Kamenskii S. B. Norkin L. E. Eľsgoľts: Some directions of investigation on the theory of differential equations with deviating arguments. (Russian). Trudy Sem. Tear. Diff. Urav. Otklon. Arg., v. 6, pp. 3-36. |

[8] | H. B. Keller: Numerical methods for two-point boundary-value problems. Blaisdel Publishing Company, Waltham 1968. · Zbl 0172.19503 |

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