Differential equations with piecewise approximate coefficients: Discrete and continuous estimation for initial and boundary value problems.

*(English)*Zbl 0763.65053Given a linear second order differential equation together with initial and boundary conditions which guarantee the existence of a unique solution, the authors study the effect of perturbations of the original coefficients of the differential equation on the solution of the corresponding initial or boundary value problems.

As remarked in the paper such a study is interesting not only in perturbation theories but also in the numerical solution of initial and boundary value problems by means of Tau and Galerkin type methods. Since it is common in some numerical methods to replace the coefficients of the differential equation by piecewise continuous approximations defined in a suitable way, a natural question is to bound or estimate the effect of such a replacement of the solution.

Here, assuming piecewise approximate coefficients, the authors derive error estimates for both the solution and its derivative of discrete and continuous initial and boundary value problems. Finally, a number of numerical examples to show the quality of the estimates are presented.

As remarked in the paper such a study is interesting not only in perturbation theories but also in the numerical solution of initial and boundary value problems by means of Tau and Galerkin type methods. Since it is common in some numerical methods to replace the coefficients of the differential equation by piecewise continuous approximations defined in a suitable way, a natural question is to bound or estimate the effect of such a replacement of the solution.

Here, assuming piecewise approximate coefficients, the authors derive error estimates for both the solution and its derivative of discrete and continuous initial and boundary value problems. Finally, a number of numerical examples to show the quality of the estimates are presented.

Reviewer: M.Calvo (Zaragoza)

##### MSC:

65L05 | Numerical methods for initial value problems |

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

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

34A30 | Linear ordinary differential equations and systems, general |

34B05 | Linear boundary value problems for ordinary differential equations |

34E10 | Perturbations, asymptotics of solutions to ordinary differential equations |

##### Keywords:

Tau method; Galerkin method; linear second order differential equation; effect of perturbations; error estimates; numerical examples
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\textit{M. K. El-Daou} et al., Comput. Math. Appl. 24, No. 4, 33--47 (1992; Zbl 0763.65053)

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