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Interval bounds on the solutions of semi-explicit index-one DAEs. Part 2: computation

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Abstract

This article presents two methods for computing interval bounds on the solutions of nonlinear, semi-explicit, index-one differential-algebraic equations (DAEs). Part 1 presents theoretical developments, while Part 2 discusses implementation and numerical examples. The primary theoretical contributions are (1) an interval inclusion test for existence and uniqueness of a solution, and (2) sufficient conditions, in terms of differential inequalities, for two functions to describe componentwise upper and lower bounds on this solution, point-wise in the independent variable. The first proposed method applies these results sequentially in a two-phase algorithm analogous to validated integration methods for ordinary differential equations (ODEs). The second method unifies these steps to characterize bounds as the solutions of an auxiliary system of DAEs. Efficient implementations of both are described using interval computations and demonstrated on numerical examples.

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Acknowledgments

This paper is based on work funded by the National Science Foundation under grant CBET-0933095.

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  1. Massachusetts Institute of Technology, 77 Massachusetts Ave 66-363, Cambridge, MA, 02139, USA

    Joseph K. Scott

  2. Massachusetts Institute of Technology, 77 Massachusetts Ave 66-464, Cambridge, MA, 02139, USA

    Paul I. Barton

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  1. Joseph K. Scott
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Correspondence to Paul I. Barton.

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Scott, J.K., Barton, P.I. Interval bounds on the solutions of semi-explicit index-one DAEs. Part 2: computation. Numer. Math. 125, 27–60 (2013). https://doi.org/10.1007/s00211-013-0532-x

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