# A verified method for solving piecewise smooth initial value problems

Ekaterina Auer; Stefan Kiel; Andreas Rauh

International Journal of Applied Mathematics and Computer Science (2013)

- Volume: 23, Issue: 4, page 731-747
- ISSN: 1641-876X

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topEkaterina Auer, Stefan Kiel, and Andreas Rauh. "A verified method for solving piecewise smooth initial value problems." International Journal of Applied Mathematics and Computer Science 23.4 (2013): 731-747. <http://eudml.org/doc/262316>.

@article{EkaterinaAuer2013,

abstract = {In many applications, there is a need to choose mathematical models that depend on non-smooth functions. The task of simulation becomes especially difficult if such functions appear on the right-hand side of an initial value problem. Moreover, solution processes from usual numerics are sensitive to roundoff errors so that verified analysis might be more useful if a guarantee of correctness is required or if the system model is influenced by uncertainty. In this paper, we provide a short overview of possibilities to formulate non-smooth problems and point out connections between the traditional non-smooth theory and interval analysis. Moreover, we summarize already existing verified methods for solving initial value problems with non-smooth (in fact, even not absolutely continuous) right-hand sides and propose a way of handling a certain practically relevant subclass of such systems. We implement the approach for the solver VAL E NC IA-IVP by introducing into it a specialized template for enclosing the first-order derivatives of non-smooth functions. We demonstrate the applicability of our technique using a mechanical system model with friction and hysteresis. We conclude the paper by giving a perspective on future research directions in this area.},

author = {Ekaterina Auer, Stefan Kiel, Andreas Rauh},

journal = {International Journal of Applied Mathematics and Computer Science},

keywords = {interval methods; non-smooth systems; initial value problems; numerical examples; friction; hysteresis},

language = {eng},

number = {4},

pages = {731-747},

title = {A verified method for solving piecewise smooth initial value problems},

url = {http://eudml.org/doc/262316},

volume = {23},

year = {2013},

}

TY - JOUR

AU - Ekaterina Auer

AU - Stefan Kiel

AU - Andreas Rauh

TI - A verified method for solving piecewise smooth initial value problems

JO - International Journal of Applied Mathematics and Computer Science

PY - 2013

VL - 23

IS - 4

SP - 731

EP - 747

AB - In many applications, there is a need to choose mathematical models that depend on non-smooth functions. The task of simulation becomes especially difficult if such functions appear on the right-hand side of an initial value problem. Moreover, solution processes from usual numerics are sensitive to roundoff errors so that verified analysis might be more useful if a guarantee of correctness is required or if the system model is influenced by uncertainty. In this paper, we provide a short overview of possibilities to formulate non-smooth problems and point out connections between the traditional non-smooth theory and interval analysis. Moreover, we summarize already existing verified methods for solving initial value problems with non-smooth (in fact, even not absolutely continuous) right-hand sides and propose a way of handling a certain practically relevant subclass of such systems. We implement the approach for the solver VAL E NC IA-IVP by introducing into it a specialized template for enclosing the first-order derivatives of non-smooth functions. We demonstrate the applicability of our technique using a mechanical system model with friction and hysteresis. We conclude the paper by giving a perspective on future research directions in this area.

LA - eng

KW - interval methods; non-smooth systems; initial value problems; numerical examples; friction; hysteresis

UR - http://eudml.org/doc/262316

ER -

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