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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