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2013 | 23 | 4 | 731-747

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A verified method for solving piecewise smooth initial value problems

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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.








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  • Department of Computer Science and Applied Cognitive Science, University of Duisburg-Essen, 47048 Duisburg, Germany
  • Department of Computer Science and Applied Cognitive Science, University of Duisburg-Essen, 47048 Duisburg, Germany
  • Chair of Mechatronics, University of Rostock, Justus-von-Liebig-Weg 6, 18059 Rostock, Germany


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