Computing Validated Solutions of Implicit Differential Equations
AbstractOrdinary differential equations (ODEs), including high-order implicit equations, describe
important problems in mechanical and chemical engineering. However, the use of selfvalidated
methods providing rigorous enclosures of the solution has mostly been limited to
explicit and weakly nonlinear problems, and no general-purpose algorithm for the validated
integration of general ODE initial value problems has been developed. Since most integration
techniques for Differential Algebraic Equations (DAEs) are based on transformation to implicit
ODEs, the integration of DAE initial value problems has traditionally been restricted to
few hand-picked problems from the relatively small class of low-index systems. The recently
developed Taylor model method combines high-order differential algebraic descriptions of
functional dependencies with intervals for verification. It has proven its power in several applications,
including verified integration of ODEs under avoidance of the wrapping effect.
Recognizing antiderivation (integration) as a natural operation on Taylor models yields methods
that treat DEs within a fully differential algebraic context as implicit equations made of
conventional functions and antiderivation. This method has the potential to be applied to highindex
DAE problems and allows the computation of guaranteed enclosures of final conditions
from large initial regions for large classes of initial value problems. In the framework of this
method, a Taylor model represents the highest derivative of the solution function occurring in
the DE and all lower derivatives are treated as antiderivatives of this Taylor model. Consequently,
one obtains a set of implicit equations involving only the highest derivative. Utilizing
methods of verified inversion of functional dependencies described by Taylor models allows
the computation of a guaranteed enclosure of the solution in the form of a Taylor model. The
performance of the method is illustrated by detailed examples.
J. Hoefkens, M. Berz, K. Makino, Advances in Computational Mathematics 19 (2003) 231-253
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