3,309 research outputs found
Periodic perturbations of constrained motion problems on a class of implicitly defined manifolds
We study forced oscillations on differentiable manifolds which are globally
defined as the zero set of appropriate smooth maps in some Euclidean spaces.
Given a T-periodic perturbative forcing field, we consider the two different
scenarios of a nontrivial unperturbed force field and of perturbation of the
zero field. We provide simple, degree-theoretic conditions for the existence of
branches of T-periodic solutions. We apply our construction to a class of
second order Differential-Algebraic Equations.Comment: 15 pages, 2 figures, to appear in Communications in Contemporary
Mathematics. arXiv admin note: substantial text overlap with arXiv:1102.156
Harmonic solutions to a class of differential-algebraic equations with separated variables
We study the set of T-periodic solutions of a class of T-periodically
perturbed Differential-Algebraic Equations with separated variables. Under
suitable hypotheses, these equations are equivalent to separated variables ODEs
on a manifold. By combining known results on Differential-Algebraic Equations,
with an argument about ODEs on manifolds, we obtain a global continuation
result for the T-periodic solutions to the considered equations. As an
application of our method, a multiplicity result is provided
Spatial Manifestations of Order Reduction in Runge-Kutta Methods for Initial Boundary Value Problems
This paper studies the spatial manifestations of order reduction that occur
when time-stepping initial-boundary-value problems (IBVPs) with high-order
Runge-Kutta methods. For such IBVPs, geometric structures arise that do not
have an analog in ODE IVPs: boundary layers appear, induced by a mismatch
between the approximation error in the interior and at the boundaries. To
understand those boundary layers, an analysis of the modes of the numerical
scheme is conducted, which explains under which circumstances boundary layers
persist over many time steps. Based on this, two remedies to order reduction
are studied: first, a new condition on the Butcher tableau, called weak stage
order, that is compatible with diagonally implicit Runge-Kutta schemes; and
second, the impact of modified boundary conditions on the boundary layer theory
is analyzed.Comment: 41 pages, 9 figure
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