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Initial Value Problems and Signature Change
We make a rigorous study of classical field equations on a 2-dimensional
signature changing spacetime using the techniques of operator theory. Boundary
conditions at the surface of signature change are determined by forming
self-adjoint extensions of the Schr\"odinger Hamiltonian. We show that the
initial value problem for the Klein--Gordon equation on this spacetime is
ill-posed in the sense that its solutions are unstable. Furthermore, if the
initial data is smooth and compactly supported away from the surface of
signature change, the solution has divergent -norm after finite time.Comment: 33 pages, LaTeX The introduction has been altered, and new work
(relating our previous results to continuous signature change) has been
include
Observability for Initial Value Problems with Sparse Initial Data
In this work we introduce the concept of -sparse observability for large
systems of ordinary differential equations. Let be such a
system. At time , suppose we make a set of observations of the
solution of the system with initial data , where is a matrix
satisfying the restricted isometry property. The aim of this paper is to give
answers to the following questions: Given the observations , is
uniquely determined knowing that is sufficiently sparse? Is there any way
to reconstruct such a sparse initial data ?Comment: Submitted to Applied Mathematics Letters in November 2009 (status:
under review)
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Multiderivative methods for periodic initial value problems
A family of two-step multiderivative methods based on Pade approximants to the exponential function is developed. The methods are analysed and periodicity intervals in PECE mode are calculated.
Two of the methods are tested on two problems from the literature and one predictor-corrector combination is tested on two further problems
Fully discrete hyperbolic initial boundary value problems with nonzero initial data
The stability theory for hyperbolic initial boundary value problems relies
most of the time on the Laplace transform with respect to the time variable.
For technical reasons, this usually restricts the validity of stability
estimates to the case of zero initial data. In this article, we consider the
class of non-glancing finite difference approximations to the hyperbolic
operator. We show that the maximal stability estimates that are known for zero
initial data and nonzero boundary source term extend to the case of nonzero
initial data in \^a 2 . The main novelty of our approach is to cover finite
difference schemes with an arbitrary number of time levels. As an easy
corollary of our main trace estimate, we recover former stability results in
the semigroup sense by Kreiss [Kre68] and Osher [Osh69b]
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