572 research outputs found
Stability analysis for combustion fronts traveling in hydraulically resistant porous media
We study front solutions of a system that models combustion in highly
hydraulically resistant porous media. The spectral stability of the fronts is
tackled by a combination of energy estimates and numerical Evans function
computations. Our results suggest that there is a parameter regime for which
there are no unstable eigenvalues. We use recent works about partially
parabolic systems to prove that in the absence of unstable eigenvalues the
fronts are convectively stable.Comment: 21 pages, 4 figure
Exponential Mixing for Retarded Stochastic Differential Equations
In this paper, we discuss exponential mixing property for Markovian
semigroups generated by segment processes associated with several class of
retarded Stochastic Differential Equations (SDEs) which cover SDEs with
constant/variable/distributed time-lags. In particular, we investigate the
exponential mixing property for (a) non-autonomous retarded SDEs by the
Arzel\`{a}--Ascoli tightness characterization of the space \C equipped with
the uniform topology (b) neutral SDEs with continuous sample paths by a
generalized Razumikhin-type argument and a stability-in-distribution approach
and (c) jump-diffusion retarded SDEs by the Kurtz criterion of tightness for
the space \D endowed with the Skorohod topology.Comment: 20 page
Multi-Adaptive Time-Integration
Time integration of ODEs or time-dependent PDEs with required resolution of
the fastest time scales of the system, can be very costly if the system
exhibits multiple time scales of different magnitudes. If the different time
scales are localised to different components, corresponding to localisation in
space for a PDE, efficient time integration thus requires that we use different
time steps for different components.
We present an overview of the multi-adaptive Galerkin methods mcG(q) and
mdG(q) recently introduced in a series of papers by the author. In these
methods, the time step sequence is selected individually and adaptively for
each component, based on an a posteriori error estimate of the global error.
The multi-adaptive methods require the solution of large systems of nonlinear
algebraic equations which are solved using explicit-type iterative solvers
(fixed point iteration). If the system is stiff, these iterations may fail to
converge, corresponding to the well-known fact that standard explicit methods
are inefficient for stiff systems. To resolve this problem, we present an
adaptive strategy for explicit time integration of stiff ODEs, in which the
explicit method is adaptively stabilised by a small number of small,
stabilising time steps
Modelling elastic structures with strong nonlinearities with application to stick-slip friction
An exact transformation method is introduced that reduces the governing
equations of a continuum structure coupled to strong nonlinearities to a low
dimensional equation with memory. The method is general and well suited to
problems with point discontinuities such as friction and impact at point
contact. It is assumed that the structure is composed of two parts: a continuum
but linear structure and finitely many discrete but strong nonlinearites acting
at various contact points of the elastic structure. The localised
nonlinearities include discontinuities, e.g., the Coulomb friction law. Despite
the discontinuities in the model, we demonstrate that contact forces are
Lipschitz continuous in time at the onset of sticking for certain classes of
structures. The general formalism is illustrated for a continuum elastic body
coupled to a Coulomb-like friction model
Self-Evaluation Applied Mathematics 2003-2008 University of Twente
This report contains the self-study for the research assessment of the Department of Applied Mathematics (AM) of the Faculty of Electrical Engineering, Mathematics and Computer Science (EEMCS) at the University of Twente (UT). The report provides the information for the Research Assessment Committee for Applied Mathematics, dealing with mathematical sciences at the three universities of technology in the Netherlands. It describes the state of affairs pertaining to the period 1 January 2003 to 31 December 2008
Linearized Stability of Partial Differential Equations with Application to Stabilization of the Kuramoto--Sivashinsky Equation
This is a final draft of a work, prior to publisher editing and production, that appears in Siam J. Control Optim. Vol. 56, No 1, pp 120-147. http://dx.doi.org/10.1137/140993417.Linearization is a useful tool for analyzing the stability of nonlinear differential equations. Unfortunately, the proof of the validity of this approach for ordinary differential equations does not generalize to all nonlinear partial differential equations. General results giving conditions for when stability (or instability) of the linearized equation implies the same for the nonlinear equation are given here. These results are applied to stability and stabilization of the Kuramoto--Sivashinsky equation, a nonlinear partial differential equation that models reaction-diffusion systems. The stability of the equilibrium solutions depends on the value of a positive parameter . It is shown that if , then the set of constant equilibrium solutions is globally asymptotically stable. If , then the equilibria are unstable. It is also shown that stabilizing the linearized equation implies local exponential stability of the equation. Stabilization of the Kuramoto--Sivashinsky equation using a single distributed control is considered and it is described how to use a finite-dimensional approximation to construct a stabilizing controller. The results are illustrated with simulations.Natural Sciences and Engineering Research Council of Canada (NSERC
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