121 research outputs found
Structures and waves in a nonlinear heat-conducting medium
The paper is an overview of the main contributions of a Bulgarian team of
researchers to the problem of finding the possible structures and waves in the
open nonlinear heat conducting medium, described by a reaction-diffusion
equation. Being posed and actively worked out by the Russian school of A. A.
Samarskii and S.P. Kurdyumov since the seventies of the last century, this
problem still contains open and challenging questions.Comment: 23 pages, 13 figures, the final publication will appear in Springer
Proceedings in Mathematics and Statistics, Numerical Methods for PDEs:
Theory, Algorithms and their Application
Regularized extremal shift in problems of stable control
We discuss a technical approach, based on the method of regularized extremal shift (RES), intended to help solve problems of stable control of uncertain dynamical systems. Our goal is to demonstrate the essence and abilities of the RES technique; for this purpose we construct feedback controller for approximate tracking a prescribed trajectory of an inaccurately observed system described by a parabolic equation. The controller is "resource-saving" in a sense that control resource spent for approximate tracking do not exceed those needed for tracking in an "ideal" situation where the current values of the input disturbance are fully observable. © 2013 IFIP International Federation for Information Processing.German Sci. Found. (DFG) Eur. Sci. Found. (ESF);Natl. Inst. Res. Comput. Sci. Control France (INRIA);DFG Research Center MATHEON;Weierstrass Institute for Applied Analysis and Stochastics (WIAS);European Patent Offic
A particle system with explosions: law of large numbers for the density of particles and the blow-up time
Consider a system of independent random walks in the discrete torus with
creation-annihilation of particles and possible explosion of the total number
of particles in finite time. Rescaling space and rates for
diffusion/creation/annihilation of particles, we obtain a stong law of large
numbers for the density of particles in the supremum norm. The limiting object
is a classical solution to the semilinear heat equation u_t =u_{xx} + f(u). If
f(u)=u^p, 1<p \le 3, we also obtain a law of large numbers for the explosion
time
Large negative velocity gradients in Burgers turbulence
We consider 1D Burgers equation driven by large-scale white-in-time random
force. The tails of the velocity gradients probability distribution function
(PDF) are analyzed by saddle-point approximation in the path integral
describing the velocity statistics. The structure of the saddle-point
(instanton), that is velocity field configuration realizing the maximum of
probability, is studied numerically in details. The numerical results allow us
to find analytical solution for the long-time part of the instanton. Its
careful analysis confirms the result of [Phys. Rev. Lett. 78 (8) 1452 (1997)
[chao-dyn/9609005]] based on short-time estimations that the left tail of PDF
has the form ln P(u_x) \propto -|u_x|^(3/2).Comment: 10 pages, RevTeX, 10 figure
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