208 research outputs found
Computability of differential equations
In this chapter, we provide a survey of results concerning the computability and computational complexity of differential equations. In particular, we study the conditions which ensure computability of the solution to an initial value problem for an ordinary differential equation (ODE) and analyze the computational complexity of a computable solution. We also present computability results concerning the asymptotic behaviors of ODEs as well as several classically important partial differential equations.info:eu-repo/semantics/acceptedVersio
Numerical homogenization of time-dependent Maxwell\u27s equations with dispersion effects
This thesis studies the propagation of electromagnetic waves in heterogeneous structures such as metamaterials. The governing equations for this problem are Maxwell\u27s equations with highly oscillatory parameters. We use an analytic homogenization result which yields an effective Maxwell system that involves a convolution integral. This convolution represents dispersive effects that result from the interaction of the wave with the (locally) periodic microscopic structure.
We discretize in space using the Finite Element Heterogeneous Multiscale Method (FE-HMM) and provide a semi-discrete error estimate. The rigorous error analysis in space is supplemented by a rather standard time discretization at the end of which an efficient, fully discrete method is proposed. This method uses a recursive approximation of the convolution that relies on the assumption that the convolution kernel is an exponential function. Eventually, we present numerical experiments both for the microscopic and the macroscopic scale
Time-fractional Cahn-Hilliard equation: Well-posedness, degeneracy, and numerical solutions
In this paper, we derive the time-fractional Cahn-Hilliard equation from
continuum mixture theory with a modification of Fick's law of diffusion. This
model describes the process of phase separation with nonlocal memory effects.
We analyze the existence, uniqueness, and regularity of weak solutions of the
time-fractional Cahn-Hilliard equation. In this regard, we consider
degenerating mobility functions and free energies of Landau, Flory--Huggins and
double-obstacle type. We apply the Faedo-Galerkin method to the system, derive
energy estimates, and use compactness theorems to pass to the limit in the
discrete form. In order to compensate for the missing chain rule of fractional
derivatives, we prove a fractional chain inequality for semiconvex functions.
The work concludes with numerical simulations and a sensitivity analysis
showing the influence of the fractional power. Here, we consider a convolution
quadrature scheme for the time-fractional component, and use a mixed finite
element method for the space discretization
Global attractor and asymptotic dynamics in the Kuramoto model for coupled noisy phase oscillators
We study the dynamics of the large N limit of the Kuramoto model of coupled
phase oscillators, subject to white noise. We introduce the notion of shadow
inertial manifold and we prove their existence for this model, supporting the
fact that the long term dynamics of this model is finite dimensional. Following
this, we prove that the global attractor of this model takes one of two forms.
When coupling strength is below a critical value, the global attractor is a
single equilibrium point corresponding to an incoherent state. Conversely, when
coupling strength is beyond this critical value, the global attractor is a
two-dimensional disk composed of radial trajectories connecting a saddle
equilibrium (the incoherent state) to an invariant closed curve of locally
stable equilibria (partially synchronized state). Our analysis hinges, on the
one hand, upon sharp existence and uniqueness results and their consequence for
the existence of a global attractor, and, on the other hand, on the study of
the dynamics in the vicinity of the incoherent and synchronized equilibria. We
prove in particular non-linear stability of each synchronized equilibrium, and
normal hyperbolicity of the set of such equilibria. We explore mathematically
and numerically several properties of the global attractor, in particular we
discuss the limit of this attractor as noise intensity decreases to zero.Comment: revised version, 28 pages, 4 figure
Rigorous derivation of nonlinear scalar conservation laws from follow-the-leader type models via many particle limit
We prove that the unique entropy solution to a scalar nonlinear conservation
law with strictly monotone velocity and nonnegative initial condition can be
rigorously obtained as the large particle limit of a microscopic
follow-the-leader type model, which is interpreted as the discrete Lagrangian
approximation of the nonlinear scalar conservation law. More precisely, we
prove that the empirical measure (respectively the discretised density)
obtained from the follow-the-leader system converges in the 1-Wasserstein
topology (respectively in ) to the unique Kruzkov entropy solution
of the conservation law. The initial data are taken in ,
nonnegative, and with compact support, hence we are able to handle densities
with vacuum. Our result holds for a reasonably general class of velocity maps
(including all the relevant examples in the applications, e.g. in the
Lighthill-Whitham-Richards model for traffic flow) with possible degenerate
slope near the vacuum state. The proof of the result is based on discrete BV
estimates and on a discrete version of the one-sided Oleinik-type condition. In
particular, we prove that the regularizing effect
for nonlinear scalar conservation laws is intrinsic of the discrete model
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