## Journals

- Advances in Mathematics of Communications
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- Discrete & Continuous Dynamical Systems - A
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- Discrete & Continuous Dynamical Systems - S
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- Journal of Computational Dynamics
- Journal of Dynamics & Games
- Journal of Geometric Mechanics
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- Journal of Modern Dynamics
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- AIMS Mathematics
- Conference Publications
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### Open Access Journals

DCDS

Our aim in this article is to study the interaction of
\textit{boundary layers} and \textit{corner singularities} in the
context of singularly perturbed convection-diffusion equations. For
the problems under consideration, we determine a simplified form of
the corner singularities and show how to use it for the numerical
approximation of such problems in the context of variational
approximations using the concept of \textit{enriched spaces}.

DCDS

We study uncertainty bounds and statistics of wave solutions
through a random waveguide which possesses certain random
inhomogeneities. The waveguide is composed of several homogeneous
media with random interfaces. The main focus is on two homogeneous
media which are layered randomly and periodically in space.
Solutions of stochastic and deterministic problems are compared. The
waveguide media parameters pertaining to the latter are the averaged
values of the random parameters of the former. We investigate the
eigenmodes coupling due to random inhomogeneities in media, i.e.
random changes of the media parameters. We present an efficient
numerical method via Legendre Polynomial Chaos expansion for
obtaining output statistics including mean, variance and probability
distribution of the wave solutions. Based on the statistical
studies, we present uncertainty bounds and quantify the robustness
of the solutions with respect to random changes of interfaces.

CPAA

In this article, we establish the asymptotic behavior, when the
viscosity goes to zero, of the solutions of the Linearized Primitive
Equations (LPEs) in space dimension $2$. More precisely, we prove
that the LPEs solution behaves like the corresponding inviscid
problem solution inside the domain plus an explicit corrector
function in the neighborhood of some parts of the boundary. Two
cases are considered, the subcritical and supercritical modes
depending on the fact that the frequency mode is less or greater
than the ratio between the reference stratified flow (around which
we linearized) and the buoyancy frequency. The problem of boundary
layers for the LPEs is of a new type since the corresponding limit
problem displays a set of (unusual) nonlocal boundary conditions.

DCDS-S

In this article, we give an asymptotic expansion, with respect to
the viscosity which is considered here to be small, of the solutions
of the $3D$ linearized Primitive Equations (EPs) in a channel with
lateral periodicity. A rigorous convergence result, in some
physically relevant space, is proven. This allows, among other
consequences, to confirm the natural choice of the

*non-local*boundary conditions for the non-viscous PEs.
DCDS

We study the asymptotic behavior of the two dimensional Helmholtz scattering problem with high wave numbers in an exterior domain, the exterior of a circle. We impose the Dirichlet boundary condition on the obstacle, which corresponds to an incidental wave. For the outer boundary, we consider the Sommerfeld conditions. Using a polar coordinates expansion, the problem is reduced to a sequence of Bessel equations. Investigating the Bessel equations mode by mode, we find that the solution of the scattering problem converges to its limit solution at a specific rate depending on *k*.

keywords:
Electromagnetism
,
acoustics
,
Helmholtz equations
,
scattering problem
,
asymptotic analysis

DCDS

In this article,
we review recent progresses in boundary layer analysis of some
singular perturbation problems.
Using the techniques of differential geometry,
an asymptotic expansion of reaction-diffusion or heat equations in a domain with curved boundary
is constructed and validated in some suitable functional spaces.
In addition, we investigate the effect of curvature
as well as that
of an ill-prepared initial data.
Concerning convection-diffusion equations, the asymptotic behavior of their solutions
is difficult and delicate to analyze because it largely depends on the characteristics of the corresponding limit problems, which are first order hyperbolic differential equations. Thus, the boundary layer analysis is performed on relatively simpler domains,
typically intervals, rectangles, or circles.
We consider also the interior transition layers at the turning point characteristics in an interval domain and classical (ordinary), characteristic (parabolic) and corner (elliptic) boundary layers in a rectangular domain
using the technique of correctors and the tools of functional analysis.
The validity of our asymptotic expansions is also established in suitable spaces.

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