Asymptotic Analysis and Applications
Org:
Dr. Chunhua Ou (Memorial University of Newfoundland) and
Dr. XiangSheng Wang (University of Louisiana at Lafayette)
[
PDF]
 JUNHO CHOI, UNIST
On Boundary Layers for the Burgers Equations in a Bounded Domain [PDF]

As a simplified model derived from the NavierStokes equations, we consider the viscous Burgers equations in a bounded domain with twopoint boundary conditions.
\begin{align}
\begin{split}
&u^\epsilon_t\epsilon u_{xx}+\frac{(u^\epsilon)^2}{2}=f(x,t),\quad x\in (0,1),\quad t\geq 0\&u^\epsilon(x,0)=u_0(x),\quad x\in(0,1),\&u^\epsilon(0,t)=g(t),\quad t\geq 0,\&u^\epsilon(1,t)=h(t),\quad t\geq 0.
\end{split}
\end{align}
We investigate the singular behaviors of their solutions $u^\epsilon$ as the viscosity parameter $\epsilon$ gets smaller. Indeed, when $\epsilon$ gets smaller, $u^\epsilon_x$ has $1/\epsilon$ order slope. So controlling the sharp slopes is one of the most important parts in this research.
The idea is constructing the asymptotic expansions in the order of the $\epsilon$ and validating the convergence of the expansions to the solutions $u^\epsilon$ as $\epsilon\rightarrow 0$ in $L^2(0,T;H^1((0,1)))$ space. In this article, we consider the case where sharp slopes occur at the boundaries, i.e. boundary layers, and we fully analyse the convergence at any order of $\epsilon$ using the socalled boundary layer correctors as follows.
In the end, we also numerically verify the convergences.
 HOWARD COHL, National Institute of Standards and Technology
Asymptotics of Fundamental Solutions for Helmholtz operators on Spaces of Constant Curvature [PDF]

We compute closedform expressions for oscillatory and damped spherically symmetric fundamental solutions of the Helmholtz equation in ddimensional hyperbolic and hyperspherical geometry. We are using the Rradius hypersphere and Rradius hyperboloid model of hyperbolic geometry. These models represent Riemannian manifolds with positive constant and negative constant sectional curvature respectively. Flatspace limits with their corresponding asymptotic representations, are used to restrict proportionality constants for these fundamental solutions. In order to accomplish this, we summarize and derive new large degree asymptotics for associated Legendre and Ferrers functions of the first and second kind. Furthermore, we prove that our fundamental solutions on the hyperboloid are unique due to their decay at infinity. To derive Gegenbauer polynomial expansions of our fundamental solutions for Helmholtz operators on hyperspheres and hyperboloids, we derive a collection of infinite series addition theorems for Ferrers and associated Legendre functions which are generalizations and extensions of the addition theorem for Gegenbauer polynomials. Using these addition theorems, in geodesic polar coordinates for dimensions greater than or equal to three, we compute Gegenbauer polynomial expansions for these fundamental solutions, and azimuthal Fourier expansions in twodimensions.
 DAN DAI, City University of Hong Kong
Gap probability at the hard edge for random matrix ensembles with pole singularities in the potential [PDF]

We study the Fredholm determinant of an integrable operator acting on the interval $(0,s)$ whose kernel is constructed out of the $\Psi$function associated with a hierarchy of higher order analogues to the Painlev\'e III equation. This Fredholm determinant describes the critical behavior of the eigenvalue gap probability at the hard edge of unitary invariant random matrix ensembles perturbed
by poles of order $k$ in a certain scaling regime. Using the RiemannHilbert method, we obtain the large $s$ asymptotics of the Fredholm determinant. Moreover, we derive a Painlev\'e type formula of the Fredholm determinant, which is expressed in terms of an explicit integral involving a solution to a coupled Painlev\'e III system.
This is a joint work with ShuaiXia Xu and Lun Zhang.
 MOURAD ISMAIL, University of Central Florida
The qNormal Distribution [PDF]

We point out some unusual properties of the weight function of the qHermite polynomials observed by P. Szablowski and X. Yang. We provide rigorous proofs and show that many other polynomials of the qAskey Scheme share the same properties. We also indicate how this leads to general questions about zeros of complex orthogonal polynomials.
 XIANGSHENG WANG, University of Louisiana at Lafayette
Asymptotic analysis of difference equations [PDF]

In this talk, we will present some preliminary results on the asymptotic analysis of Wilson polynomials and $q$orthogonal polynomials via difference equations. The first result is based on an ongoing joint work with Prof. YuTian Li and Prof. Roderick Wong. The second result is based on an ongoing joint work with Prof. Dan Dai and Prof. Mourad Ismail.
 MICHAEL WARD, University of British Columbia
The Stability of Hotspot Patterns for a Continuum Model of Urban Crime and the Effect of Police Intervention [PDF]

In a singularly perturbed limit, we analyze the existence and linear
stability of steadystate localized hotspot solutions for the 1D
threecomponent reactiondiffusion (RD) system formulated and studied
numerically in Jones et.~al.~[Math. Models. Meth. Appl. Sci.,
\textbf{20}, Suppl., (2010)], which models urban crime with police
intervention. In our model, the field variables are the
attractiveness field for burglary, the criminal density, and the
police density, and it includes a scalar parameter that determines the
strength of the police drift towards maxima of the attractiveness
field. For a special choice of this parameter, we recover the
``copsonthedots'' policing strategy of Jones et.~al., where the
police mimic the drift of the criminals towards maxima of the
attractiveness field.
For this model we develop a spectral theory based on the analysis of
of nonlocal eigenvalue problems to provide phase diagrams in parameter
space characterizing the linear stability of hotspot patterns. In one
particular parameter regime, the hotspot steadystates
are shown to be unstable to asynchronous oscillatory instabilities in
the hotspot amplitudes arising from a Hopf bifurcation. Within the
context of our model, this provides a parameter range where the effect
of a copsonthedots policing strategy is to only displace crime
temporally between neighboring spatial regions. In other parameter
regimes, we show that new hotspots of criminal activity can be
nucleated in low crime regions when the spatial extent of these
quiescent regions exceeds a critical threshold.
Both the mathematical challenges in the linear stability analysis, and the
qualitative interpretation of our results are highlighted.
 RODERICK WONG, City University of Hong Kong
Asymptotics of the associated Pollaczek polynomials (Joint with MinJie Luo) [PDF]

In this talk, we present the large$n$ behavior of the associated Pollaczek polynomials $P_{n}^{\lambda}\left(z;a,b,c\right)$. These polynomials involve four real parameters $\lambda$, $a$, $b$ and $c$, in addition to the complex variable $z$. Asymptotic formulas are derived for these polynomials, when $z$ lies in the complex plane bounded away from the interval of orthogonality $\left(1,1\right)$, as well as in the interior of the interval of orthogonality.
 SHUAIXIA XU, Sun Yatsen Univesity
Gap probability in critical unitary random matrix ensembles and the coupled Painlev\'{e} II system [PDF]

We study Fredholm determinants of the Painlev\'{e} II and Painlev\'{e} XXXIV kernels. In certain critical unitary random matrix ensembles, these determinants describe special gap probabilities of eigenvalues. We obtain TracyWidom formulas for the Fredholm determinants, which are explicitly given in terms of integrals involving a family of distinguished solutions to the coupled Painlev\'{e} II system in dimension four. Moreover, the large gap asymptotics for these Fredholm determinants are derived, where the constant terms are given explicitly in terms of the Riemann zetafunction. This talk is based on a joint work with Dan Dai.
 RUIMING ZHANG, Northwest Agriculture and Forestry University
Asymptotics of Theta Functions [PDF]

In this talk we present two asymptotic expansions of theta functions with respect to two different scalings, both of them have exponential remainders. We also apply the asymptotics to compute asymptotic behaviors of partial sums of elliptic hypergeometric series.