Showing papers by "Roderick Wong published in 2008"

Asymptotic expansions for riemann–hilbert problems

Abstract: Let Γ be a piecewise smooth contour in ℂ, which could be unbounded and may have points of self-intersection Let V(z, N) be a 2 × 2 matrix-valued function defined on Γ, which depends on a parameter N Consider a Riemann–Hilbert problem for a matrix-valued analytic function R(z, N) that satisfies a jump condition on the contour Γ with the jump matrix V(z, N) Assume that V(z, N) has an asymptotic expansion, as N → ∞, on Γ An elementary proof is given for the existence of a similar type of asymptotic expansion for the matrix solution R(z, N), as n → ∞, for z ∈ ℂ\Γ Our method makes use of only complex analysis

Global asymptotic expansions of the Laguerre polynomials—a Riemann–Hilbert approach

Abstract: By using the steepest descent method for Riemann–Hilbert problems introduced by Deift–Zhou (Ann Math 137:295–370, 1993), we derive two asymptotic expansions for the scaled Laguerre polynomial \(L^{(\alpha)}_n( u z)\) as n→∞, where ν=4n+2α+2. One expansion holds uniformly in a right half-plane \(\text{Re}\; z\geq \delta_1, 0<\delta_1<1\), which contains the critical point z=1; the other expansion holds uniformly in a left half-plane \(\text{Re}\; z\leq 1-\delta_2, 0<\delta_2<1-\delta_1\), which contains the other critical point z=0. The two half-planes together cover the entire complex z-plane. The critical points z=1 and z=0 correspond, respectively, to the turning point and the singularity of the differential equation satisfied by \(L^{(\alpha)}_n( u z)\).

On the connection formulas of the third Painlevé transcendent

Abstract: We consider the connection problem for the sine-Gordon PIII equation $u_{x x}+\frac{1}{x}u_{x}+\sin u=0,$ which is the most commonly studied case among all general third Painleve transcendents. The connection formulas are derived by the method of "uniform asymptotics" proposed by Bassom, Clarkson, Law and McLeod (Arch. Rat. Mech. Anal., 1998).

On the Number of Solutions to Carrier's Problem

Abstract: Let Nɛ denote the maximum number of spikes that a solution to Carrier's problem can have, where ɛ is a small positive parameter. We show that Nɛ is asymptotically equal to [K/ɛ], where K= 0.4725⋯, and the square brackets represent the greatest integer less than or equal to the quantity inside. If n(ɛ) stands for the number of solutions to this problem, then it is also shown that 4Nɛ− 3 ≤n(ɛ) ≤ 4Nɛ. Our approach is based on the shooting method used by Ou and Wong (Stud. Appl. Math. 111 (2003)) and on the construction of an envelope function for the minimum values of the solutions as ɛ approaches zero.

A tribute to Professor Philippe G. Ciarlet on his 70th birthday

Abstract: Professor Philippe G. Ciarlet was born on October 14, 1938, in Paris. He obtained his undergraduate degree in 1961 from the celebrated Ecole Polytechnique in Paris, followed by graduate studies (1962-1964) at the Ecole Nationale des Ponts et Chaussees in Paris. Professor Ciarlet received in 1966 his Ph.D. at the Case Institute of Technology, Cleveland, U.S.A., under the guidance of Professor Richard S. Varga. The title of his Ph.D. thesis is Variational Methods for Non-Linear Boundary-Value Problems . He continued with a Doctorat d'Etat Fonctions de Green Discretes et Principe du Maximum Discret ) at the University of Paris in 1971 and his advisor was Professor Jacques-Louis Lions. For more information please click the “Full Text” above.

Uniform Asymptotics of the Meixner Polynomials

Abstract: Using the steepest descent method of Deift-Zhou, we derive locally uniform asymptotic formulas for the Meixner polynomials. These include an asymptotic formula in a neighborhood of the origin, a result which as far as we are aware has not yet been obtained previously. This particular formula involves a special function, which is the uniformly bounded solution to a scalar Riemann-Hilbert problem, and which is asymptotically (as the polynomial degree $n$ tends to infinity) equal to the constant $"1"$ except at the origin. Numerical computation by using our formulas, and comparison with earlier results, are also given.

On a Nested Boundary-Layer Problem

Abstract: Nested boundary layers mean that one boundary layer lies inside another one. In this paper, we consider one such problem, namely, $\varepsilon^3xy''(x)+x^2y'(x)- (x^3+\varepsilon)y(x)=0$ with $0 < x <1$, $y(0) = 1$ and $y(1) = \sqrt{e}$. An asymptotic solution, which holds uniformly for $x\in [0,1]$, is constructed rigorously. This result also provides an explicit formula for the exponentially small leading term in the interval where the exact solution exhibits such behavior. This henomenon has never been mentioned in the existing literature.