Showing papers by "Roderick Wong published in 2004"
TL;DR: Two linearly independent asymptotic solutions are constructed for the second-order linear difference equation with respect to LaSalle's inequality.
Abstract: Two linearly independent asymptotic solutions are constructed for the second-order linear difference equation
34 citations
TL;DR: In this article, the authors derived asymptotic formulas for the solutions of the nonlinear differential equation ǫ + Q(ǫ) = 0 with boundary conditions u(-1) =u(1)/ǫ(1) or u′ −1 =u′(1/ǫ)) = 0.
Abstract: Asymptotic formulas, as ɛ 0+, are derived for the solutions of the nonlinear differential equation ɛu″+Q(u) = 0 with boundary conditions u(-1) =u(1) = 0 or u′(-1) =u′(1) = 0. The nonlinear term Q(u) behaves like a cubic; it vanishes at s-, 0, s+ and nowhere else in [s-, s+], where s- 0 and the integral of Q on the interval [s-, s+] is zero. Solutions to these boundary-value problems are shown to exhibit internal shock layers, and the error terms in the asymptotic approximations are demonstrated to be exponentially small. Estimates are obtained for the number of internal shocks that a solution can have, and the total numbers of solutions to these problems are also given. All results here are established rigorously in the mathematical sense.
15 citations
TL;DR: In this paper, an asymptotic expansion for the Krawtchouk polynomials was derived for a fixed number, where x is the number of vertices.
Abstract: Let \(K_{n}^{N}(x;p,q)\) be the Krawtchouk polynomials and μ = N/n. An asymptotic expansion is derived for \(K_{n}^{N}(x;p,q)\), when x is a fixed number. This expansion holds uniformly for μ in [1,∞), and is given in terms of the confluent hypergeometric functions. Asymptotic approximations are also obtained for the zeros of \(K_{n}^{N}(x;p,q)\) in various cases depending on the values of p, q and μ.
13 citations
TL;DR: In this article, an asymptotic formula for the Jacobi polynomial Pnα,β) in the interval of orthogonality [1,1] with that outside the interval is given.
Abstract: An asymptotic formula is found that links the behaviour of the Jacobi polynomial Pnα,β)(z) in the interval of orthogonality [–1,1] with that outside the interval. The two infinite series involved in this formula are shown to be exponentially improved asymptotic expansions. The method used in this paper can also be adopted in other cases of orthogonal polynomials such as Hermite and Laguerre.
12 citations