On the Exponential Convergence of Spline Approximation Methods for Wiener-Hopf Equations

This paper deals with the study of some new multidimensional operators of fractional calculus. These operators are considered in certain spaces of test and generalized functions. Some mapping properties and product rules are obtained.     

On the Prandtl Boundary Layer Equations in Presence of Corner Singularities

In this paper we prove the well-posedness of the Prandtl boundary layer equations on a periodic strip when the initial and the boundary data are not assigned to be compatible.     

Mean Convergence of Extended Lagrange Interpolation for Exponential Weights

In this paper, we complete our investigations of mean convergence of Lagrange interpolation for fast decaying even and smooth exponential weights on the line. In doing so, we also present a summary of recent related work on the line and [–1,1] by the authors, Szabados, Vertesi, Lubinsky and Matjila. We also emphasize the important and fundamental ideas, applied in our proofs, that were developed by Erds, Turan, Askey, Freud, Nevai, Szabados, Vértesi and their students and collaborators. These methods include forward quadrature estimates, orthogonal expansions, Hilbert transforms, bounds on Lebesgue functions and the uniform boundedness principle.     

Exponential Convergence of Time-Delay Systems in the Presence of Bounded Disturbances

In this paper, the problem of control design for exponential convergence of state/input delay systems with bounded disturbances is considered. Based on the Lyapunov–Krasovskii method combining with the delay-decomposition technique, a new sufficient condition is proposed for the existence of a state feedback controller, which guarantees that all solutions of the closed-loop system converge exponentially (with a pre-specified convergence rate) within a ball whose radius is minimized. The obtained condition is given in terms of matrix inequalities with one parameter needing to be tuned, which can be solved by using a one-dimensional search method with Matlab’s LMI Toolbox to minimize the radius of the ball. Two numerical examples are given to illustrate the superiority of the proposed method.     

Improved Exponential Convergence Rates by Oversampling Near the Boundary

Sampling inequalities for smooth functions bound a continuous norm in terms of a discretized norm and an error term that tends to zero exponentially as the discrete data set becomes dense. Improved estimates are derived for discrete point sets that cluster near the boundary, in particular for scattered point sets that are distributed quadratically in a boundary layer, and for tensorized Chebyshev grids. If applied to residuals of stable reconstruction processes, such inequalities yield exponential convergence orders. Our results agree with the observation that exponential deterministic approximation rates are often improved globally if the data sets are distributed more densely near the boundary.     

Generalized Spline Interpolation of Functions with Large Gradients in Boundary Layers

Computational Mathematics and Mathematical Physics - The spline interpolation of functions of one variable with large gradients in boundary layers is studied. It is well known that applying...     

On a Boundary Layer Problem

This is a continuation of our earlier article concerning the boundary-value problem     where A , B are prescribed constants, and 0 < ε ≪ 1 is a small positive parameter. In that article, we assumed the coefficients a ( x ) and b ( x ) are sufficiently smooth functions with the behavior given by a ( x ) ∼ αx and b ( x ) ∼ β as x → 0, where α > 0 and β / α ≠ 1, 2, 3,…. In the present article, we are concerned with the case α < 0 and β / α ≠ 0, −1, −2,…. An asymptotic solution is obtained for the problem, which holds uniformly for all x in [ x ₋, x ₊]. Our result is proved rigorously, and shows that a previous result in the literature is incorrect.     

On a Gradient Flow with Exponential Rate of Convergence

We present an evolution equation governed by a maximal monotone operator with exponential rate of convergence to a zero of the maximal monotone operator. When the maximal monotone operator is the subdifferential of a convex, proper, and lower semicontinuous function, we show that the trajectory of solutions of the evolution equation converges exponentially to the minimum value of the convex function.     

边界层型问题的插值摄动解法

本文在文[1]的基础上用插值摄动法研究了最高阶导数乘以小参数的二阶常微分方程的定解问题。算例表明,本文方法计算过程简单,其精度甚至比多重尺度法的一级近似结果的精度还稍高一些。     

关于边界层方法

本文指出传统的边界层方法(包括匹配法和Vi?ik—Lyusternik方法)的不足:不能作出边界层项的渐近展开式.提出多重尺度构造边界层项的方法,得到符合实情的结果.又与Levinson所用的方法比较,本方法能更简单地导出后一方法给出的边界层项的渐近展开式.又应用此方法研究现有的关于奇异摄动的某些成果,指出这些成果的局限性,并在一般情况下作出解的渐近展开式.     

On Two-Phase Flow in a Rotating Boundary Layer

The two-phase flow induced by a rotating disk in a stationary unbounded mixture is considered. The generalized similarity assumption of von Karman reduces the averaged equations of motion with a linear drag between the phases to a system of ordinary differential equations. These are investigated by asymptotic and numerical techniques. The equations display a nontrivial behavior in a sublayer near the boundary, whose thickness is of the order of the particle size. The volume fraction of the dispersed phase is singular unless a small suction is applied on the disk or a small diffusion term is added to the continuity equations. Outside this sublayer, the velocity field is quite similar to a rescaled classical von Karman flow. Good agreement between asymptotic and numerical solution is obtained, although there is considerable stiffness in the equations. The motion of a solid particle in a von Karman flow is also discussed, but the present investigation is restricted to small radii because the shear-lift force is neglected.     

On Quadrature Convergence of Extended Lagrange Interpolation

Quadrature convergence of the extended Lagrange interpolant for any continuous function is studied, where the interpolation nodes are the zeros of an orthogonal polynomial of degree and the zeros of the corresponding ``induced' orthogonal polynomial of degree . It is found that, unlike convergence in the mean, quadrature convergence does hold for all four Chebyshev weight functions. This is shown by establishing the positivity of the underlying quadrature rule, whose weights are obtained explicitly. Necessary and sufficient conditions for positivity are also obtained in cases where the nodes and interlace, and the conditions are checked numerically for the Jacobi weight function with parameters and . It is conjectured, in this case, that quadrature convergence holds for .

     

关于一类Hermite-Fejer插值算子的平均收敛

本文给出基于{xk}n+1 k=0的Hermite-Fejer插值算子平均收敛的一些新结论,这里xo=1,xn+1=-1,xk(k=1,2,…,n)是n阶Jacobi多项式的零点.     

On Thermal Boundary Layer in a Viscous Non-Newtonian Medium

Doklady Mathematics - An existence and uniqueness theorem for a classical solution to the system of equations describing thermal boundary layers in viscous media with the Ladyzhenskaya rheological...     

Exponential Convergence of Finite-dimensional Approximations to Linear Bond-based Peridynamic Boundary Value Problems

In this paper, we demonstrate that the finite-dimensional approximations to the solutions of a linear bond-based peridynamic boundary value problem converge to the exact solution exponentially with the analyticity assumption of the forcing term, therefore greatly improve the convergence rate derived in literature.     

一类三角域上的三次样条插值(英文)

In this paper, we consider spaces of cubic C^1-spline on a class of triangulations. By using the inductive algorithm, the posed Lagrange interpolation sets are constructed for cubic spline space. It is shown that the class of triangulations considered in this paper are nonsingular for S1/3 spaces. Moreover, the dimensions of those spaces exactly equal to L. L. Schuraaker＇s low bounds of the dimensions. At the end of this paper, we present an approach to construct triangulations from any scattered planar points, which ensures that the obtained triangulations for S1/3 space are nonsingular.