A HIGH ORDER METHOD FOR NON-SMOOTH FREDHOLM EQUATIONS

1.IntroductionConsidertheequationwherek(s,t)=k(f)tandf(s)aregiven,uistheunknownsolution.SinceitisrelatedcloselytoWiener-Hopfequationsandisveryimportantinpractice,therearemanynumericalresultsaboutit(e.g.[1--11]).Itiswellknownthattheaccuracyoftheapproximati…     

非光滑第二类Fredholm方程Collocation解的迭代──校正方法

本文对于一类具非光滑核第二类Ｆｒｅｄｈｏｌｍ方程的Ｃｏｌｌｏｃａｔｉｏｎ解提出一种迭代─校正方法，使得在计算量增加很少的前提下，成倍提高逼近解精度，并将此方法用于平面多角域上边界积分方程，从而给出其相应微分方程逼近解的高精度算法。此方法还是一种自适应方法。     

An error term and uniqueness for Hermite–Birkhoff interpolation involving only function values and/or first derivative values

This paper discusses various aspects of Hermite–Birkhoff interpolation that involve prescribed values of a function and/or its first derivative. An algorithm is given that finds the unique polynomial satisfying the given conditions if it exists. A mean value type error term is developed which illustrates the ill-conditioning present when trying to find a solution to a problem that is close to a problem that does not have a unique solution. The interpolants we consider and the associated error term may be useful in the development of continuous approximations for ordinary differential equations that allow asymptotically correct defect control. Expressions in the algorithm are also useful in determining whether certain specific types of problems have unique solutions. This is useful, for example, in strategies involving approximations to solutions of boundary value problems by collocation.     

高维非线性Schrdinger方程的Fourier谱方法

其中i=(-1)(1/2),△为Laplace算子,q(·)为实变量实值函数,u_0(x)和u(x,t)分别为关于x以2π为周期的已知和未知复值函数,J=(0,T](T>0),β为一实常数,e_j为R~m的第j个单位向量,x=(x_1,…,x_m)∈R~m. 方程(1.1)在非线性光学、等离子体物理、流体动力学及非相对论量子场论中用得很     

Interpolation correction for collocation solutions of Fredholm integro-differential equations

In this paper we discuss the collocation method for a large class of Fredholm linear integro-differential equations. It will be shown that, when a certain higher order interpolation operation is added to the collocation solution of this equation, the new approximations will, under suitable assumptions, admit a multiterm error expansion in even powers of the step-size . Based on this expansion, ideal multilevel correction results of this collocation solution are obtained.

     

Best uniform approximation by harmonic functions on subsets of Riemannian manifolds

We investigate best uniform approximations to bounded, continuous functions by harmonic functions on precompact subsets of Riemannian manifolds. Applications to approximation on unbounded subsets ofR ² are given.Communicated by J. Milne Anderson.     

L ∞-convergence of collocation and galerkin approximations to linear two-point parabolic problems

Two semidiscrete collocation approximations using smooth cubic splines are developed as approximations to the solution of two-point linear parabolic boundary value problems.L _∞-convergence results are presented for these two approximations as well as the piecewise linear Galerkin approximation. Several computational examples are given to illustrate the convergence results and demonstrate the applicability of the method.     

Parametric Cubic Spline Approach to the Solution of a System of Second-Order Boundary-Value Problems

We use parametric cubic spline functions to develop a numerical method for computing approximations to the solution of a system of second-order boundary-value problems associated with obstacle, unilateral, and contact problems. We show that the present method gives approximations which are better than those produced by other collocation, finite-difference, and spline methods. A numerical example is given to illustrate the applicability and efficiency of the new method.     

On rational approximations of functions

Rational analogues of Taylor and Fourier polynomials and polynomials of the Lagrange type are constructed and investigated. These analogues are shown to approximate a function with a remainder term of the same order as in the case of the aforementioned polynomials. Conditions are established under which a polynomial of the Lagrange type and its rational analogue are two-sided approximations of a function on a segment and their derivatives are two-sided approximations of the derivative of the function at collocation nodes. Bibliography: 2 titles. Translated fromObchuslyuval’na ta Prykladna Matematyka, No. 76, 1992, pp. 32–38.     

Bifurcating periodic solution for a class of first-order nonlinear delay differential equation after Hopf bifurcation

In this paper, we predict the accurate bifurcating periodic solution for a general class of first-order nonlinear delay differential equation with reflectional symmetry by constructing an approximate technique, named residue harmonic balance. This technique combines the features of the homotopy concept with harmonic balance which leads to easy computation and gives accurate prediction on the periodic solution to the desired accuracy. The zeroth-order solution using just one Fourier term is applied by solving a set of nonlinear algebraic equations containing the delay term. The unbalanced residues due to Fourier truncation are considered iteratively by solving linear equations to improve the accuracy and increase the number of Fourier terms of the solutions successively. It is shown that the solutions are valid for a wide range of variation of the parameters by two examples. The second-order approximations of the periodic solutions are found to be in excellent agreement with those obtained by direct numerical integration. Moreover, the residue harmonic balance method works not only in determining the amplitude but also the frequency of the bifurcating periodic solution. The method can be easily extended to other delay differential equations.     

Solving the Laplace equation by meshless collocation using harmonic kernels

We present a meshless technique which can be seen as an alternative to the method of fundamental solutions (MFS). It calculates homogeneous solutions of the Laplacian (i.e. harmonic functions) for given boundary data by a direct collocation technique on the boundary using kernels which are harmonic in two variables. In contrast to the MFS, there is no artificial boundary needed, and there is a fairly general and complete error analysis using standard techniques from meshless methods for the recovery of functions. We present two explicit examples of harmonic kernels, a mathematical analysis providing error bounds and convergence rates, and some illustrating numerical examples.     

Asymptotics of the eigenvalues of the rotating harmonic oscillator

The eigenenergies λ of a radial Schrödinger equation associated with the problem of a rotating harmonic oscillator are studied, these being values which admit eigensolutions which vanish at both the origin (a regular singularity of the equation) and at infinity. Asymptotic expansions, for the case where a coupling parameter α is small, are derived for λ. The approximation for λ consists of two components, an asymptotic expansion in powers of α, and a single term which is exponentially small (which can be associated with tunneling effects). The method of proof is rigorous, and utilizes three separate asymptotic approximations for the eigenfunction in the complex radial plane, involving elementary functions (WKB or Liouville-Green approximations), a modified Bessel function and a parabolic cylinder function.     

非线性积分方程迭代配置法的渐近展开及其外推

非线性积分方程迭代配置法的渐近展开及其外推韩国强（华南理工大学计算机工程与科学系）ＡＳＹＭＰＴＯＴＩＣＥＲＲＯＲＥＸＭＮＳＩＯＮＳＡＮＤＥＸＴＲＡＰＯＬＡＴＩＯＮＦＯＲＴＨＥＩＴＥＲＡＴＥＤＣＯＬＬＯＣＡＴＩＯＮＭＥＴＨＯＤＳＯＦＮＯＮＬＩＮＥＡＲＩ...     

Strongly unique spline approximations with free knots

A necessary and a sufficient alternation condition for strongly unique best spline approximations with free knots is given. In the case of simple knots these conditions coincide, and strongly unique best approximations and strongly unique local best approximations are the same. The numerical consequences are discussed.     

A note on accurate estimations of the local truncation error of polynomial methods for differential equations

We give sharp error estimations for the local truncation error of polynomial methods for the approximate solution of initial value problems. Our analysis is developed for second order differential equations with polynomial coefficients using the Tau Method as an analytical tool. Our estimates apply to approximations derived with the latter, with collocation and with techniques based on series expansions. An example on collocation is given, to illustrate the use of our estimates.     

Fourier-Padé approximations and filtering for spectral simulations of an incompressible Boussinesq convection problem

In this paper, we present rational approximations based on Fourier series representation. For periodic piecewise analytic functions, the well-known Gibbs phenomenon hampers the convergence of the standard Fourier method. Here, for a given set of the Fourier coefficients from a periodic piecewise analytic function, we define Fourier-Padé-Galerkin and Fourier-Padé collocation methods by expressing the coefficients for the rational approximations using the Fourier data. We show that those methods converge exponentially in the smooth region and successfully reduce the Gibbs oscillations as the degrees of the denominators and the numerators of the Padé approximants increase.

Numerical results are demonstrated in several examples. The collocation method is applied as a postprocessing step to the standard pseudospectral simulations for the one-dimensional inviscid Burgers' equation and the two-dimensional incompressible inviscid Boussinesq convection flow.

     

On the Choice of Segments in Piecewise Approximation

An optimal choice of segment boundaries in piecewise approximationis shown to be soluble by means of a dynamic programme in ascalar state variable. It is shown that the composite errorfunction is continuous in modulus, and that for approximationof concave or convex functions by linear segments, the compositeapproximation is continuous. Provided the residual term satisfiescertain requirements, the solution has uniqueness propertiesand can be found by means of a grid search method.     

One-sided non-linear mean approximation

This note examines one-sided (from above) nonlinear approximation with respect to a general integral (mean) “norm” on an interval. Necessary conditions for local minima are given. Interpolating properties of best approximations are given. When degenerate approximations can be best is studied for the L₁ and L_2k cases. Some comparable aspects of the discrete case are examined.     

Second-order expansion for the maximum of some stationary Gaussian sequences

We prove a second-order approximation formula for the distribution of the largest term among an infinite moving average Gaussian sequence. The second-order correction term depends on the autocovariance function only through the second largest autocovariance. Applications to Gaussian time series are discussed and a simulation study showed a substantial improvement over other approximations to the exact distribution of the maximum.     

An efficient asymptotically correct error estimator for collocation solutions to singular index-1 DAEs

A computationally efficient a posteriori error estimator is introduced and analyzed for collocation solutions to linear index-1 DAEs (differential-algebraic equations) with properly stated leading term exhibiting a singularity of the first kind. The procedure is based on a modified defect correction principle, extending an established technique from the context of ordinary differential equations to the differential-algebraic case. Using recent convergence results for stiffly accurate collocation methods, we prove that the resulting error estimate is asymptotically correct. Numerical examples demonstrate the performance of this approach. To keep the presentation reasonably self-contained, some arguments from the literature on DAEs concerning the decoupling of the problem and its discretization, which is essential for our analysis, are also briefly reviewed. The appendix contains a remark about the interrelation between collocation and implicit Runge-Kutta methods for the DAE case.