A class of semi-linear numerical differentiation formulas is designed for functions with steep gradients. A semi-linear second-order difference scheme is constructed to solve the two-point singular perturbation problem. It is shown that this semi-linear scheme has one more order of approximation precision than the central difference scheme for small $\epsilon$ and saves computation time for required accuracy. Numerical results agreeing with the above analysis are included.

In this note the exact non-local radiation condition and its local approximations at finite artificial boundary for the exterior boundary value problem of the reduced wave equation in 2 and 3 dimensions are discussed. Based on the asymptotic expansion of Hankel functions for large arguments, an approach for the construction of local approximations is suggested and gives expression of the normal derivative at spherical artificial boundary in terms of linear combination of Laplace-Beltrami operator and its iterates, i.e. tangential derivatives of even order exclusively. The resulting formalism is compatible with the usual variational principle and the finite element methodology and thus seems to be convenient in practical implementation.

This study deals with the theoretical basis of optimal control methods in primitive variable formulation and penalty function formulations of Navier-Stokes problems. Numerical examples demonstrating application are provided.

A comparison by Wang and Xu between S. Samle's cost estimation for Newton's method and that of the author's for Kuhn's algorithm, both aiming at the zero finding of complex polynomials, showed improvements the advantage of the latter in finding zeros and approximate zeros. In this paper, important on the above work are made. Furthermore, a probabilistic estimation of the monotonicity of Kuhn's algorithm is obtained.

The object of this paper is to construct a class of multivariate rational interpolation formulas that can be used to solve interpolation problems with function data given at equidistant knots of various directed lines in the higher dimensional Euclidean space. Our formulas are built up of some explicit multivariate rational functions involving three sets of free parameters so that they enjoy sufficient flexibility for interpolating functions of several variables possessing certain kinds of singularities. The method adopted is an extension and modification of that described in our previous papers.

An existence theorem of rational interpolation function for the sufficient condition has correctly been stated by Macon-Dupree in [2], but some arguments in their proof are not true. In this paper: (i) A related theorem for both the sufficient and necessary condition is asserted and proved by a new and rigorous approach, namely by introducing the notion of (m/n) quasi-rational interpolant of a given function. (ii) With use of these results thus obtained an open problem proposed by P. Turán in [4] is completely solved.

In this paper, we apply the canonical boundary reduction, suggested by Feng Kang, to the plane elasticity problems, find the expressions of canonical integral equations and Poisson integral formulas in some typical domains. We also give the numerical method for solving these equations together with their convergence and error estimates. Coupling with classical finite element method, this method can be applied to other domains.