In this paper we consider a class of singular two-point boundary value problem: $(x^αu')'=f(x,u),u(0)=A,u(1)=B,0＜α＜1$. The asymptotic expression of the spline finite difference solution for the problem is obtained. By using the asymptotic expression, Richardson's extrapolation can be done and the accuracy of numerical solution can be improved. Some numerical examples are given in illustration of this theory.  

Starting from a well known operator identity we obtain a recurrence formula, i.e., an iterative correction scheme, for the integral equations with computable kernel. From this we can increase the order of convergence step by step, say, from 4th to 8th to 12th. What is more interesting in this scheme, besides its fast acceleration, is its weak requirement on the integral kernel: the regularity of the kernel will not be strengthened during the correction procedure.  

This work presents a procedure for constructing the solution to a class of problems with application to heat transfer processes in which the energy reemission is not negligible. Such problems are characterized by a Poisson equation subjected to certain nonlinear boundary conditions. The solution is constructed from a sequence whose elements may be obtained from a minimum principle. Some practical situations are presented.  

By establishing the spectrum (matrix) function for the block Jacobi matrix, theorems of existence and uniqueness for the inverse problem and algorithms for its solution are obtained. The study takes into account all possible multiple-eigenvalue cases that are very difficult to deal with by other means.  

In this paper, we present the necessary and sufficient condition of convergence of several iterative methods for computing the generalized inverses of operators in Banach spaces. It is proved that the iterative methods converge to the generalized inverse of an Operator in Banach spaces if and only if these conditions are satisfied.  

In this paper we show that, if a problem of $(0,1,\cdots,m-2,m)$-interpolation on the zeros of the Jacobi polynomials $P^{\alpha,β}_n(x) (\alpha,β\geq -1)$ has infinite solutions, then the general form of the solutions is $f_0(x)+Cf(x)$ with an arbitrary constant $C$, where $f_0(x)$ and $f(x)$ are fixed polynomials of degree $\leq mn-1$. Moreover, the explicit form of $f(x)$ is given. A necessary and sufficient condition of quadrature regularity of the interpolation in a manageable form is also established.  

In this paper we introduce a new convergence accelerator for vector sequences — vector Padé approximation method (VPA), discuss its convergence and stability properties, and show that it is a bona fide acceleration method for some vector sequences.

A linear programming problem can be translated into an equivalent general linear complementarity problem, which can be solved by an iterative projection and contraction (PC) method [6]. The PC method requires only two matrix-vector multiplications at each iteration and the efficiency in practice usually depends on the sparsity of the constraint-matrix. The prime PC algorithm in [6] is globally convergent; however, no statement can be made about the rate of convergence. Although a variant of the PC algorithm with constant step-size for linear programming [7] has a linear speed of convergence, it converges much slower in practice than the prime method [6]. In this paper, we develop a new step-size rule for the PC algorithm for linear programming such that the resulting algorithm is globally linearly convergent. We present some numerical experiments to indicate that it also works better in practice than the prime algorithm.  

For partitioned Runge-Kutta methods, in the integration of Hamiltonian systems, a condition for symplecticness and its characterization which is based on the W-transformation of Hairer and Wanner are presented. Examples for partitioned Runge-Kutta methods which satisfy the symplecticness condition are given. A special class of symplectic partitioned Runge-Kutta methods is constructed.  

The paper considers the $L^{\infty}$ convergence for conforming finite elements, such as Argyris element, Bell element and Bogner-Fox-Schmit element, solving the boundary value problem of the biharmonic equation. The nearly optimal order $L^{\infty}$ estimates are given.