In [2-4], symplectic schemes of arbitrary order are constructed by generating functions. However, the construction of generating functions is dependent on the chosen coordinates. One would like to know that under what circumstance the construction of generating functions will be independent of the coordinates. The generating functions are deeply associated with the conservation laws, so it is important to study their properties and computations. This paper will begin with the study of Darboux transformation, then in section 2, a normalization Darboux transformation will be defined naturally. Every symplectic scheme which is constructed from Darboux transformation and compatible with the Hamiltonian equation will satisfy this normalization condition. In section 3, we will study transformation properties of generator maps and generating functions. Section 4 will be devoted to the study of the relationship between the invariance of generating functions and the generator maps. In section 5, formal symplectic energy of symplectic schemes are presented.

The numerical solutions to the nonlinear integral equations of Hammerstein-type $$ y (t)=f (t)+\int^1_0 k (t, s) g (s, y (s)) ds, \quad t\in [0,1] $$ are investigated. A degenerate kernel scheme basing on ID-wavelets combined with a new collocation-type method is presented. The Daubechies interval wavelets and their main properties are briefly mentioned. The rate of approximation solution converging to the exact solution is given. Finally we also give two numerical examples.

In this paper, some results about approximation in a norm $S$ induced by the minimax series are studied. Then a Bernstein-type theorem for the norm $S$ is established. Finally the Bernstein theorem is applied to prove the existence of certain equalities with minimax series and weighted minimax series.

In this paper, we shall study the initial boundary value problem of Schrödinger equation. The second order gradient superconvergence estimates for the problem are obtained solving by linear finite elements.

We establish the convergence theories of the symmetric relaxation methods for the system of linear equations with symmetric positive definite coefficient matrix, and more generally, those of the unsymmetric relaxation methods for the system of linear equations with positive definite matrix.

This paper extends the results by E.M. Kasenally$^{[7]}$ on a Generalized Minimum Backward Error Algorithm for nonsymmetric linear systems $Ax=b$ to the problem in which perturbations are simultaneously permitted on $A$ and $b$. The approach adopted by Kasenally has been to view the approximate solution as the exact solution to a perturbed linear system in which changes are permitted to the matrix $A$ only. The new method introduced in this paper is a Krylov subspace iterative method which minimizes the norm of the perturbations to both the observation vector $b$ and the data matrix $A$ and has better performance than the Kasenally's method and the restarted GMRES ${\rm method}^{[12]}$. The minimization problem amounts to computing the smallest singular value and the corresponding right singular vector of a low-order upper-Hessenberg matrix. Theoretical properties of the algorithm are discussed and practical implementation issues are considered. The numerical examples are also given.

By using wave splitting method the formulation of the two-dimensional potential inversion problem is set up in terms of the coupled system for downgoing and upcoming wavefields. The boundary conditions on the characteristic surface needed for solving the problem are derived by singularity analysis. Two stability theorems are given for the direct problems of the system treated as Cauchy problems in the direction of depth.

In this paper, an exterior Signorini problem is reduced to a variational inequality on a bounded inner region with the help of a coupling of boundary integral and finite element methods. We established an equivalence between the original exterior Signorini problem and the variational inequality on the bounded inner region coupled with two integral equations on an auxiliary boundary. We also introduce a finite element approximation of the variational inequality and a boundary element approximation of the integral equations. Furthermore, the optimal error estimates are given.

In this paper we prove the solution of explicit difference scheme for a semilinear parabolic equation converges to the solution of difference scheme for the relevant nonlinear stationary problem as $t \rightarrow \infty$. For nonlinear parabolic problem, we obtain the long time asymptotic behavior of its discrete solution which is analogous to that of its continuous solution. For simplicity, we discuss one-dimensional problem.