This paper is devoted to a study of stability of general linear methods for the numerical solution of nonlinear stiff initial value problems in a Hilbert space. New stability concepts are introduced. A criterion of weak algebraic stability is established, which is an improvement and extension of the existing criteria of algebraic stability.  

In this paper, we verify the conjecture in the recent paper of I. Babuška and T. Janik.

In this paper a systematical method for the construction of Poisson difference schemes with arbitrary order of accuracy for Hamiltonian systems on Poisson manifolds is considered. The transition of such difference schemes from one time-step to the next is a Poisson map. In addition, these schemes preserve all Casimir functions and, under certain conditions, quadratic first integrals of the original Hamiltonian systems. Especially, the arbitrary order centered schemes preserve all Casimir functions and all quadratic first integrals of the original Hamiltonian systems.

In the late 1970's, Wiggins proposed a minimum entropy deconvolution (MED) which has become one of the most important deconvolution methods. He gave a varimax norm $V^4_2$ and a MED iterative procedure. Fortunately, for the last ten years in the practical using, the MED algorithm has never failed to reach a maximizer of the varimax norm. But so far, no theoretical proof has been given to show the convergence of the MED procedure. In this paper, we prove the global convergence of a generalized MED iterative procedure with respect to a generalized varimax norm $V^p_q(q=2,p \gt 2)$.

This paper deals with a coefficient inverse problem of a system of ODEs whose coefficient matrix is the so-called generalized negative definite matrix. To solve the problem, an initial-boundary value problem of a hyperbolic system of PDEs is constructed. The existence and uniqueness of its solution and its asymptotic convergence with respect to one of the variables to the original inverse problem are proved. As a result, the solution of the inverse problem is reduced to the solution of the direct problem. A few numerical examples were solved to show the effectiveness of the method.

To overcome the difficulty caused by the singularity at the pitchfork bifurcation points, we introduce the homotopy parameter so that the problem of computing the pitchfork bifurcation points can be transferred to that of computing the fold points of degree 3 with respect to the homotopy parameter. An extended system for pitchfork bifurcation points is given. The regularity of the extended system is proved. Finally, the numerical examples show the effectiveness of our method.

A method for constructing the $C^1$ piecewise polynomial surface of degree four on triangles is presented in this paper. On every triangle, only nine interpolation conditions, which are function values and first partial derivatives at the vertices of the triangle, are needed for constructing the surface.

In this paper we present a method for solving initial value problems related to second order matrix differential equations. This method is based on the existence of a solution of a certain algebraic matrix equation related to the problem, and it avoids the increase of the dimension of the problem for its resolution. Approximate solutions, and their error bounds in terms of error bounds for the approximate solutions of the algebraic problem, are given.

A method is presented for obtaining recurrence relations for the coefficients in ultraspherical series of linear differential equations. This method applies Doha's method (1985) to generate polynomial approximations in terms of ultraspherical polynomials of $y(zx), -1\leq x\leq 1,z\in C,|z|\leq 1$, where y is a solution of a linear differential equation. In particular, rational approximations of $y(z)$ result if $x$ is set equal to unity. Two numerical examples are given to illustrate the application of the method to first and second order differential equations. In general, the rational approximations obtained by this method are better than the corresponding polynomial approximations, and compare favourably with Pade approximants.

In this paper, a class of integration formulas is derived from the approximation so that the first derivative can be expressed within an interval $[nh,(n+1)h]$ as $$\frac{dy}{dt}=-P(y-y_n)+f_n+Q_n(t).$$ The class of formulas is exact if the differential equation has the shown form, where $P$ is a diagonal matrix, whose elements $$-pj=\frac{\partial}{\partial y_j}f_j(t_n,y_n),j=1,\cdots,m$$ are constant in the interval $[nh,(n+1)h]$, and $Q_n(t)$ is a polynomial in $t$.  Each of the formulas derived in this paper includes only the first derivative $f$ and $$-pj=\frac{\partial}{\partial y_j}f_j(t_n,y_n).$$ It is identical with a certain Runge-Kutta method. In particular, when $Q_n(t)$ is a polynomial of degree two, one of our formulas is an extension of Treanor's method, and possesses better stability properties. Therefore the formulas derived in this paper can be regarded as a modified or an extended form of the classical Runge-Kutta methods. Preliminary numerical results indicate that our fourth order formula is superior to Treanor's in stability properties.