In this paper we use the spectral method to analyse the generalized Kuramoto-Sivashinsky equations. We prove the existence and uniqueness of global smooth solution of the equations. Finally, we obtain the error estimation between spectral approximate solution and exact solution on large time.  

The paper considers the $L^\infty$ convergence for TRUNC finite elements solving the boundary value problems of the biharmonic equation. The nearly optimal $L^\infty$ estimates for the error of first order derivatives are given.

Nonlinear Galerkin methods are numerical schemes adapted well to the long time integration of evolution partial differential equations. The aim of this paper is to discuss such schemes for reaction diffusion equations. The convergence results are proved.  

In this paper, a truncated hybrid method is proposed and developed for solving sparse large-scale nonlinear programming problems. In the hybrid method, a symmetric system of linear equations, instead of the usual quadratic programming subproblems, is solved at iterative process. In order to ensure the global convergence, a method of multiplier is inserted in iterative process. A truncated solution is determined for the system of linear equations and the unconstrained subproblems are solved by the limited memory BFGS algorithm such that the hybrid algorithm is suitable to the large-scale problems. The local convergence of the hybrid algorithm is proved and some numerical tests for medium-sized truss problem are given.

We use a combination of both algebraic and numerical techniques to construct a $C^1$-continuous, piecewise $(m,n)$ rational $\epsilon$-approximation of a real algebraic plane curve of degree $d$. At singular points we use the classical Weierstrass Preparation Theorem and Newton power series factorizations, based on the technique of Hensel lifting. These, together with modified rational Padé approximations, are used to efficiently construct locally approximate, rational parametric representations for all real branches of an algebraic plane curve. Besides singular points we obtain an adaptive selection of simple points about which the curve approximations yield a small number of pieces yet achieve $C^1$ continuity between pieces. The simpler cases of $C^{-1}$ and $C^0$ continuity are also handled in a similar manner. The computation of singularity, the approximation error bounds and details of the implementation of these algorithms are also provided.

In this paper, the semi-discrete entropy conditions with so called the proper discrete entropy flux of a class of high resolution MUSCL type schemes are discussed for genuinely nonlinear scalar conservation laws. It is shown that the high resolution schemes satisfying such semi-discrete entropy conditions cannot preserve second order accuracy in the rarefaction region.

This paper gives a kind of series representation of the scaling functions $\phi_N$ and the associated wavelets $\psi _N$ constructed by Daubechies. Based on Poission summation formula, the functions $\phi_N(x+N-1)$, $\phi_N(x+N)$, $\cdots$, $\phi_N(x+2N-2) (0 \leq x\leq 1)$ are linearly represented by $\phi_N(x)$, $\phi_N(x+1)$, $\cdots$, $\phi_N(x+2N-2)$ and some polynomials of order less than $N$, and $\Phi _0(x):=(\phi_N(x), \phi_N(x+1), \cdots,\phi_N(x+N-2))^t$ is translated into a solution of a nonhomogeneous vector-valued functional equation $$f(x)=A_d f(2x-d)+P _d(x),x\in [\frac{d}{2},\frac{d+1}{2}], d=0,1,$$ where $A_0,A_1$ are $(N-1)\times (N-1)$-dimensional matrices, the components of $P_0(x), P_1(x)$ are polynomials of order less than $N$. By iteration, $\Phi_0(x)$ is eventually represented as an $(N-1)$-dimensional vector series $\sum\limits_{k=0} ^\infty u _k(x)$ with vector norm $\|U _k(x)\|\leq C\beta ^k$, where $\beta =\beta_N <1$ and $\beta_N \searrow 0$ as $N \rightarrow \infty$.