Weak local residuals as smoothness indicators for the shallow water equations

The system of shallow water equations admits infinitely many conservation laws. This paper investigates weak local residuals as smoothness indicators of numerical solutions to the shallow water equations. To get a weak formulation, a test function and integration are introduced into the shallow water equations. We use a finite volume method to solve the shallow water equations numerically. Based on our numerical simulations, the weak local residual of a simple conservation law with a simple test function is identified to be the best as a smoothness indicator.     

Well posedness of a linearized fractional derivative fluid model

The one-dimensional fractional derivative Maxwell model (e.g. Palade, et al., Rheol. Acta 35 (1996) 265), of importance in modeling the linear viscoelastic response in the glass transition region, has been generalized in Palade, et al., Internat. J. Engrg. Sci. 37 (1999) 315, to objective three-dimensional constitutive equations (CEs) for both fluids and solids. Regarding the rest state stability of the fluid CE, in Heibig and Palade, J. Math. Phys. 49 (2008) 043101, we gave a proof for the existence of weak solutions to the corresponding boundary value problem. The aim of this work is to achieve the study of the existence and uniqueness of the aforementioned solutions and to present smoothness results.     

Efficient curve fitting: An application of multiobjective programming

Curve fitting is an interesting and important subject in mathematics and engineering. It has been studied extensively and a number of approaches, mostly based on polynomials and piecewise polynomials, have been employed. In the usual setting, some data points are given and one wants to find a polynomial function with the minimum violations measured by a norm in the given data points. In these approaches, norms are applied to aggregate all violations as a scalar.     

Regularities of the solutions to the Fokker-Planck-Boltzmann equation

In this paper, we consider the regularities of the solutions to the Fokker-Planck-Boltzmann equation. In particular, we prove that for hard sphere case, the strong solution constructed by Li and Matsumura [H. Li, A. Matsumura, Behavior of the Fokker-Planck-Boltzmann equation near a Maxwellian, Arch. Ration. Mech. Anal. (2008), in press] near Maxwellian becomes immediately smooth with respect to all variables as long as t>0.     

Pseudo-stable and Pseudo-unstable Fiber Bundles for Dynamic Equations on Measure Chains

Invariant fiber bundles are the generalization of invariant manifolds from classical discrete or continuous dynamical systems to non-autonomous dynamic equations on measure chains. In this paper, we present a self-contained proof of their existence and smoothness. Our main result generalizes the so-called "Hadamard-Perron-Theorem" for hyperbolic finite-dimensional diffeomorphisms to pseudo-hyperbolic time-dependent non-regressive dynamic equations in Banach spaces. The proof of their smoothness uses a fixed point theorem of Vanderbauwhede-Van Gils.     

Dimension of -splines on type-6 tetrahedral partitions

We consider a linear space of piecewise polynomials in three variables which are globally smooth, i.e. trivariate C¹-splines of arbitrary polynomial degree. The splines are defined on type-6 tetrahedral partitions, which are natural generalizations of the four-directional mesh. By using Bernstein–Bézier techniques, we analyze the structure of the spaces and establish formulae for the dimension of the smooth splines on such uniform type partitions.     

有限链上的左(右)零模

为了更好地描述信息的聚合,有限链上的左(右)nullnorm的概念被引入。然后,各种光滑的左(右)nullnorm的结构定理被给出,即给出了一个二元运算是光滑的左(右)nullnorm的充要条件。     

Szàsz-Mirakian Kantorovich算子拟中插式的逼近等价定理

本文利用高阶光滑模ω■2r(f,t)p(1≤P≤∞)和ω■λ2r(f,t)∞(0≤λ≤1)得到了Szasz-Mirakian Kantorovich算子对于函数f∈Lp[0,00)(1≤P≤∞)的逼近等价定理.     

On the stability of the high-order Higdon Absorbing Boundary Conditions

The Higdon sequence of Absorbing Boundary Conditions (ABCs) for the linear wave equation is considered. Building on a previous work of Ha-Duong and Joly, which related to other forms of boundary conditions, the Higdon ABCs are proved to be energy-stable (on the continuous level) up to any order. This type of stability is stronger than the more standard notion of stability of initial boundary value problems in the sense of Kreiss; in particular it leads to stability estimates which are uniform in time. In consequence to the theorem proved here, energy-stability is immediately implied for the high-order Givoli-Neta and Hagstrom-Warburton ABCs, which are reformulations of the Higdon ABCs using auxiliary variables. A weakness of this theory is that it requires sufficiently smooth data, and that the required smoothness level increases with the order of the ABC. This issue and its implications are discussed.