ON SOLVABILITY AND WAVEFORM RELAXATION METHODS FOR LINEAR VARIABLE-COEFFICIENT DIFFERENTIAL-ALGEBRAIC EQUATIONS

This paper is concerned with the solvability and waveform relaxation methods of linear variable-coefficient differential-algebraic equations （DAEs）. Most of the previous works have been focused on linear variable-coefficient DAEs with smooth coefficients and data, yet no results related to the convergence rate of the corresponding waveform relaxation methods has been obtained. In this paper, we develope the solvability theory for the linear variable-coefficient DAEs on Legesgue square-integrable function space in both traditional and least squares senses, and determine the convergence rate of the waveform relaxation methods for solving linear variable-coefficient DAEs.     

一类复差分方程组的亚纯解(英文)

本文研究了一类复差分方程组的增长性的问题.利用亚纯函数的Nevanlinna值分布理论,得到了有关复差分方程组的亚纯解的一个重要结果,将复差分方程的某些结果推广到复差分方程组中.     

一类非线性chemotaxis系统的局部零能控性

本文研究了一类非线性chemotaxis系统的能控性问题.利用这类线性化结合不动点的方法,获得了非线性系统的局部零能控性结果,推广了非线性抛物-椭圆型方程能控性的结果.     

On the selection of a good value of shape parameter in solving time-dependent partial differential equations using RBF approximation method

Radial basis function method is an effective tool for solving differential equations in engineering and sciences. Many radial basis functions contain a shape parameter c which is directly connected to the accuracy of the method. Rippa [1] proposed an algorithm for selecting good value of shape parameter c in RBF-interpolation. Based on this idea, we extended the proposed algorithm for selecting a good value of shape parameter c in solving time-dependent partial differential equations.     

Remarks on non-linear noise excitability of some stochastic heat equations

We consider nonlinear parabolic SPDEs of the form ${\partial }_{t}u=\Delta u+\lambda \sigma \left(u\right)\dot{w}$ on the interval $(0,L)$, where $\dot{w}$ denotes space–time white noise, $\sigma$ is Lipschitz continuous. Under Dirichlet boundary conditions and a linear growth condition on $\sigma$, we show that the expected $L^2$-energy is of order $exp[\text{const}\times {\lambda }^4]$ as $\lambda \to \infty$. This significantly improves a recent result of Khoshnevisan and Kim. Our method is very different from theirs and it allows us to arrive at the same conclusion for the same equation but with Neumann boundary condition. This improves over another result in Khoshnevisan and Kim.     

An axisymmetric problem of an embedded mixed-mode crack in a functionally graded magnetoelectroelastic infinite medium

This paper investigates the problem of an axisymmetric penny shaped crack embedded in an infinite functionally graded magneto electro elastic medium. The loading consists of magnetoelectromechanical loads applied on the crack surfaces assumed to be magneto electrically impermeable. The material’s gradient is parallel to the axisymmetric direction and is perpendicular to the crack plane. An anisotropic constitutive law is adopted to model the material behavior. The governing equations are converted analytically using Hankel transform into coupled singular integral equations, which are solved numerically to yield the crack tip stress, electric displacement and magnetic induction intensity factors. A similar problem but with a different crack morphology, that is a plane crack embedded in an infinite functionally graded magneto electro elastic medium, was considered by the authors in a previous work (Rekik et al., 2012) [25]. While the overall solution schemes look similar, the axisymmetric problem resulted in more mathematical complexities and let to different conclusions with respect to the influence of coupling between elastic, electric and magnetic effects. The main focus of this paper is to study the effect of material non-homogeneity on the fields’ intensity factors to understand further the behavior of graded magnetoelectroelastic materials containing penny shaped cracks and to inspect the effect of varying the crack geometry.     

Convergence results for a class of spectrally hyperviscous models of 3-D turbulent flow

We consider the spectrally hyperviscous Navier–Stokes equations (SHNSE) which add hyperviscosity to the NSE but only to the higher frequencies past a cutoff wavenumber _m0$m_0$. In Guermond and Prudhomme (2003) [18], subsequence convergence of SHNSE Galerkin solutions to dissipative solutions of the NSE was achieved in a specific spectral-vanishing-viscosity setting. Our goal is to obtain similar results in a more general setting and to obtain convergence to the stronger class of Leray solutions. In particular we obtain subsequence convergence of SHNSE strong solutions to Leray solutions of the NSE by fixing the hyperviscosity coefficient μ$\mu$ while the spectral hyperviscosity cutoff _m0$m_0$ goes to infinity. This formulation presents new technical challenges, and we discuss how its motivation can be derived from computational experiments, e.g. those in Borue and Orszag (1996, 1998)  and . We also obtain weak subsequence convergence to Leray weak solutions under the general assumption that the hyperviscous coefficient μ$\mu$ goes to zero with no constraints imposed on the spectral cutoff. In both of our main results the Aubin Compactness Theorem provides the underlying framework for the convergence to Leray solutions.     

Remarks on partial regularity for suitable weak solutions of the incompressible magnetohydrodynamic equations

In this paper, we are concerned with the partial regularity for suitable weak solutions of the tri-dimensional magnetohydrodynamic equations. With the help of the De Giorgi iteration method, we obtain the results proved by He and Xin (C. He, Z. Xin, Partial regularity of suitable weak solutions to the incompressible magnetohydrodynamic equations, J. Funct. Anal. 227 (2005) 113–152), namely, the one dimensional parabolic Hausdorff measure of the possible singular points of the velocity field and the magnetic field is zero.     

A note on weakly hyperbolic equations with analytic principal part

In this note we show how to include low order terms in the C^∞$C^{\infty }$ well-posedness results for weakly hyperbolic equations with analytic time-dependent coefficients. This is achieved by doing a different reduction to a system from the previously used one. We find the Levi conditions such that the C^∞$C^{\infty }$ well-posedness continues to hold.