Alternating direction implicit schemes for the two-dimensional fractional sub-diffusion equation

New numerical techniques are presented for the solution of a two-dimensional anomalous sub-diffusion equation with time fractional derivative. In these methods, standard central difference approximation is used for the spatial discretization, and, for the time stepping, two new alternating direction implicit (ADI) schemes based on the L₁ approximation and backward Euler method are considered. The two ADI schemes are constructed by adding two different small terms, which are different from standard ADI methods. The solvability, unconditional stability and H¹ norm convergence are proved. Numerical results are presented to support our theoretical analysis and indicate the efficiency of both methods.     

Sixth moment optimization of multi-channel Bragg filters

The design of multichannel optical filter is studied theoretically. New efficient two-step method based on combination of 6-th moment minimization and straightforward numerical minimization is developed. It is applied to find a fiber Bragg grating profile with minimal refractive index variation for 5 ÷ 32 channels. Obtained results are closer to the theoretical limit than previously reported.     

求解非线性P_0互补问题的非单调磨光算法

提出了一种新的磨光函数,在分析它与已有磨光函数不同特性的基础上,研究了将它用于求解非线性P_0互补问题时,其磨光路径的存在性和连续性,进而设计了求解一类非线性P_0互补问题的非单调磨光算法.在适当的假设条件下,证明了该算法的全局收敛性和局部超线性收敛性.数值算例验证了算法的有效性.     

Spectral collocation method for linear fractional integro-differential equations

In this paper, we propose and analyze a spectral Jacobi-collocation method for the numerical solution of general linear fractional integro-differential equations. The fractional derivatives are described in the Caputo sense. First, we use some function and variable transformations to change the equation into a Volterra integral equation defined on the standard interval [-1,1]$[-1\text{,}1]$. Then the Jacobi–Gauss points are used as collocation nodes and the Jacobi–Gauss quadrature formula is used to approximate the integral equation. Later, the convergence order of the proposed method is investigated in the infinity norm. Finally, some numerical results are given to demonstrate the effectiveness of the proposed method.     

Quasi-static Maxwell's equations with a dissipative non-linear boundary condition: Full discretization

We study a time dependent eddy current equation for the magnetic field H accompanied with a nonlinear boundary condition, which is a generalization of the classical Silver–Müller condition for a non-perfect conductor. More exactly, the relation between the normal components of electric (E) and magnetic (H  ) fields obeys the following power law (linearized for small and large values) ν×E=ν×(|H×ν|^α−1H×ν)$\mathit{\nu }\times \mathit{E}=\mathit{\nu }\times ({|\mathit{H}\times \mathit{\nu }|}^{\alpha -1}\mathit{H}\times \mathit{\nu })$ for some α∈(0,1]$\alpha \in (0,1]$. We design a linear fully discrete approximation scheme to solve this nonlinear problem. The convergence of the approximations to a weak solution is proved, error estimates describing the dependence of the error on discretization parameters are derived as well. The efficiency of the proposed method is supported by numerical experiments.     

On the global exponential stability of impulsive functional differential equations with infinite delays or finite delays

This paper considers the impulsive functional differential equations with infinite delays or finite delays. Some new sufficient conditions are obtained to guarantee the global exponential stability by employing the improved Razumikhin technique and Lyapunov functions. The result extends and improves some recent works. Moreover, the obtained Razumikhin condition is very simple and effective to implement in real problems and it is helpful to investigate the stability of delayed neural networks and synchronization problems of chaotic systems under impulsive perturbation. Finally, a numerical example and its simulation is given to show the effectiveness of the obtained result in this paper.     

Convergence of densities of some functionals of Gaussian processes

The aim of this paper is to establish the uniform convergence of the densities of a sequence of random variables, which are functionals of an underlying Gaussian process, to a normal density. Precise estimates for the uniform distance are derived by using the techniques of Malliavin calculus, combined with Stein?s method for normal approximation. We need to assume some non-degeneracy conditions. First, the study is focused on random variables in a fixed Wiener chaos, and later, the results are extended to the uniform convergence of the derivatives of the densities and to the case of random vectors in some fixed chaos, which are uniformly non-degenerate in the sense of Malliavin calculus. Explicit upper bounds for the uniform norm are obtained for random variables in the second Wiener chaos, and an application to the convergence of densities of the least square estimator for the drift parameter in Ornstein–Uhlenbeck processes is discussed.     

带有多项式非线性项的高维反应扩散方程有限差分格式的长时间行为

本文考虑了一类带有多项式非线性项的高维反应扩散方程．建立了一个全离散的有限差分格式,并证明了差分解的存在唯一性．分析了由差分格式生成的离散系统的动力性质,在对差分解先验估计的基础上得到了离散动力系统的整体吸引子的存在性．最后证明了差分格式的长时间稳定性和收敛性．