Euler-Maclaurin expansions and approximations of hypersingular integrals
Chaolang Hu Xiaoming He Tao Lü
Discrete & Continuous Dynamical Systems - B 2015, 20(5): 1355-1375 doi: 10.3934/dcdsb.2015.20.1355
This article presents the Euler-Maclaurin expansions of the hypersingular integrals $\int_{a}^{b}\frac{g(x)}{|x-t|^{m+1}}dx$ and $\int_{a}^{b}% \frac{g(x)}{(x-t)^{m+1}}dx$ with arbitrary singular point $t$ and arbitrary non-negative integer $m$. These general expansions are applicable to a large range of hypersingular integrals, including both popular hypersingular integrals discussed in the literature and other important ones which have not been addressed yet. The corresponding mid-rectangular formulas and extrapolations, which can be calculated in fairly straightforward ways, are investigated. Numerical examples are provided to illustrate the features of the numerical methods and verify the theoretical conclusions.
keywords: mid-rectangular quadrature formula Euler-Maclaurin expansion arbitrary singular point extrapolation. Hypersingular integral
Numerical approximations for a smectic-A liquid crystal flow model: First-order, linear, decoupled and energy stable schemes
Qiumei Huang Xiaofeng Yang Xiaoming He
Discrete & Continuous Dynamical Systems - B 2018, 23(6): 2177-2192 doi: 10.3934/dcdsb.2018230

In this paper, we consider numerical approximations for a model of smectic-A liquid crystal flows in its weak flow limit. The model, derived from the variational approach of the de Gennes free energy, is consisted of a highly nonlinear system that couples the incompressible Navier-Stokes equations with two nonlinear order parameter equations. Based on some subtle explicit-implicit treatments for nonlinear terms, we develop an unconditionally energy stable, linear and decoupled time marching numerical scheme for the reduced model in the weak flow limit. We also rigorously prove that the numerical scheme obeys the energy dissipation law at the discrete level. Various numerical simulations are presented to demonstrate the accuracy and the stability of the scheme.

keywords: Smectic-A liquid crystal unconditional stability Ginzburg-Landau decoupled linear
The Nehari manifold for fractional systems involving critical nonlinearities
Xiaoming He Marco Squassina Wenming Zou
Communications on Pure & Applied Analysis 2016, 15(4): 1285-1308 doi: 10.3934/cpaa.2016.15.1285
We study the combined effect of concave and convex nonlinearities on the number of positive solutions for a fractional system involving critical Sobolev exponents. With the help of the Nehari manifold, we prove that the system admits at least two positive solutions when the pair of parameters $(\lambda,\mu)$ belongs to a suitable subset of $R^2$.
keywords: Nehari manifold. concave-convex nonlinearities Fractional systems

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