Euler-Maclaurin expansions and approximations of hypersingular integrals
Chaolang Hu Xiaoming He Tao Lü
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
The Nehari manifold for fractional systems involving critical nonlinearities
Xiaoming He Marco Squassina Wenming Zou
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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