Asymptotic error expansions for hypersingular integrals
The authors consider the hypersingular integral \[ I(g)=f.p.\displaystyle\int_{a}^{b}\frac{g(x)}{|x-t|^{1+\alpha}}dx, \, 0<\alpha\leq 1,\,t\in(a,b), \tag{1} \] where \(f.p.\) denotes the Hadamard finite part of the hypersingular integral. They present quadrature formulae to evaluate (1) for \(0\leq \alpha\leq 1\), which are obtained from the Euler-Maclaurin expansion for the error of the trapezoidal rule. If \(g(x)\) is \(2m\) times differentiable on \([a,b]\), the accuracy order of the algorithms is \(O(h^{2\mu})\) for \(\alpha=1\) and \(O(h^{2\mu-\alpha})\) for \(0<\alpha<1\). Here \(\mu\) is a positive integer determined by the integrand. More accurate results can be achieved, especially, for the hypersingular integration with a periodic integrand. Furthermore, the accuracy order can be greatly improved by applying the extrapolation method. At the same time, this method can be easily extended to evaluate hypersingular integrals with a higher-order singularity. The quadratures only use values of \(g(x)\), but do not depend on its derivatives, which make them easy to use in solving the hypersingular boundary integral equation. Numerical experiments verify the theoretical analysis.
scientific article; zbMATH DE number 6153902
scientific article