Multi-Parameter Asymptotic Expansions With Errors for Multi-Dimensional Hypersingular Integrals With Product Type and Splitting Extrapolation

Abstract

This paper presents the modified quadrature rules for 1-D hypersingular integrals, and then constructs the quadrature formulas to numerically evaluate multi-dimensional hypersingular integrals in the form of $f.p.\int _{\Omega }~g(x)/\left({\prod _{i=1}^{s}|x_{i}-t_{i}|^{1+\gamma _{i}}}\right)~\prod _{i=1}^{s} dx_{i}~(s\geq 2)$ with $\Omega =\prod _{i=1}^{s}[a_{i},b_{i}]$ , $0 and $t_{i}\in (a_{i},b_{i})$ . The multi-parameter asymptotic error estimates are derived for three different situations. The error estimates illustrate that, if $g(x)$ is $2l+1~(l\geq (\gamma _{0}-1)/2)$ times differentiable on the $\Omega $ , the order of convergence is $\mathcal {O}(h_{0}^{2k})$ for $\gamma _{i} = 1~(i=1,\cdots ,s)$ or $\mathcal {O}(h_{0}^{2k-\gamma _{0}})$ for some $0 , $(i=1,\cdots ,p, p\leq s)$ and $\gamma _{p+j}=1~(j=1,\cdots ,s-p)$ with $\gamma _{0}=\max \{\gamma _{1},\cdots ,\gamma _{p}\}$ , $h_{0}=\max \{h_{1},\cdots ,h_{s}\}$ , where $k$ is a positive integer determined by the integrand. To obtain more accurate approximate solution, the splitting extrapolation algorithms are proposed. Numerical experiments are provided to verify the theoretical estimates.