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.
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.