| Summary: | 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. f.<sub>Ω</sub> g(x)/(Π<sub>i=1</sub><sup>s</sup> |x<sub>i</sub>-t<sub>i</sub>|<sup>1+γ</sup><sub>i</sub>) Π<sub>i=1</sub><sup>s</sup> dx<sub>i</sub> (s ≥ 2) with Ω = Π<sub>i=1</sub><sup>s</sup>[a<sub>i</sub>, b<sub>i</sub>], 0 <; γ<sub>i</sub> ≤ 1 and t<sub>i</sub> ∈ (a<sub>i</sub>, bi<sub>)</sub>. The multi-parameter asymptotic error estimates are derived for three different situations. The error estimates illustrate that, if g(x) is 2l + 1 (l ≥ (γ<sub>0</sub> - 1)/2) times differentiable on the Ω, the order of convergence is O(h<sub>0</sub><sup>2k</sup> ) for γ<sub>i</sub> = 1 (i = 1, · · · , s) or O(h<sub>0</sub><sup>2k-γ0</sup>) for some 0 <; γ<sub>i</sub> <; 1, (i = 1, · · · , p, p 0 s) and γ<sub>p+j</sub> = 1 (j = 1, · · · , s - p) with γ<sub>0</sub> = max{γ<sub>1</sub>, · · · , γ<sub>p</sub>}, h<sub>0</sub> = max{h<sub>1</sub>, · · · , h<sub>s</sub>}, 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.
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