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

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.
_Ω
g(x)/(Π
_i=1
^s
|x
_i
-t
_i
|
^1+γ
_i
) Π
_i=1
^s
dx
_i
(s ≥ 2) with Ω = Π
_i=1
^s
[a
_i
, b
_i
], 0 <; γ
_i
≤ 1 and t
_i
∈ (a
_i
, bi
₎
. 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
₀
- 1)/2) times differentiable on the Ω, the order of convergence is O(h
₀
^2k
) for γ
_i
= 1 (i = 1, · · · , s) or O(h
₀
^2k-γ0
) for some 0 <; γ
_i
<; 1, (i = 1, · · · , p, p 0 s) and γ
_p+j
= 1 (j = 1, · · · , s - p) with γ
₀
= max{γ
₁
, · · · , γ
_p
}, h
₀
= max{h
₁
, · · · , 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.