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. f.Ω g(x)/(Πi=1s |xi-ti|1+γi) Πi=1s dxi (s ≥ 2) with Ω = Πi=1s[ai, bi], 0 <; γi ≤ 1 and ti ∈ (ai, 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 ≥ (γ0 - 1)/2) times differentiable on the Ω, the order of convergence is O(h02k ) for γi = 1 (i = 1, · · · , s) or O(h02k-γ0) for some 0 <; γi <; 1, (i = 1, · · · , p, p 0 s) and γp+j = 1 (j = 1, · · · , s - p) with γ0 = max{γ1, · · · , γp}, h0 = max{h1, · · · , hs}, 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.

Keywords

• Multi-dimensional hypersingular integrals
• euler-maclaurin error expansion
• quadrature rule
• splitting extrapolation