PVTSI^(m): A Novel Approach to Computation of Hadamard Finite Parts of Nonperiodic Singular Integrals
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We consider the numerical computation of I[f]=^b_a f(x) dx, the Hadamard Finite Part of the finite-range singular integral ∫^b_a f(x) dx, f(x)=g(x)/(x-t)^m with a<t<b and m∈{1,2,…}, assuming that (i) g∈ C^∞(a,b) and (ii) g(x) is allowed to have arbitrary integrable singularities at the endpoints x=a and x=b. We first prove that ^b_a f(x) dx is invariant under any suitable variable transformation x=ψ(ξ), ψ:[α,β]→[a,b], hence there holds ^β_α F(ξ) dξ=^b_a f(x) dx, where F(ξ)=f(ψ(ξ)) ψ'(ξ). Based on this result, we next choose ψ(ξ) such that the transformed integrand F(ξ) is sufficiently periodic with period =β-α, and prove, with the help of some recent extension/generalization of the Euler–Maclaurin expansion, that we can apply to ^β_α F(ξ) dξ the quadrature formulas derived for periodic singular integrals developed in an earlier work of the author. We give a whole family of numerical quadrature formulas for ^β_α F(ξ) dξ for each m, which we denote T^(s)_m,n[ F], where F(ξ) is the -periodic extension of F(ξ).
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