Spectrally accurate numerical quadrature formulas for a class of algebraically singular periodic Hadamard finite part integrals by regularization

Abstract

We consider the numerical computation of Hadamard Finite Part (HFP) integrals Hσ(t;u)=⨎0T|sinπ(x−t)T|σu(x)dx,0<t<T,σ<−1,σ⁄∈Z,u(x) being sufficiently differentiable and T-periodic on R. Thus σ=−(m+δ), m∈{1,2,3,…}, 0<δ<1. For each such σ, we regularize Hσ(t;u), and show that Hσ(t;u)=Hσ+2r(t;Uσ),r=⌊(m+1)/2⌋, where Uσ(x)=∑k=0raku(2k)(x) for some constants ak, Hσ+2r(t;Uσ) being a regular integral. We then propose to approximate Hσ(t;u) by the quadrature formula Qσ,n(t;u)≡Hσ(t;ϕn), where ϕn(x) is the nth-order balanced trigonometric polynomial that interpolates u(x) on [0,T] at the 2n equidistant points xn,k=kT2n, k=0,1,…,2n−1. The implementation of Qσ,n(t;u) is simple, the only input needed for this being the 2n function values u(xn,k), k=0,1,…,2n−1. Using Fourier analysis techniques, we develop a complete convergence theory for Qσ,n(t;u) as n→∞ and prove that it enjoys spectral convergence when u∈C∞(R).We also show that the theoretical developments and numerical quadrature formulas developed for the HFP integrals Hσ(t;u) with σ<−1 and σ⁄∈Z apply, with no changes, to the regular singular integrals Hσ(t;u) with σ>−1 and σ⁄∈Z.We illustrate the effectiveness of Qσ,n(t;u) with numerical examples both for σ<−1 and σ>−1.Finally, we show that the HFP or regular integral ⨎0Tf(x)dx of any T-periodic integrand f(x) that has algebraic singularities of the form |x−t+kT|σ, 0<t<T, k=0,±1,±2,…, with σ⁄∈Z, but is sufficiently differentiable in x on R∖{t±kT}k=0∞, can be expressed as Hσ(t;u), where u(x) is a T-periodic and sufficiently differentiable function of x on R that can be computed from f(x). Therefore, ⨎0Tf(x)dx can be computed efficiently using our new numerical quadrature formulas Qσ,n(t;u) on the individual Hσ(t;u). Again, only 2n function evaluations, namely, u(xn,k), k=0,1,…,2n−1, are needed for the whole process.