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

We consider the numerical computation of Hadamard Finite Part (HFP) integrals H σ (t ; u) = ⨎ 0 T | sin π (x − t) T | σ u (x) d x , 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 H σ (t ; u) , and show that H σ (t ; u) = H σ + 2 r (t ; U σ) , r = ⌊ (m + 1) / 2 ⌋ , where U σ (x) = ∑ k = 0 r a k u (2 k) (x) for some constants a k , H σ + 2 r (t ; U σ) being a regular integral. We then propose to approximate H σ (t ; u) by the quadrature formula Q σ , n (t ; u) ≡ H σ (t ; ϕ n) , where ϕ n (x) is the n th -order balanced trigonometric polynomial that interpolates u (x) on [ 0 , T ] at the 2 n equidistant points x n , k = k T 2 n , k = 0 , 1 , ... , 2 n − 1. The implementation of Q σ , n (t ; u) is simple, the only input needed for this being the 2 n function values u (x n , k) , k = 0 , 1 , ... , 2 n − 1. Using Fourier analysis techniques, we develop a complete convergence theory for Q σ , 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 H σ (t ; u) with σ < − 1 and σ ⁄ ∈ Z apply, with no changes, to the regular singular integrals H σ (t ; u) with σ > − 1 and σ ⁄ ∈ Z. We illustrate the effectiveness of Q σ , n (t ; u) with numerical examples both for σ < − 1 and σ > − 1. Finally, we show that the HFP or regular integral ⨎ 0 T f (x) d x of any T -periodic integrand f (x) that has algebraic singularities of the form | x − t + k T | σ , 0 < t < T , k = 0 , ± 1 , ± 2 , ... , with σ ⁄ ∈ Z , but is sufficiently differentiable in x on R ∖ { t ± k T } k = 0 ∞ , can be expressed as H σ (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, ⨎ 0 T f (x) d x can be computed efficiently using our new numerical quadrature formulas Q σ , n (t ; u) on the individual H σ (t ; u). Again, only 2 n function evaluations, namely, u (x n , k) , k = 0 , 1 , ... , 2 n − 1 , are needed for the whole process. [ABSTRACT FROM AUTHOR]