Unified compact numerical quadrature formulas for Hadamard finite parts of singular integrals of periodic functions.

We consider the numerical computation of finite-range singular integrals <graphic href="MediaObjects/10092_2021_407_Equ83_HTML.png"></graphic> that are defined in the sense of Hadamard Finite Part, assuming that g ∈ C ∞ [ a , b ] and f (x) ∈ C ∞ (R t) is T-periodic with f ∈ C ∞ (R t) , R t = R \ { t + k T } k = - ∞ ∞ , T = b - a . Using a generalization of the Euler–Maclaurin expansion developed in [A. Sidi, Euler–Maclaurin expansions for integrals with arbitrary algebraic endpoint singularities. Math. Comp., 81:2159–2173, 2012], we unify the treatment of these integrals. For each m, we develop a number of numerical quadrature formulas T ^ m , n (s) [ f ] of trapezoidal type for I[f]. For example, three numerical quadrature formulas of trapezoidal type result from this approach for the case m = 3 , and these are T ^ 3 , n (0) [ f ] = h ∑ j = 1 n - 1 f (t + j h) - π 2 3 g ′ (t) h - 1 + 1 6 g ′ ′ ′ (t) h , h = T n , T ^ 3 , n (1) [ f ] = h ∑ j = 1 n f (t + j h - h / 2) - π 2 g ′ (t) h - 1 , h = T n , T ^ 3 , n (2) [ f ] = 2 h ∑ j = 1 n f (t + j h - h / 2) - h 2 ∑ j = 1 2 n f (t + j h / 2 - h / 4) , h = T n. <graphic href="10092_2021_407_Article_Equ51.gif"></graphic> For all m and s, we show that all of the numerical quadrature formulas T ^ m , n (s) [ f ] have spectral accuracy; that is, T ^ m , n (s) [ f ] - I [ f ] = o (n - μ) as n → ∞ ∀ μ > 0. <graphic href="10092_2021_407_Article_Equ52.gif"></graphic> We provide a numerical example involving a periodic integrand with m = 3 that confirms our convergence theory. We also show how the formulas T ^ 3 , n (s) [ f ] can be used in an efficient manner for solving supersingular integral equations whose kernels have a (x - t) - 3 singularity. A similar approach can be applied for all m. [ABSTRACT FROM AUTHOR]