A unified approach with spectral convergence for the evaluation of hypersingular and supersingular integrals with a periodic kernel

In this paper, a general class of methods is proposed for the evaluation of hypesingular/supersingular integrals with a periodic integrand, of singularity higher than or equal to 2. The method is based on regularizing the singular integrals into proper regular integrals when we can apply the classical trapezoidal rule. Error analysis is presented for hypesingular/supersingular integrals with a singularity of order p+1p+1, where pp is a positive integer. We prove that the convergence rate of the method is of O(lnn/nk+α)O(lnn/nk+α) if the integrand uu satisfies u∈C̃k+p,α[0,2π] (for any positive integer kk), with exponential convergence rate if the integrand uu is analytic. Several numerical examples are provided to support the theoretical error analysis.