Interpolation based formulation of the oscillatory finite Hilbert transforms

Abstract

A novel quadrature based on interpolation is proposed for approximation of the finite Hilbert transforms with high-frequency parameter. The transforms are arduous to evaluate due to the existence of Cauchy-type singularity and oscillatory kernel. Accordingly, we fixate the Lagrange interpolation technique with asymptotic-meshfree collocation method to tackle the two challenges. Applying Lagrange’s interpolation and root-coefficient relation of a polynomial to avoid Cauchy-type singularity in the domain. After a large manipulation, the finite Hilbert transform is reduced to a sum of finite Fourier transforms and a singular part integral. The meshfree collocation method is implemented to evaluate the oscillatory Fourier transforms, while the singular part integral is evaluated analytically. Numerical examples confirm the accuracy and efficiency of the proposed algorithm. Results of the new procedure are compared with some state of the art methods and assure better performance of the proposed procedure.