A fast solver for the Hilbert-type singular integral equations based on the direct Fourier spectral method

In this paper, we develop a fast and direct Fourier spectral method for solving the Hilbert-type singular integral equation. This method leads to a fully discrete linear system, whose coefficient matrix is expressed as the sum of a sparse matrix and a quasi-circulant matrix. We show that it requires a nearly linear computational cost to obtain and then solve the fully discrete linear system. We also prove that the proposed algorithm preserves the optimal convergent order. One numerical experiment is presented to demonstrate its approximate accuracy and computational efficiency, verifying the theoretical estimates.