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

Gaussian Fibonacci Circulant Type Matrices

“…The reason why we focus our attentions on circulant type matrices is to explore the application of it in the related field. On the basis of existing application situation [1][2][3][4][5][6][7][8], we will develop solving integrable system, Hamiltonian structure, and integral equations.…”

“…Circulant matrices have been used in solving integrable system [1], Hamiltonian structure [2,3], and integral equations [4][5][6][7][8]. By using the KdV and Boussinesq systems, the circulant forward shift matrix, and the antisymmetric circulant matrix, Weiss in [1] constructed a cosymplectic form and presents the factorization of the BLP equation by the periodic fixed points of its Bäcklund transformations.…”

“…Cai developed a fast and direct Fourier spectral method for solving the Hilbert type singular integral equation. When the direct Fourier spectral method is used to solve (1.1) in [7], Cai observed that the matrix representation of operator under the Fourier basis is a quasicirculant matrix. Abramyan proved the solvability of the system of (5) [8] for all ∈ , beginning with some 0 via that any circulant matrix is a normal matrix and its spectral norm is equal to the maximal modulus of its eigenvalues.…”