On the collocation methods for singular integral equations with Hilbert kernel

Abstract

In the present paper, we introduce some singular integral operators, singular quadrature operators and discretization matrices of singular integral equations with Hilbert kernel. These results both improve the classical theory of singular integral equations and develop the theory of singular quadrature with Hilbert kernel. Then by using them a unified framework for various collocation methods of numerical solutions of singular integral equations with Hilbert kernel is given. Under the framework, it is very simple and obvious to obtain the coincidence theorem of collocation methods, then the existence and convergence for constructing approximate solutions are also given based on the coincidence theorem.

Computational Techniques for Solving Mixed (1 + 1) Dimensional Integral Equations with Strongly Symmetric Singular Kernel

“…Gabdulkhaev and Tikhonov [5] applied general projection and projection-iteration techniques for solving SIE with the Cauchy kernel in the real line. Jinyuan [6] used the collocation method to obtain the solution of SIE with the Hilbert kernel.…”