On the collocation methods for singular integral equations with Hilbert kernel

Du, Jinyuan

doi:10.1090/S0025-5718-08-02182-0

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ISSN 1088-6842(online) ISSN 0025-5718(print)

On the collocation methods for singular integral equations with Hilbert kernel

Author: Jinyuan Du
Journal: Math. Comp. 78 (2009), 891-928
MSC (2000): Primary 65E05, 65J10, 41A55, 42A10
Posted: December 10, 2008
MathSciNet review: 2476564
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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.

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