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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
Published electronically: 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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Additional Information

Jinyuan Du
Affiliation: Department of Mathematics, Wuhan University, Wuhan 430072, People’s Republic of China

Keywords: Singular integral equations with Hilbert kernel, Hilbert singular integral operators, Hilbert singular quadrature operators, discretization matrices, collocation methods.
Received by editor(s): June 20, 2007
Received by editor(s) in revised form: May 11, 2008
Published electronically: December 10, 2008
Additional Notes: This project was supported by NNSF of China (#10471107) and RFDP of Higher Eduction of China (#20060486001).
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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