Chebyshev collocation method for solving singular integral equation with cosecant kernel

Abstract

A collocation method based on Chebyshev polynomials is proposed for solving cosecant-type singular integral equations (SIE). For solving SIE, difficulties lie in its singular term. In order to remove singular term, we introduce Gauss–Legendre integration and integral properties of the cosecant kernel. An advantage of this method is to approximate the best uniform approximation by the best square approximation to obtain the unknown coefficients in the method. On the other hand, the convergence is fast and the accuracy is high, which is verified by the final numerical experiments compared with the existing references.

Keywords:

• collocation method
• Chebyshev polynomials
• best uniform approximation
• best square approximation

2000 AMS Subject Classifications::

• 45E99
• 45A05

Acknowledgements

The authors thank the unknown referees for their careful reading, helpful comments and valuable suggestions. The work was supported by the Scientific Research Foundation of Harbin Institute of Technology at Weihai (grant no. HIT(WH)XBQD201011), Shandong Province Natural Science Foundation (grant no. IMEQ03140001) and the Natural Scientific Research Innovation Foundation in Harbin Institute of Technology (grant nos. HIT.NSRIF.2011102 and HIT.NSRIF.2011103).