Robust projection method for Fredholm integral equation of the second kind

Abstract

The projection method based on the Schauder basis expansion for solving the Fredholm integral equation of the second kind is investigated. (1) It is shown that any given finite linearly independent function set in L ^p [0, 1] can be extended as a Schauder basis. The truncation of the dual Schauder bases is then used to form the projection in approximating the kernel. (2) With the finite rank approximation, the use of the Sherman–Morrison–Woodbury formula achieves efficient implementation. The computation is successive when dimension of projection space increases. Numerical experiments demonstrate the robustness of the Schauder basis expansion.

Keywords:

• Fredholm integral equation
• Schauder basis
• projection method
• degenerate kernel method
• Sherman–Morrison–Woodbury formula

2000 AMS Subject Classifications :

• 45B05
• 65R20

Acknowledgements

The authors would like to thank Professors Shengzhi Xu and Jixiu Chen at Fudan University for their helpful discussions. The authors were supported by sgst 09DZ2272900. The work of W. Gao is partly supported by the 111 Project and the National Science Foundation of China under Grant No. 10571031.