Numerical results for linear Fredholm integral equations of the first kind over surfaces in three dimensions

Abstract

Linear Fredholm integral equations of the first kind over surfaces are less familiar than those of the second kind, although they arise in many applications like computer tomography, heat conduction and inverse scattering. This article emphasizes their numerical treatment, since discretization usually leads to ill-conditioned linear systems. Strictly speaking, the matrix is nearly singular and ordinary numerical methods fail. However, there exists a numerical regularization method – the Tikhonov method – to deal with this ill-conditioning and to obtain accurate numerical results.

Keywords:

• Fredholm integral equations
• ill-posed problems
• regularization
• Tikhonov method
• Nyström interpolation
• boundary element method

2000 AMS Subject Classifications :

• 45B05
• 65F22
• 65N38

Acknowledgements

The author thanks Professor Kendall E. Atkinson for his source code that solves the Laplace equation by a boundary element method (BIEPACK). Special thanks goes to Professor Tzu-Chu Lin for numerous fruitful discussions.