On the stability of piecewise polynomial collocation methods for solving weakly singular integral equations of the second kind

Abstract

Piecewise polynomial collocation methods on special nonuniform grids are efficient methods for solving weakly singular Fredholm and Volterra integral equations but there is a widespread belief that those methods are numerically unstable in the case of large values of the nonuniformity parameter r. We show that this method by itself is stable and discuss some implementation problems that may lead to unstable behavior of numerical results.

Key words:

• weakly singular
• Fredholm integral equation
• Volterra integral equation
• nonuniform grid
• collocation method

Notes

The work of Raul Kangro was partially supported by EstSF grant no. 7044. The work of Inga Kangro has been partially supported by EstSF Postdoc grant No. JD05–35.