Application of Lanczos' minimized iterations to non-homogeneous linear integral equations with weak singularities.

Abstract

In this thesis, Lanczos' Method of Minimized Iterations is applied in a Hilbert space framework to solve a non-homogeneous linear integral equation of the second kind. The kernel of the integral equation is real, non-symmetric and has a weak singularity of a type frequently occurring in Potential Theory. In Chapter 1, the given operator is symmetrized. Theorem 2 shows that this symmetrization process does not affect the solution, and Theorem 5 shows that the symmetrized operator is completely continuous and self-adjoint. In Chapter 2, we use the concept of invariant (closed) subspace and Lanczos' Method of Minimized Iterations to find the approximate solution. Theorem 9 shows that this converges to the required solution faster than any geometrical progression.