Holyhead, Peter Alan William
(1976)
Direct methods for the numerical solution of volterra integral equations of the first kind.
*University of Southampton, Doctoral Thesis*.

## Abstract

This thesis examines the conditions for convergence of direct methods for the numerical solution of the Volterra and generalized Abel integral equations of the first kind. A direct method produces its approximation on the points o£ a mesh through the solution of a triangular or near triangular matrix equation. The dominant problem is its likely instability, unbounded deterioration of the condition of the matrix as the meshlength is reduced. We prove the stability properties of a method are independent of the kernel of the integral equation within classes of kernel functions. Particular kernels in these classes are of convolution type. lie point out that in such a case the matrix for any meshlength can be embedded in an infinite matrix U. The method is stable, for the particular integral equation, when flu 1 f w. The structure of the method gives UW a quality of repetition in its columns to show U -1 can be described in terms of generating functions of the distinct columns of U, and obtain a necessary and a sufficient condition forr stability. The conditions are generalizations of the root condition for multistep methods in ordinary differential equations. The practical aspects of finding stable, convergent methods are studied. An alternative expression is given to the conditions on methods for the Volterra integral equation and two third order convergent methods are constructed. The basic series of the generating functions associated with a method for the Abel equation have to be summed, and an example is given. Other results are on the continuity of the conditions, on an idea of weak stability, and on an extension to a nonlinear integral equation.

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