Stability in the Numerical Treatment of Volterra Integral and Integro-Differential Equations with emphasis on Finite Recurrence Relations.

Abstract

In the last two decades the theory of Volterra integral equations and of integro-differential equations has developed extensively. New classes of methods for the numerical solution of such equations have been developed and at the same time there have been advances in the qualitative theory of these equations. More frequent use is being made of Volterra equations to model various physical and biological phenomenon as new insight has occurred into the asymptotic behaviour of solutions. In consequence, there has emerged a need for reliable and efficient methods for the numerical treatment of such equations. This thesis is concerned with an aspect of numerical solution of Volterra integral and integro-differential equations. In Chapters 1 and 2 we are concerned with background material. We provide results on the classical theory of Volterra equations in Chapter 1 and on numerical methods in Chapter 2. The original material is contained in Chapters 3, 4 and 5. Here, stability results which involve the construction and analysis of finite-term recurrence relations are presented. The techniques relate to the treatment of Volterra integral and integro-differential equations. They permit the analysis of classical and 7-modified numerical methods. The results presented should be viewed as a contribution towards an understanding of numerical stability for the methods considered. The area is one in which further work (subsequent to the present investigation and involving advanced techniques) has been performed and where open questions still remain. The techniques which are employed in this thesis are applicable in other areas of numerical analysis and therefore have intrinsic interest