Automatic Integration of Functional Differential Equations: An Approach

Abstract

Heretofore, the automatic numerical integration of functional differential equations (FDEs) has received little attention. However, in recent years the use of such equations in the mathematical modeling of various biological, environmental, and societal processes has increased markedly. The computer solution of FDEs of the following form are considered. [formula omitted] (f a vector-valued function) where [formula omitted] for i ≤ n and where the required initial functions are [formula omitted]. An approach to the conversion of automatic ordinary differential equation (ODE) solvers to automatic FDE solvers is considered. The approach emphasizes the preservation of the essential characteristics of the original ODE solver, such as error estimation and step changing The use of such a converted algorithm, which is capable of integrating even more general FDEs than (*), is discussed The ability of the alogrlthm to overcome the problem of derivative discontinuities in the solutmn is also discussed. Test problems presented include systems of FDEs with multiple and vanishing lags, Volterra integro-differential equations, and FDEs with nonsmooth solutions. © 1975, ACM. All rights reserved.