The Numerical Solution of Volterra Functional Differential Equations by Euler’s Method

Let $\alpha \leqq a < b$ be real numbers and let $C([t_1 ,t_2 ] \to E^n )$ denote the space of continuous functions on $[t_1 ,t_2 ]$ into $E^n $ (n-dimensional Euclidean space). We consider the Cauchy problem for Volterra functional differential equations: \[y'(t) = F(y,t),\quad ,t \in [a,b];\qquad ,y(t) = g(t),\quad, t[\alpha ,b].\] Here, $F:C([\alpha ,b] \to E^n ]) \times [a,b] \to E^n $ is a Volterra functional, that is, $F(y,t)$ depends on t and on $y(s)$ for $s \in [\alpha ,t]$, but is independent of $y(s)$ for $s > t$; and $g \in C([\alpha ,a] \to E^n )$ is a specified initial function. This problem includes as special cases the initial value problems for ordinary differential equations, retarded ordinary differential equations, and Volterra integro-differential equations. We develop two methods for numerically solving the Cauchy problem for Volterra functional equations; these methods are generalizations of the methods of Euler and Heun for ordinary differential equations. It is proved that these methods converge and a numerical example is given.