A new error analysis for a cubic spline approximate solution of a class of Volterra integro-differential equations

Abstract

In this paper a third-order numerical method is considered which utilizes a twice continuously differentiable third degree spline to approximate the solution of \[ x ˙ ( t ) = F ( t , x ( t ) , ∫ a t K ( t , u , x ( u ) ) d u ) , x ( a ) = x 0 , \begin {array}{*{20}{c}} {\dot x(t) = F\left ( {t,x(t),\int _a^t {K(t,u,x(u))\;du} } \right ),} \hfill \\ {x(a) = {x_0},} \hfill \\ \end {array} \] at discrete points in the interval [a, b]. The error analysis uses a technique usually associated with linear multistep methods.