Numerical Methods for Nonlinear Volterra Integro-Differential Equations

Abstract

Let $u'(x) = f(x,u(x),Iu(x))$ and $u(a) = \mu $, the initial value, where I is the Volterra functional $Iu(x) = \int_a^x {k(x,s,u(s))ds} $. Call this the general nonlinear Volterra integro-differential equation (VIDE). Approximations are given for the functional I and its derivatives. These approximations are applied to create one-step algorithms for the constructive solution of the VIDE problem. Let $x \in [a,b]$, integer $N \geqq 1$, and ${{h = (b - a)} / N}$. Define $x_n = a - nh$ for $n = 0(1)N$. A class of algorithms (each depending upon an order parameter $p \geqq 1$ and upon the step size h) are defined. For each fixed p and h, each algorithm recursively generates a sequence $\{ u_n \} $ which approximates $\{ u(x_n) \} $. Two error propagation theorems are presented. THEOREM 1. Under suitable smoothness conditions, $u_n = u(x_n ) + O(h^p )$, where$p \geqq 1$is the order parameter. The convergence is uniform on closed intervals. THEOREM 2. Under further smoothness conditions, $u_n = u(x_n ) + h^p e(x_n ) + O(h^{p + 1} )$, where the principal error function$e(x)$is defined as the solution of another Volterra integro-differential equation. The results of various numerical experiments, including 4th order Runge–Kutta type algorithms, are presented. These experiments serve to confirm the validity of our results.