High Order Methods for a Class of Volterra Integral Equations with Weakly Singular Kernels

The solution of the Volterra integral equation, \[ ( * )\qquad x(t) = g_1 (t) + \sqrt {t}g_2 (t) + \int _0^t \frac {K(t,s,x(s))} {\sqrt {t - s} } ds, \quad 0 \leqq t \leqq T,\] where $g_1 (t)$, $g_2 (t)$ and $K(t,s,x)$ are smooth functions, can be represented as $x(t) = u(t) + \sqrt {t}v(t) $,$0 \leqq t \leqq T$, where $u(t)$, $v(t)$ are, smooth and satisfy a system of Volterra integral equations. In this paper, numerical schemes for the solution of (*) are suggested which calculate $x(t)$ via $u(t)$, $v(t)$ in a neighborhood of the origin and use (*) on the rest of the interval $0 \leqq t \leqq T$. In this way, methods of arbitrarily high order can be derived. As an example, schemes based on the product integration analogue of Simpson's rule are treated in detail. The schemes are shown to be convergent of order $h^{{7 / 2}} $. Asymptotic error estimates are derived in order to examine the numerical stability of the methods.