Regions of stability in the Numerical treatment of Volterra integro-differential equations

Conditions are determined for the stability of certain numerical methods for the approximate solution of integro-differential equations of the form \[ y'(x) = G\left( {x,y(x),\int_0^x {K(x,t,y(t))dt} } \right)\quad (x \geqq 0)\] (given $y(0)$) when applied to the test equation $y'(x) = \xi y(x) + \eta \int_0^x {y(t)dt + d(x)} $.The simplest results are those obtained (in § 4) for a class of methods which may be derived on applying an appropriate method to a system of integral equations derived from the integro-differential equation. These results are analogous to those obtained for integral equations by Baker and Keech (1978), from which they may be derived, and they are complementary to or consistent with earlier results of Brunner and Lambert (1974). A more general result is given for stability polynomials of other methods, and we indicate that similar results also obtain in the stability analysis of methods for the initial-value problem for a second-order differential equation.