Extended explicit Belʹtyukov pairs of orders 4 and 5 for Volterra integral equations of the second kind Original Research Article

The derivation of extended explicit Belʹtyukov pairs of methods for Volterra integral equations of the second kind is related to that of explicit Runge–Kutta pairs, but is more intricate. Techniques previously developed for deriving families of explicit Runge–Kutta pairs and a family of Belʹtyukov pairs of orders 3 and 4 are adapted to construct a parametric family of pairs of orders 4 and 5. While ten stages (of which one is reused) are required in this construction, it is an open question whether pairs of this type and order with fewer stages exist. A range of parameters for which the propagating method satisfies the kernel condition is given, and numerical performance of a representative pair is compared with that of a near-optimal 3,4 pair.