In this paper, we are concerned with the deverlopment
of an explicit Runge-Kutta scheme for the numerical solution of delay differential
equations (DDEs) where one or more delay lies in the current Runge-Kutta
interval. The scheme presented is also applicable to the numerical solution
of Volterra functional equations (VFEs), although the theory is
not covered in this paper. We also derive the stability equations of the
scheme for the ODE
y'(t) = l y(t),
and the DDE
y'(t) = l y(t) + u y(t-T),
where the delay T and the Runge-Kutta stepsize
H are both constant. In the case of the DDE, we consider the two
distinct cases: (i) T >= H, and (ii) T < H. We evaluate
the performance of the scheme by solving several types of singular
DDE and VFE.
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