Runge-Kutta Methods for Retarded Functional Differential Equations

We introduce Runge–Kutta (RK) methods for Retarded Functional Differential Equations (RFDEs).With respect to RK methods (A, b, c) for Ordinary Differential Equations the weights vector b ∈ Rs and the coefficients matrix A ∈ Rs×s are replaced by Rsvalued and Rs×s-valued polynomial functions b(·) and A(·) respectively. Such methods for RFDEs are different from Continuous RK (CRK) methods where only the weights vector is replaced by a polynomial function. We develop order conditions and construct explicit methods up to the convergence order four.