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 ε dRs and the coefficients matrix A ε dRs×s are replaced by ℝs-valued and dRs×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.