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In the paper explicit functional continuous Runge–Kutta and Runge–Kutta–Nystrom methods for retarded functional differential equations are considered. New methods for first order equations as well as for second order equations of the special form are constructed with the reuse of the last stage of the step. The order conditions for Runge–Kutta–Nystrom methods are derived. Methods of orders three, four and five which require less computations than the known methods are presented. Numerical solution of the test problems confirm the convergence order of the new methods and their lower computational cost is performed.

An attempt to construct fast Runge–Kutta methods for delay differential equations with distributed delays is made. As a starting point functional continuous methods for general retarded functional differential equations are considered. New explicit methods of order three and four are constructed, which are more effective than functional continuous Runge–Kutta methods of the same order. Numerical tests confirm the convergence order of the new methods. Comparison to closely related Runge–Kutta methods for Volterra integro-differential equations is made.

Structural partitioning of systems of ordinary differential equations is made on base of right-hand side dependencies on the unknown variables. It is used to construct fully explicit Runge–Kutta methods with several computational schemes applied to different parts of the system. The constructed structural methods require fewer right-hand side evaluations (stages) per step for some parts of the system than classic explicit Runge–Kutta methods of the same order. The full structural form of the system is presented, which after permutation of variables can be applied to any system of ordinary differential equation. For such structure a multischeme method is formulated and conditions of the sixth order are written down. We present simplifying conditions and reduce the system to a solvable smaller system. A particular computational scheme, that requires seven stages for a group without special structure and only six stages for other equations, is presented. Its sixth order is confirmed by a numerical convergence test.

Numerical methods for solving retarded functional differential equations of the second order with right-hand side independent of the function derivative are considered. The approach used by E. Nyström for second-order ordinary differential equations with the mentioned property is applied for construction of effective functional continuous methods. Order conditions are formulated, and example methods are constructed. They have fewer stages than Runge–Kutta type methods of the same order. Application of the constructed methods to test problems confirms their declared orders of convergence.

The so-called structural methods for systems of partitioned ordinary differential equations studied by Olemskoy are considered. An ODE system partitioning is based on special structure of right-hand side dependencies on the unknown functions. The methods are generalization of Runge–Kutta–Nystrom methods and as the latter are more efficient than classical Runge–Kutta schemes for a wide range of systems. Polynomial interpolants for structural methods that can be used for dense output and in standard approach to solve delay differential equations are constructed. The proposed methods take fewer stages than the existing most general continuous Runge–Kutta methods. The orders of the constructed methods are checked with constant step integration of test delay differential equations. Also the global error to computational costs ratios are compared for new and known methods by solving the problems with variable time-step.

We construct here an embedded Dormand–Prince pair of explicit methods of orders 6 and 4 for systems of ordinary differential equations with special structure, namely with two parts, in which the right-hand sides are dependent only on the unknown functions from the other group. The number of stages is six, which is fewer than for general explicit Runge–Kutta methods. The comparison to Dormand–Prince method of the same computation cost is made showing the higher accuracy of the suggested method.

We consider here numerical methods for systems of retarded functional differential equations of two equations in which the right-hand sides are cross-dependent of the unknown functions, i.e. the derivatives of unknowns don’t depend on the same unknowns. It is shown that using the special structure of the system one can construct functional continuous methods of Runge–Kutta type with fewer stages than it is necessary in case of general Runge–Kutta functional continuous methods. Order conditions and example methods of orders three and four are presented. Test problems are solved, demonstrating the declared convergence order of the new methods.

An explicit Runge-Kutta type method for systems of ordinary differential equations with special structure is considered. For partitioned systems a family of explicit methods of order six with just six stages is constructed, which makes them more efficient than classic Runge-Kutta methods of order six. It is shown that second order differential equations, which right-hand side doesn't depend on the first derivative, can be rewritten as the considered partitioned systems. Direct application of the constructed methods to them generates two different families of Runge-Kutta-Nystrom methods. The comparison of constructed methods with known methods of order six is held.

An explicit embedded pair of methods for systems of second order ordinary equations with special structure is considered. Two-parametric families of methods of orders four and three with automatic step-size control are constructed. The numeric comparison to known embedded Runge-Kutta pairs of the same order is held.