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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.

We propose stabilized explicit stochastic Runge–Kutta methods of strong order one half for Ito stochastic delay differential equations with one fixed delay. The family of the methods is constructed by embedding Runge–Kutta–Chebyshev methods of order one for ordinary differential equations. The values of a damping parameter of the methods are determined appropriately in order to obtain excellent mean square stability properties. Numerical experiments are carried out to confirm their order of convergence and stability properties.

Structure based partitioning of a system of ordinary differential equations is considered. A general form of the explicit multischeme Runge–Kutta type method for such systems is presented. Order conditions and simplifying conditions are written down. An algorithm of derivation of the sixth order method with seven stages and reuse with two free parameters is given. It embeds a fourth order error estimator. Numerical comparison to the Dormand–Prince method with the same computation cost but of lower order 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.

A theorem was originally proposed to deal with the stochastic theta methods when they are applied to a linear test equation with delay and non-delay in a diffusion term. We extend the theorem to a proposition in a general form, and use it for stability analysis of stochastic orthogonal Runge-Kutta-Chebyshev methods when the methods are applied to the test equation., International Conference on Numerical Analysis and Applied Mathematics 2019 (ICNAAM-2019), 23–28 September, 2019, Rhodes, Greece

In this paper we study a system of delay differential equations from the viewpoint of a finite time blow-up of the solution. We prove that the system admits blow-up solutions, no matter how small the length of the delay is. In the non-delay system every solution approaches to a stable unit circle in the plane, thus time delay induces blow-up of solutions, which we call “delay-induced blow-up” phenomenon. Furthermore, it is shown that the system has a family of infinitely many periodic solutions, while the non-delay system has only one stable limit cycle. The system studied in this paper is an example that arbitrary small delay can be responsible for a drastic change of the dynamics. We show numerical examples to illustrate our theoretical results.

Two families of structural four-stage Runge–Kutta methods of order five are considered. The choice of the scheme with the best numerical properties in each family is justified. The comparison of classical and Runge–Kutta–Nystrom fifth order methods is shown and the better performance for some test problems is demonstrated.Two families of structural four-stage Runge–Kutta methods of order five are considered. The choice of the scheme with the best numerical properties in each family is justified. The comparison of classical and Runge–Kutta–Nystrom fifth order methods is shown and the better performance for some test problems is demonstrated.

Diagonally implicit Runge–Kutta type methods are considered for retarded functional differential equations. Methods are constructed to be L-stable when applied to ordinary differential equations. Two approached are used: complete diagonally implicit methods and methods with first explicit stage. It is shown that the latter lets construction of methods of desired orders with fewer stages.

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.

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.

When solving delay differential equations (DDEs) with state-dependent delays the problem of breaking point detection is important. Points where the solution is not smooth enough to provide the order of the method must be included into the computational mesh, otherwise a reduction in the order of the solution will result. The problem, however is to detect and compute such points efficiently. Breaking points arise every time a delay falls on a previous breaking point (either of the calculated solution or in the history function). In the case of retarded DDEs the new breaking point is (at least) one order smoother than the previous breaking point that gave rise to it. For fixed or time-dependent delays the breaking points can be precomputed independent of the solution, but for state-dependent delays the positions of the breaking points depend on the computed solution. If a breaking point is detected and the step-size is changed in order to incorporate the point into the mesh, then the new step-size generates a new solution and the breaking point moves. Consequently, breaking point detection is traditionally performed iteratively, and is computationally expensive. The same breaking point can also be detected multiple times. In the current work we propose a fast non-iterative method for finding breaking points with sufficient precision to preserve the order of up to third or fourth order methods. Our method makes use of analytic continuation of the solution across breaking points (including possible breaking points in the initial history function), and we explain how we handle this carefully to attain the desired order. Test results are presented for Explicit Functional Continuous Runge–Kutta methods, showing that they retain their order of convergence when the solutions have breaking points.