I. Th. Famelis

On modified Runge–Kutta trees and methods

Modifled Runge{Kutta (mRK) methods can have interesting properties as their coe‐cients may depend on the step-length. By a simple perturbation of very few coe‐cients we may produce various function-fltted methods and avoid the overload of evaluating all the coe‐cients in every step. It is known that, for Runge{Kutta methods, each order condition corresponds to a rooted tree. When we expand this theory to the case of mRK methods, some of the rooted trees produce additional trees, called mRK rooted trees, and so additional conditions of order. In this work we present the relative theory including a theorem for the generating function of these additional mRK trees and explain the procedure to determine the extra algebraic equations of condition generated for a major subcategory of these methods. Moreover, e‐cient symbolic codes are provided for the enumeration of the trees and the generation of the additional order conditions. Finally, phase-lag and phase-fltted properties are analyzed for this case and speciflc phase-fltted pairs of orders 8(6) and 6(5) are presented and tested.

Phase-fitted RungeKutta pairs of orders 8(7)

A new phase fitted RungeKutta pair of orders 8(7) which is a modification of a well known explicit RungeKutta pair for the integration of periodic initial value problems is presented. Numerical experiments show the efficiency of the new pair in a wide range of oscillatory problems.

Runge-Kutta methods for fuzzy differential equations

Abstract Fuzzy differential equations (FDEs) generalize the concept of crisp initial value problems. In this article, we deal with the numerical solution of FDEs. The notion of convergence of a numerical method is defined and a category of problems which is more general than the one already found in the numerical analysis literature is solved. Efficient s-stage Runge–Kutta methods are used for the numerical solution of these problems and the convergence of the methods is proved. Several examples comparing these methods with the previously developed Euler method are displayed.

EXPLICIT NUMEROV TYPE METHODS FOR SECOND ORDER IVPs WITH OSCILLATING SOLUTIONS

New explicit hybrid Numerov type methods are presented in this paper. These efficient methods are constructed using a new approach, where we do not need the use of the intermediate high accuracy interpolatory nodes, since only the Taylor expansion of the internal points is needed. The methods share sixth algebraic order at a cost of five stages per step while their phase-lag order is 14 and partly satisfy the dissipation order conditions. It has be seen that the property of phase-lag is more important than the nonempty interval in constructing numerical methods for the solution of Schrodinger equation and related problems.1–3 Numerical results over some well known problems in physics and mechanics indicate the superiority of the new methods.

Symbolic derivation of Runge-Kutta order conditions

Tree theory, partitions of integer numbers, combinatorial mathematics and computer algebra are the basis for the construction of a powerful and efficient symbolic package for the derivation of Runge-Kutta order conditions and principal truncation error terms.

A P-stable singly diagonally implicit Runge–Kutta–Nyström method

A five-stage fifth-order singly diagonally implicit Runge–Kutta–Nyström method for the integration of second order differential equations possessing an oscillatory solution, is presented in this article. This method is P-stable, which is recommended for problems with a theoretical solution consisting of a periodic part of moderate frequency with a high frequency oscillation with small amplitude superimposed. It also attains an order which is one higher than existing methods of this type. Numerical comparisons with existing methods of this type show its clear advantage.

EXPLICIT EIGHTH ORDER NUMEROV-TYPE METHODS WITH REDUCED NUMBER OF STAGES FOR OSCILLATORY IVPs

Using a new methodology for deriving hybrid Numerov-type schemes, we present new explicit methods for the solution of second order initial value problems with oscillating solutions. The new methods attain algebraic order eight at a cost of eight function evaluations per step which is the most economical in computational cost that can be found in the literature. The methods have high amplification and phase-lag order characteristics in order to suit to the solution of problems with oscillatory solutions. The numerical tests in a variety of problems justify our effort.

On using explicit Runge-Kutta-Nyström methods for the treatment of retarded differential equations with periodic solutions

In this work we dial with the treatment of second order retarded differential equations with periodic solutions by explicit Runge-Kutta-Nystrom methods. In the past such methods have not been studied for this class of problems. We refer to the underline theory and study the behavior of various methods proposed in the literature when coupled with Hermite interpolants. Among them we consider methods having the characteristic of phase-lag order. Then we consider continuous extensions of the methods to treat the retarded part of the problem. Finally we construct scaled extensions and high order interpolants for RKN pairs which have better characteristics compared to analogous methods proposed in the literature. In all cases numerical tests and comparisons are done over various test problems.

NUMEROV-TYPE METHODS FOR OSCILLATORY LINEAR INITIAL VALUE PROBLEMS

We present a new explicit Numerov-type method for the solution of second-order linear initial value problems with oscillating solutions. The new method attains algebraic order seven at a cost of six function evaluations per step. The method has the characteristic of zero dissipation and high phase-lag order making it suitable for the solution of problems with oscillatory solutions. The numerical tests in a variety of problems justify our effort.

A new eighth order exponentially fitted explicit Numerov-type method for solving oscillatory problems

In this work we derive a new family of seven stages, eighth order, exponential fitted, Numerov (i.e. two-step) type methods. A particular variable coefficient method is constructed. The numerical results show that it outperforms exponential fitted two step hybrid method found in the literature sharing similar characteristics. Moreover, the presented method outperforms Numerov type methods with minimized phase-lag for problems in consideration share a non dominant linear part.