Abstract
Two exponentially fitted and trigonometrically fitted explicit two-derivative Runge-Kutta-Nyström (TDRKN) methods are being constructed. Exponentially fitted and trigonometrically fitted TDRKN methods have the favorable feature that they integrate exactly second-order systems whose solutions are linear combinations of functions {exp(wx),exp(-wx)} and {sin(wx),cos(wx)} respectively, when w∈R, the frequency of the problem. The results of numerical experiments showed that the new approaches are more efficient than existing methods in the literature.
Highlights
We consider the numerical solution of the general second-order initial value problems (IVPs) of the following form: y (x) = f (x, y, y), y (x0) = y0, (1)
These lead to our new method, an explicit exponentially fitted two-derivative Runge-Kutta-Nystrom (TDRKN) two-stage fourth-order method denoted as EFTDRKN4(2)
These lead to our new explicit exponentially fitted explicit TDRKN three-stage fifth-order method denoted as EFTDRKN5(3)
Summary
Many well-known researchers have constructed Runge-Kutta-Nystrom methods (RKN) for solving (1) by using numerous techniques, for instance, exponentially fitted and trigonometrically fitted techniques. Van de Vyver [14] constructed the embedded pair of exponentially fitted RKN methods to solve orbital problems. Mathematical Problems in Engineering fitted RKN methods based on Garcia and Hairer methods of orders four and five, respectively, for periodic IVPs. Demba et al [21] constructed a 5(4) pair of embedded trigonometrically fitted RKN method for solving second-order IVPs where the solutions are oscillatory.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
