Two derivative and three derivative Runge-Kutta- Nyström methods for second-order ordinary differential equations - Salama, Tahani Mohamed (2019)
This thesis focuses mainly on deriving special two derivative and three derivative Runge-Kutta-Nyström (STDRKN, SThDRKN) methods for solving general secondorder ordinary differential equations (ODEs). The derivation of the explicit STDRKN methods by including the second and third derivatives which involves only one evaluation of second derivative and many evaluations of third derivative per step and explicit SThDRKN methods by including the second, third and fourth derivatives which involve only one evaluation of second derivative, one evaluation of third derivative, and many evaluations of fourth derivative per step has been presented. The regions of stability are presented. The implementation of STDRKN and SThDRKN methods in variable step size is also discussed. The numerical results are shown in terms of function evaluation and accuracy. The mathematical formulation of exponentially-fitted and trigonometrically-fitted for modified explicit STDRKN and SThDRKN methods and exponentially-fitted and trigonometrically-fitted for explicit general two derivative Runge-Kutta-Nyström (TDRKN) methods for solving the general second-order ODEs whose solutions involving exponential or trigonometric form has been described. The numerical results show that the new methods are more accurate and efficient than several existing methods in the literature. The semi-implicit STDRKN and SThDRKN methods are derived. The stability properties are investigated. Some numerical examples are given to illustrate the efficiency of the methods. As a whole, the two and three derivative Runge-Kutta- Nyström methods for solving general second-order ordinary differential equations have been presented. The illustrative examples demonstrate the accuracy advantage of the new methods.