# Two Step Runge-Kutta-Nyström method for solving second order ordinary differential equations

## Citation

Md Ariffin, Latifah (2016) Two Step Runge-Kutta-Nyström method for solving second order ordinary differential equations. Doctoral thesis, Universiti Putra Malaysia.

## Abstract

In this research, methods that will be able to solve the second order initial value problem (IVP) directly are developed. These methods are in the scheme of a multi-step method which is known as the two-step method. The two-step method has an advantage as it can estimate the solution with less function evaluations compared to the one-step method. The selection of step size is also important in obtaining more accurate and efficient results. Smaller step sizes will produce a more accurate result, but it lengthens the execution time. Two-Step Runge-Kutta (TSRK) method were derived to solve first-order Ordinary Differential Equations (ODE). The order conditions of TSRK method were obtained by using Taylor series expansion. The explicit TSRK method was derived and its stability were investigated. It was then analyzed experimentally. The numerical results obtained were analyzed by making comparisons with the existing methods in terms of maximum global error, number of steps taken and function evaluations. The explicit Two-Step Runge-Kutta-Nyström (TSRKN) method was derived with reference to the technique of deriving the TSRK method. The order conditions of TSRKN method were also obtained by using Taylor series expansion. The strategies in choosing the free parameters were also discussed. The stability of the methods derived were also investigated. The explicit TSRKN method was then analyzed experimentally and comparisons of the numerical results obtained were made with the existing methods in terms of maximum global error, number of steps taken and function evaluations. Next, we discussed the derivation of an embedded pair of the TSRKN (ETSRKN) methods for solving second order ODE. Variable step size codes were developed and numerical results were compared with the existing methods in terms of maximum global error, number of steps taken and function evaluations. The ETSRKN were then used to solve second-order Fuzzy Differential Equation (FDE). We observe that ETSRKN gives better accuracy at the end point of fuzzy interval compared to other existing methods. In conclusion, the methods developed in this thesis are able to solve the system of second-order differential equation (DE) which consists of ODE and FDE directly.