A Runge-Kutta type method for directly solving special fourth-order ordinary differential equations (ODEs) which is denoted by RKFD method is constructed. The order conditions of RKFD method up to order five are derived; based on the order conditions, three-stage fourth- and fifth-order Runge-Kutta type methods are constructed. Zero-stability of the RKFD method is proven. Numerical results obtained are compared with the existing Runge-Kutta methods in the scientific literature after reducing the problems into a system of first-order ODEs and solving them. Numerical results are presented to illustrate the robustness and competency of the new methods in terms of accuracy and number of function evaluations.

We present two pairs of embedded Runge-Kutta type methods for direct solution of fourth-order ordinary differential equations (ODEs) of the formy(iv)=f(x,y)denoted as RKFD methods. The first pair, which we will call RKFD5(4), has orders 5 and 4, and the second one has orders 6 and 5 and we will call it RKFD6(5). The techniques used in the derivation of the methods are that the higher order methods are very precise and the lower order methods give the best error estimate. Based on these pairs, we have developed variable step codes and we have used them to solve a set of special fourth-order problems. Numerical results show the robustness and the efficiency of the new RKFD pairs as compared with the well-known embedded Runge-Kutta pairs in the scientific literature after reducing the problems into a system of first-order ordinary differential equations (ODEs) and solving them.

This paper provides a four-stage Trigonometrically Fitted Improved Runge-Kutta (TFIRK4) method of four orders to solve oscillatory problems, which contains an oscillatory character in the solutions. Compared to the traditional Runge-Kutta method, the Improved Runge-Kutta (IRK) method is a natural two-step method requiring fewer steps. The suggested method extends the fourth-order Improved Runge-Kutta (IRK4) method with trigonometric calculations. This approach is intended to integrate problems with particular initial value problems (IVPs) using the set functions and for trigonometrically fitted. To improve the method's accuracy, the problem primary frequency is used. The novel method is more accurate than the conventional Runge-Kutta method and IRK4. Several test problems for the system of first-order ordinary differential equations carry out numerically to demonstrate the effectiveness of this approach. The computational studies show that the TFIRK4 approach is more efficient than the existing Runge-Kutta methods.