Solving directly special fourth-order ordinary differential equations using Runge-Kutta type method

In this paper, an explicit Runge–Kutta method for solving directly fourth-order ordinary differential equations (ODEs) is constructed and denoted as (RKFD). We present a relevant-colored tree theory and the associated B-series theory for the order conditions. Based on the order conditions a three-stage fourth-order RKFD method and a three-stage fifth-order RKFD method are constructed. Numerical illustrations are presented to show the efficiency of the new RKFD methods by comparing them with other existing Runge–Kutta Nyström (RKN) and Runge–Kutta (RK) methods in the scientific literature after converting the problem into a system of second order ODEs and a system of first order ODEs respectively

A sixth-order RKFD method with four-stage for directly solving special fourth-order ODEs

In this article, the general form of Runge-Kutta method for directly solving a special fourth- order ordinary differential equations denoted as RKFD method is given. The order conditions up to order seven are derived, based on the order conditions, we construct a new explicit four-stage sixth-order RKFD method denoted as RKFD6 method. Zero-stability of the method is proven. Comparisons are made using the existing Runge–Kutta methods after the problems are reduced to a system of first order ordinary differential equations. Numerical results are presented to illustrate the efficiency and competency of the new method

Guidance, flight mechanics and trajectory optimization. Volume 6 - The N-body problem and special perturbation techniques

Analytical formulations and numerical integration methods for many body problem and special perturbative technique