Derivation of the Reduction Formula of Sixth Order and Seven Stages Runge-Kutta Method for the Solution of an Ordinary Differential Equation  

作　　者：Georgios D. Trikkaliotis Maria Ch. Gousidou-Koutita Georgios D. Trikkaliotis;Maria Ch. Gousidou-Koutita(Department of Mathematics, Aristotle University of Thessaloniki, Thessaloniki, Greece)

机构地区：[1]Department of Mathematics, Aristotle University of Thessaloniki, Thessaloniki, Greece

出　　处：《Applied Mathematics》2022年第4期338-355,共18页应用数学（英文）

摘　　要：This paper is describing in detail the way we define the equations which give the formulas in the methods Runge-Kutta 6^th order 7 stages with the incorporated control step size in the numerical solution of Ordinary Differential Equations (ODE). The purpose of the present work is to construct a system of nonlinear equations and then by solving this system to find the values of all set parameters and finally the reduction formula of the Runge-Kutta (6,7) method (6^th order and 7 stages) for the solution of an Ordinary Differential Equation (ODE). Since the system of high order conditions required to be solved is complicated, all coefficients are found with respect to 7 free parameters. These free parameters, as well as some others in addition, are adjusted in such a way to furnish more efficient R-K methods. We use the MATLAB software to solve several of the created subsystems for the comparison of our results which have been solved analytically. Some examples for five different choices of the arbitrary values of the systems are presented in this paper.This paper is describing in detail the way we define the equations which give the formulas in the methods Runge-Kutta 6^th order 7 stages with the incorporated control step size in the numerical solution of Ordinary Differential Equations (ODE). The purpose of the present work is to construct a system of nonlinear equations and then by solving this system to find the values of all set parameters and finally the reduction formula of the Runge-Kutta (6,7) method (6^th order and 7 stages) for the solution of an Ordinary Differential Equation (ODE). Since the system of high order conditions required to be solved is complicated, all coefficients are found with respect to 7 free parameters. These free parameters, as well as some others in addition, are adjusted in such a way to furnish more efficient R-K methods. We use the MATLAB software to solve several of the created subsystems for the comparison of our results which have been solved analytically. Some examples for five different choices of the arbitrary values of the systems are presented in this paper.

关 键 词：Initial Value Problem Runge-Kutta Methods Ordinary Differential Equations 

分 类 号：O241.8[理学—计算数学]