Abstract
The main contribution of this paper is the development of direct explicit methods of Runge-Kutta (RK) type for solving special fifth-Order Ordinary Differential Equations (ODEs). For this purpose, we have generalized RKD and RKT methods of special third and fourth-order ODEs. Using Taylor expansion, we have derived the algebraic equations of algebraic equations of order conditions for the proposed RKM integrators up to the eighth order. Based on these conditions, two RKM methods of orders five and six with three and four-stage are derived. Numerical implementation shows that the new methods agree well with existing RK methods, but requires less function evaluations. This is so due to the fact that RKM methods are direct; hence, they save considerable amount of computational time.
Highlights
The mathematical modeling of many real-life problems in physics, engineering and economics can be written as higher-order differential equations, ordinary or partial, (DEs) model
In this study, using the same technique, we have derived the algebraic equations of order conditions of RKM methods for solving special fifth-order Ordinary Differential Equations (ODEs)
We have derived the algebraic equations of order conditions for direct integrators of RKM for special fifth-order ordinary differential equations
Summary
The mathematical modeling of many real-life problems in physics, engineering and economics can be written as higher-order differential equations, ordinary or partial, (DEs) model. A derivation for nonlinear equations (order conditions) of direct explicit RKM methods for solving a special fifth-order ODEs is presented, along with two numerical methods for ODEs with orders five and six. We concerned with fifth-order ordinary differential equation with no appearance for the first, second, third and fourth derivatives w(i)(x), for i = 1,2,3,4.
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