Explicit Direct Integrators of RK Type for Solving Special Fifth-Order Ordinary Differential Equations

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.