Derivation of embedded explicit RK type methods for directly solving class of seventh-order ordinary differential equations

In this paper, a pair of embedded integrator for solving class of 7th-order ordinary differential equations (ODEs) of the form w((7))(t) = g(t, w(t)) denoted by RKM6(5), which is 5th-order integrator embedded in 6th -order integrator, is presented. The proposed method has FSAL property, it mean that the vector c and matrix A of RKM5, RKM6 are identical. The derivation strategy of the of the embedded integrator is that the higher-order integrator is used to approximate accurate solutions and the lower-order integrator should give the best error estimates. Variables step-size algorithm is developed based on the property of absolute errors should be minimum. The proposed method used to solve a class of 7th-order ODEs Numerical solutions of the implementations are compared with the classical embedded RK integrator in which require the problems of the study to be reduced into a system of 1th-order ODEs. They have proven the efficiency and the accurate of the proposed RKM6(5) integrator.