Derivation of direct explicit methods of RKM-type for solving special class of tenth-order ordinary differential equations

The objective of this study is centered around of the modification and improvement of computational efficiency of the directly method of RK-type for solving special class of ordinary differential equations (ODEs) of tenth-order. As a first step in our work, RK, RKN, RKT, RKD, and RKM-integrators have generalized for solving the special classes of ODEs in which have orders less than orders-tenth. The non-linear algebraic equations of the order conditions (OCs) have been derived for constructing of proposed RKM-integrators up to the eighteenth-order using Taylor expansion approach. RKM-method of tenth-order with five-stages is derived based on these OCs. The numerical results of a set of special tenth-order ODEs have clearly shown the efficiency and advantage of the new method and the exact and numerical solutions of these examples are identical since the fact that the developed method is more accurate and efficient method.