Construction of Numerical RKM-Method for Solving A Class of Twelves-Order Ordinary Differential Equations

Abstract

     This paper constructs and generalizes the numerical Runge-Kutta-Mohammed (RKM) method for solving twelve-order ordinary differential equations (ODEs). The novel contribution of this study is the development and generalization of the numerical RKM methods for solving ODEs of the order of less than a tenth. The algebraic order conditions (OCs) for the proposed RKM method are derived up to order thirteen using Taylor expansion. Then, the constructed method has been derived from these order conditions. However, the proposed numerical RKM method has been evaluated at some implementations and compared to an existing Runge-Kutta (RK) method to determine the method's viability. Moreover, this comparison demonstrates the proposed direct method is more efficient than the classical method in terms of efficiency and accuracy. Also, numerical implementations are used to prove the efficiency and time complexity of function evaluations. This direct RKM method is a suggested technique for solving ODEs of twelve orders which has great features like a direct and efficient method. Consequently, the proposed method requires less time complexity of computation than other methods.