Generally, the higher-order ordinary differential equation (ODE) is used to solve by converting the ODE, nth - times, to a system of first-order ODEs to be nth dimensions system of ODEs. However, it is a lot more efficient in terms of accuracy, number of function evaluations as well as computational time if the problem of ODEs can be solved directly using direct numerical methods. In this thesis, we are focus on the derivations, adaptation and modifications of the direct numerical methods based on RKM methods for solving some classes of ODEs, PDEs, FDEs in addition to fractional partial differential equations (FPDEs). Also, we improve these methods to be more efficient for solving special or general classes of these DEs. As a result, solving FDEs is an essential component of scientific inquiry
