Different types of DEs have a wide range of applications in both engineering and science. Typically, when modelling a physical quantity's variation, Recently, it has been found that models based on the theory of fractional-order derivatives and integrals provide an exceptionally good description of a wide range of scientific phenomena. The FDE is a differential equation that contains some derivatives of non-integer powers order. FDEs have become increasingly significant in the theoretical and applied parts of a wide variety of scientific and technical disciplines in recent years. The high-order ODE can be reduced to systems of first order ODEs, which can then be solved. Directly attacking the issue with numerical methods, however, is much more efficient in terms of accuracy, number of function evaluations, and processing time. In this article, the RKM method for solving ordinary differential equations has been introduced. This numerical approach has been generalized to be suitable for solving a class of fractional differential equations (FDEs). However, the developed RKM approach with three- and four-stages for solving sixth-order FODEs is developed. Moreover, this technique was used to solve various test problems, these examples to which the developed method was applied were for various functions with different values of ⍺ and then, the solutions of the developed numerical method were compared with the exact solution, the numerical results proved the efficiency and accuracy of the modified technique.
