Mediterr. J. Math. DOI 10.1007/s00009-016-0768-7 c  Springer International Publishing 2016 Numerical Solutions for Systems of Nonlinear Fractional Ordinary Differential Equations Using the FNDM Mahmoud S. Rawashdeh and Hadeel Al-Jammal Abstract. A new technique has been developed for analytical solutions of fractional order nonlinear ODE system. We propose a reliable method called the fractional natural decomposition method (FNDM). The FNDM is based on the natural transform method (NTM) and the Adomian decomposition method. We use the FNDM to con- struct new analytical approximate and exact solutions to systems of non- linear fractional ordinary differential equation (NLFODEs). The frac- tional derivatives are described in the Caputo sense. Mathematics Subject Classification. 35Q61, 44A10, 44A15, 44A20, 44A30, 44A35, 81V10. Keywords. Fractional natural decomposition method, System of fractional differential equations, Caputo fractional derivative. 1. Introduction Differential equations with fractional order have recently proved to be valu- able tools to the modeling of many physical phenomena and started to attract much more attention of Physicists and Mathematicians [4–6, 8–10, 18, 20]. These equations are represented by linear and nonlinear ODEs and solving such fractional differential equations (FDEs) is very important. So it is very important to find efficient methods for solving FDEs. Most of the fractional differential equations do not have exact analytical solutions; hence consider- able effort has been focused on approximate and numerical solutions of these equations. Recently, various researchers have introduced new methods in the literature. These methods include fractional Sumudu Transform [12, 16], fractional ma- trix method [6], fractional Adomian decomposition method (FADM) [7, 19, 27], the fractional reduced differential transform method (FRDTM) [25, 26],