Solutions of the initial value problem for nonlinear fractional ordinary differential equations by the Rach–Adomian–Meyers modified decomposition method

In this paper we present the generalized Adomian–Rach theorem and the generalized Rach–Adomian–Meyers modified decomposition method for solving multi-order nonlinear fractional ordinary differential equations. We consider different classes of initial value problems for nonlinear fractional ordinary differential equations, including the case of real-valued orders and another case of rational-valued orders, which are solved by the present method. This method can treat any analytic nonlinearity. The coefficients of the solution in the form of a generalized power series are determined by a convenient recurrence scheme, which does not involve integration operations compared with the classic Adomian decomposition method.