Analytical solution with existence and uniqueness conditions of non-linear initial value multi-order fractional differential equations using Caputo derivative

The main focus of this study is to apply the two-step Adomian decomposition method (TSADM) for finding a solution of a fractional-order non-linear differential equation by using the Caputo derivative. We are interested in obtaining an analytical solution with two main constraints, that are, without converting the non-linear fractional differential equation to a system of linear algebraic fractional equation, and secondly, with less number of iterations. Moreover, we have investigated conditions for the existence and uniqueness of a solution with the help of some fixed point theorems. Furthermore, the method is demonstrated with the help of some examples. We also compare the results with the Adomian decomposition method (ADM), the modified Adomian decomposition method, and the combination of the ADM and a spectral method. It is concluded that the TSADM provides an analytical solution of fractional-order non-linear differential equation, while the other methods furnish an approximate solution.