On Failed Methods of Fractional Differential Equations: The Case of Multi-step Generalized Differential Transform Method

Numerical treatment of fractional differential equations, in a reliable and accurate way, is very challenging in comparison with classical integer-order differential equations. This difficulty is primarily related to the effect of non-local structure of fractional differentiation operators, to the solution of nonlinear equations involved in implicit methods and so forth. In this paper, a so-called method for fractional differential equations (FDEs) is briefly described: the multi-step generalized differential transform method (MSGDTM). It is shown that the method takes the incorrect approach in dealing with FDEs. The goal is to demonstrate that the MSGDTM is based on failed assumptions and therefore unfit for FDEs. For further verification, an illustrative example is given, in which the MSGDTM is compared with other effective and accurate methods such as fractional linear multi-step methods and predictor–corrector method of Adams–Bashforth–Moulton type. The obtained results show that the MSGDTM is unfit for FDEs.