ON NEW SOLUTIONS OF THE NORMALIZED FRACTIONAL DIFFERENTIAL EQUATIONS

The normalized fractional derivatives (NFDs) are a normalization of the existing fractional derivatives (FDs) which admit geometrical meanings. They have finite ordinary derivatives at the initial point, which cause smoothness of solutions for related fractional differential equations (FDEs). For FDs with non-singular kernels, the NFDs do not vanish at the starting point, in general, and therefore related FDEs admit solutions without the need to impose extra conditions. In this paper, we consider FDEs with the normalized Caputo-Fabrizio derivative of Caputo type. We show in closed forms the solutions of related FDEs and show that the Cauchy problem with the NFD admits a nontrivial solution. We also define the higher-order NFDs and show that related FDEs can be solved by transforming them to integro-differential equations with integer orders.