Solution of Linear Caputo Fractional Differential Equations with Fractional Initial Conditions

The computation of solutions of Caputo fractional differential equations is paramount in modeling to establish its benefits over the corresponding integer order models. In the literature so far, in order to compute the solution of Caputo fractional differential equations, the solution is typically assumed to be a Cn function, which is a sufficient condition for the Caputo derivative to exist. In this work, we assume the necessary condition for the Caputo derivative of order nq,(n−1)<nq<n, to exist, which means that we assume it to be a Cnq function. Recently, it has been established that the Caputo derivative of order nq is sequential of order q. As such, we assume the fractional initial conditions. In our work, we have obtained an analytical solution for the Caputo fractional differential equation of order nq with fractional initial conditions by two different methods. Namely, the approximation method and the Laplace transform method. The application of our main results is illustrated with examples.