  Initialization of fractional differential equations remains an ongoing
problem. In recent years, the initialization function approach and the infinite
state approach provide two effective ways to deal with this problem. The
purpose of this paper is to prove the equivalence of the initialized
Riemann-Liouville derivatives and the initialized Caputo derivatives with
arbitrary orders. By synthesizing the above two initialization theories, the
diffusive representations of the two initialized derivatives with arbitrary
orders are derived. Laplace transforms of the two initialized derivatives are
shown to be equal. As a result, the two most commonly used derivatives are
proved to be equivalent when initial conditions are properly imposed.
