Equivalence of the Initialized Riemann-Liouville Derivatives and the Initialized Caputo Derivatives

Abstract: 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.