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Equivalence of Initialized Riemann-Liouville and Caputo Derivatives
Jian Yuan,Song Gao,Bao Shi,Guozhong Xiu
Keywords:Fractional calculus, initialized fractional derivatives, diffusive representation, equivalence of fractional derivatives
      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 equivalence of initialized Riemann-Liouville derivatives and initialized Caputo derivatives with arbitrary orders. By synthesizing the above two initialization theories, diffusive representations of the two initialized derivatives with arbitrary orders are derived. Laplace transforms of the two initialized derivatives are shown to be identical. Thus, the two most commonly used derivatives are proved to be equivalent when initial conditions are properly imposed.