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Volume 10, Number 5, 2020, Pages 2008-2023                                                                DOI:10.11948/20190317
equivalence of initialized Riemann-Liouville and caputo derivatives
Jian Yuan,Song Gao,Guozhong Xiu,Bao Shi
Keywords:Fractional calculus, initialized fractional derivatives, diffusive representation, equivalence of fractional derivatives.
      Initialization of fractional differential equations remains an ongoing problem. The initialization function approach and the infinite state approach provide two effective ways of dealing with this issue. The purpose of this paper is to prove the equivalence of the initialized Riemann-Liouville derivative and the initialized Caputo derivative with arbitrary order. By synthesizing the above two initialization theories, diffusive representations of the two initialized derivatives with arbitrary order are derived. The Laplace transforms of the two initialized derivatives are shown to be identical. Therefore, the two most commonly used derivatives are proved to be equivalent as long as initial conditions are properly imposed.
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