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

2018/11/14 by Jian Yuan, Yuan, Jian, Guozhong Xiu +3
Mathematics · #FOS: Mathematics #Fractional Differential Equations Solutions #General Mathematics (math.GM) #Mathematical functions and polynomials #Nonlinear Differential Equations Analysis

paper · pdf · doi:10.48550/arxiv.1811.11537

openalex publication_date 2018/11/14 · openalex created_date 2018/12/11 · openalex updated_date 2026/07/28

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.

Citations

• Dynamics and Control of Initialized Fractional-Order Systems

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