EQUIVALENCE OF INITIALIZED RIEMANN-LIOUVILLE AND CAPUTO DERIVATIVES

Jian Yuan; Song Gao; Guozhong Xiu; Bao Shi

doi:10.11948/20190317

2020 Volume 10 Issue 5

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Jian Yuan, Song Gao, Guozhong Xiu, Bao Shi. EQUIVALENCE OF INITIALIZED RIEMANN-LIOUVILLE AND CAPUTO DERIVATIVES[J]. Journal of Applied Analysis & Computation, 2020, 10(5): 2008-2023. doi: 10.11948/20190317

Citation:

Jian Yuan, Song Gao, Guozhong Xiu, Bao Shi. EQUIVALENCE OF INITIALIZED RIEMANN-LIOUVILLE AND CAPUTO DERIVATIVES[J]. Journal of Applied Analysis & Computation, 2020, 10(5): 2008-2023. doi: 10.11948/20190317

EQUIVALENCE OF INITIALIZED RIEMANN-LIOUVILLE AND CAPUTO DERIVATIVES

1.
School of Transportation and Vehicle Engineering, Shandong University of Technology, Zibo 255000, China
2.
School of Basic Science for Aviation, Naval Aviation University, Yantai 264001, China

Corresponding author: Email address:gaosong@sdut.edu.cn(S. Gao)

Abstract

Abstract

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.
- Fractional calculus /
- initialized fractional derivatives /
- diffusive representation /
- equivalence of fractional derivatives

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References

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