Discussion on the Leibniz rule and Laplace transform of fractional derivatives using series representation

ABSTRACT

Taylor series is a useful mathematical tool when describing and constructing a function. With the series representation, some properties of fractional calculus can be revealed clearly. On this basis, the Lebiniz rule and Laplace transform of fractional calculus is investigated. It is analytically shown that the commonly used Leibniz rule cannot be applied for Caputo derivative. Similarly, the well-known Laplace transform of Riemann–Liouville derivative is doubtful for n-th continuously differentiable function. After pointing out such problems, the exact formula of Caputo Leibniz rule and the explanation of Riemann–Liouville Laplace transform are presented. Finally, three illustrative examples are revisited to confirm the obtained results.

KEYWORDS:

• Fractional calculus
• Taylor series
• Leibniz rule
• Laplace transform
• non-zero initial instant

AMS CLASSIFICATIONS:

• Primary: 26A33
• Secondary: 30K05
• 44A10
• 65L05

Acknowledgments

The authors would like to thank the Associate Editor and the anonymous reviewers for their keen and insightful comments which greatly improved the contents and presentation.

Disclosure statement

No potential conflict of interest was reported by the authors.

ORCID

Yiheng Wei  http://orcid.org/0000-0002-0080-5365

Da-Yan Liu  http://orcid.org/0000-0003-2853-0129

Peter W. Tse  http://orcid.org/0000-0002-6796-7617

Yong Wang  http://orcid.org/0000-0002-6773-6544

Additional information

Funding

The work described in this paper was supported by the National Natural Science Foundation of China [61601431, 61573332, 61973291], the Anhui Provincial Natural Science Foundation [1708085QF141] and the fund of China Scholarship Council [201806345002].