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Equivalence of higher order linear Riemann-Liouville fractional differential and integral equations


Author: Kunquan Lan
Journal: Proc. Amer. Math. Soc. 148 (2020), 5225-5234
MSC (2010): Primary 34A08; Secondary 26A33, 34A12, 45D05
DOI: https://doi.org/10.1090/proc/15169
Published electronically: September 18, 2020
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Abstract: A linear $ n$th order (Riemann-Liouville) fractional differential equation with $ m+1$ initial values, together with a suitable assumption, is proved to be equivalent to a Volterra integral equation of the second kind involving an $ n$th order (Riemann-Liouville) fractional integral operator. Two special cases of the result are given: one shows that a well-known result on the solution of an $ n$th order fractional differential equation needs an additional condition to hold, and another strengthens a previous result on an $ n$th order fractional integral operator composed with an $ n$th order fractional differential operator.


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Additional Information

Kunquan Lan
Affiliation: Department of Mathematics, Ryerson University, Toronto, Ontario, M5B 2K3 Canada
MR Author ID: 256493
Email: klan@ryerson.ca

DOI: https://doi.org/10.1090/proc/15169
Keywords: Riemann-Liouville fractional integral operator, Riemann-Liouville fractional differential operator, linear fractional differential equation, Volterra integral equation of the second kind, initial value problems, equivalence
Received by editor(s): March 3, 2020
Received by editor(s) in revised form: April 16, 2020
Published electronically: September 18, 2020
Additional Notes: The author was supported in part by the Natural Sciences and Engineering Research Council (NSERC) of Canada under grant no. 135752-2018.
Communicated by: Wenxian Shen
Article copyright: © Copyright 2020 American Mathematical Society