Comparison principles of fractional differential equations with non-local derivative and their applications

Mohammed Al-Refai; Dumitru Baleanu; Mohammed Al-Refai; Dumitru Baleanu

doi:10.3934/math.2021088

AIMS Mathematics

2021, Volume 6, Issue 2: 1443-1451. doi: 10.3934/math.2021088

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Research article

Comparison principles of fractional differential equations with non-local derivative and their applications

Mohammed Al-Refai ^{1
,
,},
Dumitru Baleanu ^2,3,4

1.
Department of Mathematics, College of Science, Yarmouk University, Irbed, Jordan
2.
Department of Mathematics and Computer Science, Cankaya University, Angara, Turkey
3.
Institute of Space Sciences, Magurele-Bucharest, Romania
4.
Department of Medical Research, China Medical University Hospital, China Medical University, Taichung, Taiwan

Received: 17 September 2020 Accepted: 10 November 2020 Published: 20 November 2020
MSC : 34A08, 35B50, 26A33

In this paper, we derive and prove a maximum principle for a linear fractional differential equation with non-local fractional derivative. The proof is based on an estimate of the non-local derivative of a function at its extreme points. A priori norm estimate and a uniqueness result are obtained for a linear fractional boundary value problem, as well as a uniqueness result for a nonlinear fractional boundary value problem. Several comparison principles are also obtained for linear and nonlinear equations.
- fractional differential equations,
- maximum principle,
- fractional derivatives
Citation: Mohammed Al-Refai, Dumitru Baleanu. Comparison principles of fractional differential equations with non-local derivative and their applications[J]. AIMS Mathematics, 2021, 6(2): 1443-1451. doi: 10.3934/math.2021088

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Abstract

In this paper, we derive and prove a maximum principle for a linear fractional differential equation with non-local fractional derivative. The proof is based on an estimate of the non-local derivative of a function at its extreme points. A priori norm estimate and a uniqueness result are obtained for a linear fractional boundary value problem, as well as a uniqueness result for a nonlinear fractional boundary value problem. Several comparison principles are also obtained for linear and nonlinear equations.

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