Generalized fractional differential equations for past dynamic

Dumitru Baleanu; Babak Shiri; Dumitru Baleanu; Babak Shiri

doi:10.3934/math.2022793

AIMS Mathematics

2022, Volume 7, Issue 8: 14394-14418. doi: 10.3934/math.2022793

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Research article

Generalized fractional differential equations for past dynamic

Dumitru Baleanu ^1,2,3,
Babak Shiri ^{4
,
,}

1.
Cankaya University, Department of Mathematics, 06530 Balgat, Ankara, Turkey
2.
Institute of Space Sciences, Magurele-Bucharest, Romania
3.
Department of Medical Research, China Medical University Hospital, China Medical, University, Taichung, Taiwan
4.
Data Recovery Key Laboratory of Sichuan Province, College of Mathematics and Information Science, Neijiang Normal University, Neijiang 641100, China

Received: 15 March 2022 Revised: 21 May 2022 Accepted: 23 May 2022 Published: 06 June 2022
MSC : 34A08, 45G05

Well-posedness of the terminal value problem for nonlinear systems of generalized fractional differential equations is studied. The generalized fractional operator is formulated with a classical operator and a related weighted space. The terminal value problem is transformed into weakly singular Fredholm and Volterra integral equations with delay. A lower bound for the well-posedness of the corresponding problem is introduced. A collocation method covering all problems with generalized derivatives is introduced and analyzed. Illustrative examples for validation and application of the proposed methods are supported. The effects of various fractional derivatives on the solution, well-posedness, and fitting error are studied. An application for estimating the population of diabetes cases in the past is introduced.
Citation: Dumitru Baleanu, Babak Shiri. Generalized fractional differential equations for past dynamic[J]. AIMS Mathematics, 2022, 7(8): 14394-14418. doi: 10.3934/math.2022793

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Abstract

Well-posedness of the terminal value problem for nonlinear systems of generalized fractional differential equations is studied. The generalized fractional operator is formulated with a classical operator and a related weighted space. The terminal value problem is transformed into weakly singular Fredholm and Volterra integral equations with delay. A lower bound for the well-posedness of the corresponding problem is introduced. A collocation method covering all problems with generalized derivatives is introduced and analyzed. Illustrative examples for validation and application of the proposed methods are supported. The effects of various fractional derivatives on the solution, well-posedness, and fitting error are studied. An application for estimating the population of diabetes cases in the past is introduced.

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