Qualitative results and numerical approximations of the $ (k, \psi) $-Caputo proportional fractional differential equations and applications to blood alcohol levels model

Weerawat Sudsutad; Chatthai Thaiprayoon; Aphirak Aphithana; Jutarat Kongson; Weerapan Sae-dan; Weerawat Sudsutad; Chatthai Thaiprayoon; Aphirak Aphithana; Jutarat Kongson; Weerapan Sae-dan

doi:10.3934/math.20241622

AIMS Mathematics

2024, Volume 9, Issue 12: 34013-34041. doi: 10.3934/math.20241622

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Research article

Qualitative results and numerical approximations of the $ (k, \psi) $-Caputo proportional fractional differential equations and applications to blood alcohol levels model

1.
Department of Statistics, Faculty of Science, Ramkhamhaeng University, Bangkok 10240, Thailand
2.
Research Group of Theoretical and Computation in Applied Science, Department of Mathematics, Faculty of Science, Burapha University, Chonburi 20131, Thailand
3.
Department of Computer Engineering, Faculty of Engineering, Ramkhamkaeng University, Bangkok 10240, Thailand

Received: 30 August 2024 Revised: 13 November 2024 Accepted: 26 November 2024 Published: 03 December 2024
MSC : 26A33, 26D10, 34A08, 34B10, 33E12

The initial value problem in Cauchy-type under the $ (k, \psi) $-Caputo proportional fractional operators was our focus in this paper. An extended Gronwall inequality and its properties were analyzed. The existence and uniqueness results were proven utilizing the fixed point theory of Banach's and Leray-Schauder's types. The qualitative analysis included results for Ulam-Mittag-Leffler stability, which was also investigated. Using a decomposition principle, a novel numerical technique was presented for the $ (k, \psi) $-Caputo proportional fractional derivative operator. Finally, theoretical results were supported with numerical examples to demonstrate their practical application, especially to blood alcohol level problems.
Citation: Weerawat Sudsutad, Chatthai Thaiprayoon, Aphirak Aphithana, Jutarat Kongson, Weerapan Sae-dan. Qualitative results and numerical approximations of the $ (k, \psi) $-Caputo proportional fractional differential equations and applications to blood alcohol levels model[J]. AIMS Mathematics, 2024, 9(12): 34013-34041. doi: 10.3934/math.20241622

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Abstract

The initial value problem in Cauchy-type under the $ (k, \psi) $-Caputo proportional fractional operators was our focus in this paper. An extended Gronwall inequality and its properties were analyzed. The existence and uniqueness results were proven utilizing the fixed point theory of Banach's and Leray-Schauder's types. The qualitative analysis included results for Ulam-Mittag-Leffler stability, which was also investigated. Using a decomposition principle, a novel numerical technique was presented for the $ (k, \psi) $-Caputo proportional fractional derivative operator. Finally, theoretical results were supported with numerical examples to demonstrate their practical application, especially to blood alcohol level problems.

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