Analytical results for positivity of discrete fractional operators with approximation of the domain of solutions

Pshtiwan Othman Mohammed; Donal O'Regan; Dumitru Baleanu; Y. S. Hamed; Ehab E. Elattar; Pshtiwan Othman Mohammed; Donal O'Regan; Dumitru Baleanu; Y. S. Hamed; Ehab E. Elattar

doi:10.3934/mbe.2022343

Mathematical Biosciences and Engineering

2022, Volume 19, Issue 7: 7272-7283. doi: 10.3934/mbe.2022343

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Analytical results for positivity of discrete fractional operators with approximation of the domain of solutions

1.
Department of Mathematics, College of Education, University of Sulaimani, Sulaimani 46001, Kurdistan Region, Iraq
2.
School of Mathematical and Statistical Sciences, National University of Ireland, Galway, Ireland
3.
Department of Mathematics, Cankaya University, Balgat 06530, Ankara, Turkey
4.
Institute of Space Sciences, R76900 Magurele-Bucharest, Romania
5.
China Medical University Hospital, China Medical University, Taichung 40402, Taiwan
6.
Department of Mathematics and Statistics, College of Science, Taif University, Taif 21944, Saudi Arabia
7.
Department of Electrical Engineering, College of Engineering, Taif University, Taif 21944, Saudi Arabia

Academic Editor: Feliz Manuel Barrão Minhós

Received: 08 April 2022 Revised: 05 May 2022 Accepted: 11 May 2022 Published: 18 May 2022

We study the monotonicity method to analyse nabla positivity for discrete fractional operators of Riemann-Liouville type based on exponential kernels, where $ \left({}_{{c_0}}^{C{F_R}}\nabla^{\theta} \mathtt{F}\right)(t) > -\epsilon\, \Lambda(\theta-1)\, \bigl(\nabla \mathtt{F}\bigr)(c_{0}+1) $ such that $ \bigl(\nabla \mathtt{F}\bigr)(c_{0}+1)\geq 0 $ and $ \epsilon > 0 $. Next, the positivity of the fully discrete fractional operator is analyzed, and the region of the solution is presented. Further, we consider numerical simulations to validate our theory. Finally, the region of the solution and the cardinality of the region are discussed via standard plots and heat map plots. The figures confirm the region of solutions for specific values of $ \epsilon $ and $ \theta $.
- discrete fractional calculus,
- Caputo-Fabrizio fractional difference,
- nabla positivity,
- numerical analysis
Citation: Pshtiwan Othman Mohammed, Donal O'Regan, Dumitru Baleanu, Y. S. Hamed, Ehab E. Elattar. Analytical results for positivity of discrete fractional operators with approximation of the domain of solutions[J]. Mathematical Biosciences and Engineering, 2022, 19(7): 7272-7283. doi: 10.3934/mbe.2022343

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Abstract

We study the monotonicity method to analyse nabla positivity for discrete fractional operators of Riemann-Liouville type based on exponential kernels, where $ \left({}_{{c_0}}^{C{F_R}}\nabla^{\theta} \mathtt{F}\right)(t) > -\epsilon\, \Lambda(\theta-1)\, \bigl(\nabla \mathtt{F}\bigr)(c_{0}+1) $ such that $ \bigl(\nabla \mathtt{F}\bigr)(c_{0}+1)\geq 0 $ and $ \epsilon > 0 $. Next, the positivity of the fully discrete fractional operator is analyzed, and the region of the solution is presented. Further, we consider numerical simulations to validate our theory. Finally, the region of the solution and the cardinality of the region are discussed via standard plots and heat map plots. The figures confirm the region of solutions for specific values of $ \epsilon $ and $ \theta $.

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