Qualitative analytical results of complex order nonlinear fractional differential equations with robust control scheme

Abdelatif Boutiara; Jehad Alzabut; Hasib Khan; Saim Ahmed; Ahmad Taher Azar; Abdelatif Boutiara; Jehad Alzabut; Hasib Khan; Saim Ahmed; Ahmad Taher Azar

doi:10.3934/math.20241006

AIMS Mathematics

2024, Volume 9, Issue 8: 20692-20720. doi: 10.3934/math.20241006

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Qualitative analytical results of complex order nonlinear fractional differential equations with robust control scheme

1.
Department of Mathematics and Computer Science, University of Ghardaia, Ghardaia 47000, Algeria
2.
Department of Mathematics and Sciences, Prince Sultan University, Riyadh 11586, Saudi Arabia
3.
Department of Industrial Engineering, OSTIM Technical University, Ankara 06374, Turkey
4.
Department of Mathematics, Shaheed Benazir Bhutto University, Sheringal, Dir Upper 18000, Khyber Pakhtunkhwa, Pakistan
5.
Automated Systems and Soft Computing Lab (ASSCL), Prince Sultan University, Riyadh 11586, Saudi Arabia
6.
College of Computer and Information Sciences, Prince Sultan University, Riyadh 11586, Saudi Arabia
7.
Faculty of Computers and Artificial Intelligence, Benha University, Benha, Egypt

Received: 13 March 2024 Revised: 29 May 2024 Accepted: 17 June 2024 Published: 26 June 2024
MSC : 34A08

In this manuscript, our work was about a qualitative study for a class of multi-complex orders nonlinear fractional differential equations (FDEs). Our methodology utilized the topological degree theory and studied a novel operator tailored for non-singular FDEs with $ \mathrm{T} $-Riemann-Liouville (T-RL) fractional order derivatives. The primary objective was to prove the existence and uniqueness of solutions for both initial and boundary value problems within the intricated framework. With the help of topological degree theory, we contributed to a wider understanding of the structural aspects governing the behavior of the considered FDE. The novel operator proposing for non-singular FDEs added a unique dimension to our analytical problem, offering a versatile and effective means of addressing the challenges posed by these complex systems for their theoretical analysis. For the practical implications of our theoretical framework, we presented two concrete examples that reinforced and elucidated the key concepts introduced. These examples underscored our approach's viability and highlighted its potential applications in diverse scientific and engineering domains. Through this comprehensive exploration, we aimed to contribute significantly to advancing the theoretical foundation related to the study of multi-complex nonlinear FDEs. Moreover, a fixed-time terminal sliding mode control (TSMC) has been developed. This proposed control strategy for eliminating leukemic cells while maintaining normal cells was based on a chemotherapeutic treatment that was not only effective but also widely acknowledged to be safe. This strategy was evaluated using the fixed-time Lyapunov stability theory, and simulations were included to illustrate its performance in terms of tracking and convergence.
- T-Riemann-Liouville fractional derivatives,
- topological degree theory,
- fixed point theorem,
- control scheme,
- leukemia model
Citation: Abdelatif Boutiara, Jehad Alzabut, Hasib Khan, Saim Ahmed, Ahmad Taher Azar. Qualitative analytical results of complex order nonlinear fractional differential equations with robust control scheme[J]. AIMS Mathematics, 2024, 9(8): 20692-20720. doi: 10.3934/math.20241006

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Abstract

In this manuscript, our work was about a qualitative study for a class of multi-complex orders nonlinear fractional differential equations (FDEs). Our methodology utilized the topological degree theory and studied a novel operator tailored for non-singular FDEs with $ \mathrm{T} $-Riemann-Liouville (T-RL) fractional order derivatives. The primary objective was to prove the existence and uniqueness of solutions for both initial and boundary value problems within the intricated framework. With the help of topological degree theory, we contributed to a wider understanding of the structural aspects governing the behavior of the considered FDE. The novel operator proposing for non-singular FDEs added a unique dimension to our analytical problem, offering a versatile and effective means of addressing the challenges posed by these complex systems for their theoretical analysis. For the practical implications of our theoretical framework, we presented two concrete examples that reinforced and elucidated the key concepts introduced. These examples underscored our approach's viability and highlighted its potential applications in diverse scientific and engineering domains. Through this comprehensive exploration, we aimed to contribute significantly to advancing the theoretical foundation related to the study of multi-complex nonlinear FDEs. Moreover, a fixed-time terminal sliding mode control (TSMC) has been developed. This proposed control strategy for eliminating leukemic cells while maintaining normal cells was based on a chemotherapeutic treatment that was not only effective but also widely acknowledged to be safe. This strategy was evaluated using the fixed-time Lyapunov stability theory, and simulations were included to illustrate its performance in terms of tracking and convergence.

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