Page 92 - IJOCTA-15-3

An International Journal of Optimization and Control: Theories & Applications
ISSN: 2146-0957 eISSN: 2146-5703
Vol.15, No.3, pp.464-482 (2025)
https://doi.org/10.36922/IJOCTA025120054

RESEARCH ARTICLE

Application of Jumarie-Stancu Collocation Series Method and
Multi-Step Generalized Differential Transform Method to fractional

glucose-insulin

3
3
Sayed Saber 1,2* , Brahim Dridi , Abdullah Alahmari , and Mohammed Messaoudi 4
1
Department of Mathematics, Faculty of Science, Al-Baha University, Al-Baha, Saudi Arabia
2
Department of Mathematics and Computer Science, Faculty of Science, Beni-Suef University, Egypt
3
Department of Mathematics, Faculty of Sciences, Umm Al-Qura University, Saudi Arabia
4
Department of Mathematics and Statistics, College of Science, Imam Mohammad Ibn Saud Islamic
University (IMSIU), Riyadh, Saudi Arabia
Sayed011258@science.bsu.edu.eg, iodridi@uqu.edu.sa, aaahmari@uqu.edu.sa, mmessaoudi@imamu.edu.sa
ARTICLE INFO ABSTRACT
Article History:
Received: March 19, 2025 This study applies the Multi-Step Generalized Differential Transform Method
1st revised: April 20, 2025 (MSGDTM) and the Jumarie-Stancu Collocation Series Method (JSCSM) to
2nd revised: April 24, 2025 analyze a fractional-order Model (1). The model incorporates Caputo frac-
Accepted: April 29, 2025 tional derivatives to capture the nonlocal and memory-dependent characteris-
Published Online: May 20, 2025 tics of glucose-insulin interactions, considering physiological factors such as β-
cell activity and external glucose intake. Stability analysis reveals bifurcations
Keywords:
and chaotic attractors, demonstrating the system’s sensitivity to fractional or-
Fractional calculus
ders. Numerical simulations compare MSGDTM and JSCSM accuracy and
Glucose-insulin model
efficiency, highlighting MSGDTM’s superior convergence and lower approxi-
Numerical methods
mation error. The results show that fractional-order modeling provides a more
Chaos control
accurate framework for understanding glucose-insulin dynamics and predict-
Numerical simulation
MSGDTM ing metabolic behavior. Furthermore, control mechanisms are introduced to
JSCSM mitigate chaos, offering potential strategies for managing diabetes. This work
emphasizes the robustness of MSGDTM in solving complex fractional biological
AMS Classification: models. It provides insights into fractional calculus applications in biomedical
46C05; 49J20; 93C20; 49K20; research.
34K05; 34A12; 26A33

1. Introduction the complex physiological memory and hereditary
properties inherent in biological systems. To ad-
Insulin regulation is a cornerstone of human meta- dress this limitation, fractional-order differential
bolic processes, with imbalances leading to dia- equations (FDEs) have emerged as a powerful
betes mellitus, a chronic condition affecting mil- tool, incorporating memory effects and nonlocal
lions worldwide. Understanding the glucose- dynamics that better represent real-world biolog-
insulin interaction is crucial for developing ef- ical phenomena.
fective treatment strategies. Traditional integer-
order models are widely used to describe these Fractional calculus, characterized by its non-
interactions; however, they often fail to capture local operators and ability to retain memory
*Corresponding Author
464