article · Ain Shams Engineering Journal
This research investigates a coupled nonlinear system of hybrid fractional differential equations involving Riemann-Liouville fractional derivatives and series operators. Theoretical foundations are established using fixed-point techniques to prove the existence, uniqueness, and stability of the solutions, alongside a numerical scheme created via Lagrange interpolation. To demonstrate a real-world use case, the mathematical framework is applied to model leukaemia infection dynamics and chemotherapy delivery. A fractional-order fixed-time terminal sliding mode controller is designed to eradicate cancerous cells while preserving healthy cells. The stability of this controller is confirmed through fixed-time Lyapunov stability theory. Comparative computational simulations show that this control approach delivers enhanced tracking accuracy and faster convergence performance.
Designing safe chemotherapy regimens requires maintaining a delicate balance between eliminating cancer cells and protecting healthy tissue. By applying advanced mathematical modeling and control theory, this approach helps predict infection progression and optimize drug delivery timing, which could ultimately support more effective and safer cancer treatment protocols.
This work represents early-stage theoretical and computational research. It could eventually inform algorithms used in clinical decision-support software or automated drug-delivery systems for oncology. Potential end users include medical device developers and pharmaceutical researchers, but considerable translational research and clinical validation would be required before real-world use.
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In this article, a coupled nonlinear problem of hybrid fractional differential equations (HFDEs) is presented for the qualitative work and numerical results. Two types of operators are involved in the research problem. One of them is DβrR which represent Riemann–Liouville's (RL) fractional derivatives while the operator L(DϱrR) is a series operator and DϱrR's are RL operators such that βr,ϱr∈(0,1]. These operators are joined by Φp operator. As a result, we have a nonlinear coupled system of FDEs. The newly established nonlinear system is studied for the existence, uniqueness criteria, stability of the solutions, and numerical computations. For the theoretical results, we take help from the available literature about the fixed point (FP) techniques. Then a computational scheme is developed with the help of Lagrange's interpolation technique. An application of the problem as a particular case is presented in the sense of the Leukemia mathematical model. The model presents the infection propagation. Leukemia can be managed by providing a chemotherapeutic treatment generally accepted to be safe, and a fractional-order fixed-time terminal sliding mode control has been developed to achieve this goal of removing Leukemic cells while keeping a sufficient number of normal cells. In order to evaluate the proposed controller stability, the fixed-time Lyapunov stability theory is employed. To better illustrate the study, comparison simulations are shown, demonstrating that the suggested control approach has higher tracking and convergence performance.
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DOI: 10.1016/j.asej.2023.102566
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