A nonlinear system of hybrid fractional differential equations with application to fixed time sliding mode control for Leukemia therapy

Abstract

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.

Keywords

• Riemann-Louville derivative
• Fixed point theorems
• Fractional-order sliding mode control
• Fixed-time control
• Lagrange's interpolation polynomial