A generalized coupled system of fractional differential equations with application to finite time sliding mode control for Leukemia therapy

Abstract

In this article, a general nonlinear system of functional differential equations for two types of operators is considered. One of them includes RDβi which are n-operators in the Riemann–Liouville’s sense of derivative while the second is based on a series operator L(RDϱi) where all the RDϱi’s are Riemann–Liouville’s fractional operators with the assumption of βi,ϱi∈(0,1]. These two types of operators are combined with the help of a nonlinear Φp-operator. This novel construction is more useful in scientific problems. The novel system of FDEs is studied for the solution existence, uniqueness analysis, stability analysis and numerical computation is also carried out. For an application of the presumed general coupled system, a fractional order Leukemia therapy model with its numerical simulations and optimization, is given. The Leukemia model describes the propagation of infected cells. A fractional-order finite-time terminal sliding mode control is designed to control Leukemia for the fractional-order dynamics for eliminating Leukemic cells while maintaining an adequate number of normal cells by the application of a chemotherapeutic agent that is considered as safe. The Lyapunov stability theory has been employed to analyze the controllers. The comparative simulations are presented for a better illustration of the work and show the superior tracking and convergence performance of the proposed control method.