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In this paper, we present a new fractional-order mathematical model for a tumor-immune surveillance mechanism. We analyze the interactions between various tumor cell populations and immune system via a system of fractional differential equations (FDEs). An efficient numerical procedure is suggested to solve these FDEs by considering singular and nonsingular derivative operators. An optimal control strategy for investigating the effect of chemotherapy treatment on the proposed fractional model is also provided. Simulation results show that the new presented model based on the fractional operator with Mittag–Leffler kernel represents various asymptomatic behaviors that tracks the real data more accurately than the other fractional- and integer-order models. Numerical simulations also verify the efficiency of the proposed optimal control strategy and show that the growth of the naive tumor cell population is successfully declined.

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In this study, a harmonic oscillator with position-dependent mass is investigated. Firstly, as an introduction, we give a full description of the system by constructing its classical Lagrangian; thereupon, we derive the related classical equations of motion such as the classical Euler–Lagrange equations. Secondly, we fractionalize the classical Lagrangian of the system, and then we obtain the corresponding fractional Euler–Lagrange equations (FELEs). As a final step, we give the numerical simulations corresponding to the FELEs within different fractional operators. Numerical results based on the Caputo and the Atangana-Baleanu-Caputo (ABC) fractional derivatives are given to verify the theoretical analysis.

This paper deals with a general form of fractional optimal control problems involving the fractional derivative with singular or non‐singular kernel. The necessary conditions for the optimality of these problems are derived and a new numerical method is designed to solve these equations effectively. Simulation results indicate that the proposed method works well and provides satisfactory results with regard to accuracy and computational effort. Comparative results also verify that a particular case with Mittag‐Leffler kernel improves the performance of the controlled system in terms of the transient response compared to the other fractional‐ and integer‐order derivatives.

The main objective of this research is to investigate a new fractional mathematical model involving a nonsingular derivative operator to discuss the clinical implications of diabetes and tuberculosis coexistence. The new model involves two distinct populations, diabetics and nondiabetics, while each of them consists of seven tuberculosis states: susceptible, fast and slow latent, actively tuberculosis infection, recovered, fast latent after reinfection, and drug-resistant. The fractional operator is also considered a recently introduced one with Mittag–Leffler nonsingular kernel. The basic properties of the new model including non-negative and bounded solution, invariant region, and equilibrium points are discussed thoroughly. To solve and simulate the proposed model, a new and efficient numerical method is established based on the product-integration rule. Numerical simulations are presented, and some discussions are given from the mathematical and biological viewpoints. Next, an optimal control problem is defined for the new model by introducing four control variables reducing the number of infected individuals. For the control problem, the necessary and sufficient conditions are derived and numerical simulations are given to verify the theoretical analysis.

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