Based on tempered fractional derivatives, in this paper, the prediction-corrector scheme of tempered fractional differential equation is studied by using the characteristics of chaotic systems, and it is used to simulate the chaotic behaviors of tempered fractional Lorenz systems. The method works well for tempered fractional differential equation of arbitrary order. A striking finding of the simulation is that the dynamical behaviors of tempered fractional Lorenz system become stronger as $\lambda(\lambda$ is a temper parameter) decreases. Finally, the synchronization of the classical Lorenz systems and tempered fractional Lorenz systems is discussed.
Through a combination of hesitant fuzzy sets with rough sets, this study develops a single-granulation hesitant fuzzy rough set model from the perspective of granular computing. In the multi-granulation framework, we propose two types of multi-granulation rough set model, called the optimistic mult i-granulation hesitant fuzzy rough sets and pessimistic multi-granulation hesitant fuzzy rough sets. In the models, the multi-granulation hesitant fuzzy lower and upper approximations are defined based on multiple hesitant fuzzy tolerance relations. The relationships among the single-granulation hesitant fuzzy rough sets, optimistic multi-granulation hesitant fuzzy rough sets and pessimistic multi-granulation hesitant fuzzy rough sets are also investigated. Finally, we develop an approximation reduction approach of multi-granulation hesitant fuzzy rough sets to eliminate redundant hesitant fuzzy granulations with a detailed example.
As the COVID‐19 continues to mutate, the number of infected people is increasing dramatically, and the vaccine is not enough to fight the mutated strain. In this paper, a SEIR‐type fractional model with reinfection and vaccine inefficacy is proposed, which can successfully capture the mutated COVID‐19 pandemic. The existence, uniqueness, boundedness, and nonnegativeness of the fractional model are derived. Based on the basic reproduction number , locally stability and globally stability are analyzed. The sensitivity analysis evaluate the influence of each parameter on the and rank key epidemiological parameters. Finally, the necessary conditions for implementing fractional optimal control are obtained by Pontryagin's maximum principle, and the corresponding optimal solutions are derived for mitigation COVID‐19 transmission. The numerical results show that humans will coexist with COVID‐19 for a long time under the current control strategy. Furthermore, it is particularly important to develop new vaccines with higher protection rates.
A weighted finite difference scheme was proposed in order to solve initialboundary value problems of space-time fractional diffusion equations.Their stability was analyzed by means of discrete energy method.Using mathematical induction,we proved that the scheme was convergent under the same condition.Illustrative example was included to demonstrate the validity and applicability of the scheme.
This paper studies the outer synchronization problem of discrete fractional complex networks (DFCNs) with and without the presence of unknown topology. A discrete complex network with a fractional difference is first established and analyzed. By constructing a suitable Lyapunov function and utilizing properties of the fractional difference, outer synchronization criteria for the DFCNs with and without unknown topology are established based on linear matrix inequalities. Meanwhile, the unknown parameters in the topology structure of the network can be identified by adaptive update laws. In the end, two numerical examples are given to exemplify the validity and applicability of the obtained results.
In the famous continuous time random walk (CTRW) model, because of the finite lifetime of biological particles, it is sometimes necessary to temper the power law measure such that the waiting time measure has a convergent first moment. The CTRW model with tempered waiting time measure is the so‐called tempered fractional derivative. In this article, we introduce the tempered fractional derivative into complex networks to describe the finite life span or bounded physical space of nodes. Some properties of the tempered fractional derivative and tempered fractional systems are discussed. Generalized synchronization in two‐layer tempered fractional complex networks via pinning control is addressed based on the auxiliary system approach. The results of the proposed theory are used to derive a sufficient condition for achieving generalized synchronization of tempered fractional networks. Numerical simulations are presented to illustrate the effectiveness of the methods.
This paper studies the problem of partial topology identification of tempered fractional complex networks. By tempered fractional calculus theory and pinning controlling techniques of network synchronization, we propose a strategy that can greatly reduce the expense of the partial topology identification. This method can also identify the whole topology of tempered fractional complex network. Sufficient conditions are given to guarantee partial topology identification of tempered fractional complex networks by designing suitable controllers and parameters update law. This method can also achieve synchronization between the drive network and the response network. Finally, some numerical experiments are presented to verify the validity of the method.
By combining the interval-valued hesitant fuzzy set and soft set models, the purpose of this paper is to introduce the concept of interval-valued hesitant fuzzy soft sets. Further, some operations on the interval-valued hesitant fuzzy soft sets are investigated, such as complement, “AND,” “OR,” ring sum, and ring product operations. Then, by means of reduct interval-valued fuzzy soft sets and level hesitant fuzzy soft sets, we present an adjustable approach to interval-valued hesitant fuzzy soft sets based on decision making and some numerical examples are provided to illustrate the developed approach. Finally, the weighted interval-valued hesitant fuzzy soft set is also introduced and its application in decision making problem is shown.
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