一类分数阶微积分的中值定理及应用

Riemann-Liouville分数阶微积分算子是一类带有一个函数的分数阶微积分算子的特殊情形，以Riemann-Liouville分数阶微积分算子的积分中值定理和微分中值定理为基础，我们得到了一类带有一个函数的分数阶微积分算子的积分中值定理和微分中值定理，并给出其在计算方面的一些应用。     

Elliptical distribution-based weight-determining method for ordered weighted averaging operators

The ordered weighted averaging (OWA) operators play a crucial role in aggregating multiple criteria evaluations into an overall assessment supporting the decision makers’ choice. One key point steps is to determine the associated weights. In this paper, we first briefly review some main methods for determining the weights by using distribution functions. Then we propose a new approach for determining OWA weights by using the regular increasing monotone quantifier. Motivated by the idea of normal distribution-based method to determine the OWA weights, we develop a method based on elliptical distributions for determining the OWA weights, and some of its desirable properties have been investigated.     

Some q-rung orthopair uncertain linguistic aggregation operators and their application to multiple attribute group decision making

q-Rung orthopair fuzzy sets (q-ROFSs), originally presented by Yager, are a powerful fuzzy information representation model, which generalize the classical intuitionistic fuzzy sets and Pythagorean fuzzy sets and provide more freedom and choice for decision makers (DMs) by allowing the sum of the $q\mathrm{t}\mathrm{h}$ power of the membership and the $q\mathrm{t}\mathrm{h}$ power of the nonmembership to be less than or equal to 1. In this paper, a new class of fuzzy sets called q-rung orthopair uncertain linguistic sets (q-ROULSs) based on the q-ROFSs and uncertain linguistic variables (ULVs) is proposed, and this can describe the qualitative assessment of DMs and provide them more freedom in reflecting their belief about allowable membership grades. On the basis of the proposed operational rules and comparison method of q-ROULSs, several q-rung orthopair uncertain linguistic aggregation operators are developed, including the q-rung orthopair uncertain linguistic weighted arithmetic average operator, the q-rung orthopair uncertain linguistic ordered weighted average operator, the q-rung orthopair uncertain linguistic hybrid weighted average operator, the q-rung orthopair uncertain linguistic weighted geometric average operator, the q-rung orthopair uncertain linguistic ordered weighted geometric operator, and the q-rung orthopair uncertain linguistic hybrid weighted geometric operator. Then, some desirable properties and special cases of these new operators are also investigated and studied, in particular, some existing intuitionistic fuzzy aggregation operators and Pythagorean fuzzy aggregation operators are proved to be special cases of these new operators. Furthermore, based on these proposed operators, we develop an approach to solve the multiple attribute group decision making problems, in which the evaluation information is expressed as q-rung orthopair ULVs. Finally, we provide several examples to illustrate the specific decision-making steps and explain the validity and feasibility of two methods by comparing with other methods.     

Multicriteria decision-making using Archimedean aggregation operators in Pythagorean hesitant fuzzy environment

In multicriteria decision-making (MCDM), the existing aggregation operators are mostly based on algebraic t-conorm and t-norm. But, Archimedean t-conorms and t-norms are the generalized forms of t-conorms and t-norms which include algebraic, Einstein, Hamacher, Frank, and other types of t-conorms and t-norms. From that view point, in this paper the concepts of Archimedean t-conorm and t-norm are introduced to aggregate Pythagorean hesitant fuzzy information. Some new operational laws for Pythagorean hesitant fuzzy numbers based on Archimedean t-conorm and t-norm have been proposed. Using those operational laws, Archimedean t-conorm and t-norm-based Pythagorean hesitant fuzzy weighted averaging operator and weighted geometric operator are developed. Some of their desirable properties have also been investigated. Afterwards, these operators are applied to solve MCDM problems in Pythagorean hesitant fuzzy environment. The developed Archimedean aggregation operators are also applicable in Pythagorean fuzzy contexts also. To demonstrate the validity, practicality, and effectiveness of the proposed method, a practical problem is considered, solved, and compared with other existing method.     

改进的区间犹豫算子应用于物流企业选择决策

针对现有区间犹豫模糊Hamacher算子存在的缺陷，构建了一种基于改进的区间犹豫模糊Hamacher加权算子的群决策方法。在分析现有区间犹豫模糊Hamacher算子不能满足幂等性的基础上，定义新的区间犹豫模糊Hamacher四则运算；提出两种改进的区间犹豫模糊Hamacher加权算子，包括改进的区间犹豫模糊Hamacher有序加权平均（I-IVHFHOWA）算子和改进的区间犹豫模糊Hamacher有序加权几何（I-IVHFHOWG）算子，并详细探究它们的常用算子形式以及算子之间的内在联系；建立基于I-IVHFHOWA算子和I-IVHFHOWG算子的物流企业选择决策模型，并通过实例说明模型的有效性。     

Weighted power means of q-rung orthopair fuzzy information and their applications in multiattribute decision making

Weighted power means with weights and exponents serving as their parameters are generalizations of arithmetic means. Taking into account decision makers' flexibility in decision making, each attribute value is usually expressed by a $q$-rung orthopair fuzzy value ($q$-ROFV, $q\ge 1$), where the former indicates the support for membership, the latter support against membership, and the sum of their $q$th powers is bounded by one. In this paper, we propose the weighted power means of $q$-rung orthopair fuzzy values to enrich and flourish aggregations on $q$-ROFVs. First, the $q$-rung orthopair fuzzy weighted power mean operator is presented, and its boundedness is precisely characterized in terms of the power exponent. Then, the $q$-rung orthopair fuzzy ordered weighted power mean operator is introduced, and some of its fundamental properties are investigated in detail. Finally, a novel multiattribute decision making method is explored based on developed operators under the $q$-rung orthopair fuzzy environment. A numerical example is given to illustrate the feasibility and validity of the proposed approach, and it is shown that the power exponent is an index suggesting the degree of the optimism of decision makers.     

Early fault feature extraction of bearings based on Teager energy operator and optimal VMD

As the fault shock component in vibration signals is extremely sparse and weak, it is difficult to extract the fault features when large-scale, low-speed and heavy-duty mechanical equipment is in the early stage of failure. To solve this problem, an early fault feature extraction method based on the Teager energy operator, combined with optimal variational mode decomposition (VMD) is presented in this study. First, the Teager energy operator was used to strengthen the weak shock component of the original signal. Next, a logistic–sine complex chaotic mapping with variable dimensions was constructed to enhance the global search ability and convergence speed of the pigeon-inspired optimization (PIO) algorithm, which is named the variable dimension chaotic pigeon-inspired optimization (VDCPIO) algorithm. Then, the VDCPIO algorithm is used to search for the optimal combination value of key parameters of VMD. The enhanced vibration signal is decomposed into a set of intrinsic mode functions (IMFs) by the optimized VMD, and then kurtosis for every IMF and mean kurtosis of all IMFs are extracted. According to the average kurtosis, several IMFs, whose kurtosis value is greater than the average kurtosis value, are selected to reconstruct a new signal. Then, envelope spectrum analysis of the reconstructed signal is carried out to extract the early fault features. Finally, experimental verification of the method was performed using the simulated signal and measured signal from a rolling bearing; the experimental results indicate that the method presented in this paper is more effective to extract the early fault features of this kind of mechanical equipment.     

Optimal interval type-2 fuzzy fractional order super twisting algorithm: A second order sliding mode controller for fully-actuated and under-actuated nonlinear systems

In this paper, a novel interval type-2 fuzzy fractional order super twisting algorithm (IT2FFOSTA) which is essentially a second order sliding mode controller is presented. The proposed IT2FFOSTA enhances fractional order super twisting algorithm (FOSTA) by taking advantage of an interval type-2 fuzzy fractional order sliding surface (IT2FFOSS) for some classes of fully-actuated and under-actuated nonlinear systems in presence of uncertainty. The FOSTA significantly reduces the amount of chattering and the IT2FFOSS results in decreasing the tracking error, control effort, and chattering level. In order to control under-actuated systems, a hierarchical sliding surface is employed. The multi-tracker optimization algorithm is utilized to adjust the controller’s parameters; this leads to an optimal performance for the IT2FFOSTA. To examine the performance of the IT2FFOSTA, some simulation and experimental tests on three examples of different classes of fully-actuated and under-actuated systems, including ball and plate, inverted pendulum, and ball and beam systems are carried out. The simulation and experimental results demonstrate the superiority of the IT2FFOSTA in reducing the amount of chattering, tracking error, and control effort compared to those of the other control methods.     

Global weak solutions to a spatio-temporal fractional Landau–Lifshitz–Bloch equation

The present paper deals with global existence of weak solutions of a time-space fractional Landau–Lifshitz–Bloch equation involving the weak Caputo derivative and a fractional Laplacian. We use Faedo–Galerkin method with some commutator estimates in order to prove global existence of weak solutions for the model. The uniqueness is also discussed in a special one dimensional case.