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Fractional order nonlinear evolution equations (FNLEEs) pertaining to conformable fractional derivative are considered to be revealed for well-furnished analytic solutions due to their importance in the nature of real world. In this article, the autors suggest a productive technique, called the rational fractional (DξαG/G)-expansion method, to unravel the nonlinear space-time fractional potential Kadomtsev–Petviashvili (PKP) equation, the nonlinear space-time fractional Sharma–Tasso–Olver (STO) equation and the nonlinear space-time fractional Kolmogorov–Petrovskii–Piskunov (KPP) equation. A fractional complex transformation technique is used to convert the considered equations into the fractional order ordinary differential equation. Then the method is employed to make available their solutions. The constructed solutions in terms of trigonometric function, hyperbolic function and rational function are claimed to be fresh and further general in closed form. These solutions might play important roles to depict the complex physical phenomena arise in physics, mathematical physics and engineering.

The rational fractional (DξαG/G)-expansion method shows high performance and might be used as a strong tool to unravel any other FNLEEs. This method is of the form U(ξ)=∑i=0nai(DξαG/G)i/∑i=0nbi(DξαG/G)i.

Achieved fresh and further abundant closed form traveling wave solutions to analyze the inner mechanisms of complex phenomenon in nature world which will bear a significant role in the of research and will be recorded in the literature.

The rational fractional (DξαG/G)-expansion method shows high performance and might be used as a strong tool to unravel any other FNLEEs. This method is newly established and productive.

In this paper, the author presents a hybrid method along with its error analysis to solve (1+2)-dimensional non-linear time-space fractional partial differential equations (FPDEs).

The proposed method is a combination of Sumudu transform and a semi-analytc technique Daftardar-Gejji and Jafari method (DGJM).

The author solves various non-trivial examples using the proposed method. Moreover, the author obtained the solutions either in exact form or in a series that converges to a closed-form solution. The proposed method is a very good tool to solve this type of equations.

The present work is original. To the best of the author's knowledge, this work is not done by anyone in the literature.

This study describes the applicability of the a priori estimate method on a nonlocal nonlinear fractional differential equation for which the weak solution's existence and uniqueness are proved. The authors divide the proof into two sections for the linear associated problem; the authors derive the a priori bound and demonstrate the operator range density that is generated. The authors solve the nonlinear problem by introducing an iterative process depending on the preceding results.

The functional analysis method is the a priori estimate method or energy inequality method.

The results show the efficiency of a priori estimate method in the case of time-fractional order differential equations with nonlocal conditions. Our results also illustrate the existence and uniqueness of the continuous dependence of solutions on fractional order differential equations with nonlocal conditions.

The authors’ work can be considered a contribution to the development of the functional analysis method that is used to prove well-positioned problems with fractional order.

The authors confirm that this work is original and has not been published elsewhere, nor is it currently under consideration for publication elsewhere.

The purpose of this paper is to alleviate the problem of poor robustness and over-fitting caused by large-scale data in collaborative filtering recommendation algorithms.

Interpreting user behavior from the probabilistic perspective of hidden variables is helpful to improve robustness and over-fitting problems. Constructing a recommendation network by variational inference can effectively solve the complex distribution calculation in the probabilistic recommendation model. Based on the aforementioned analysis, this paper uses variational auto-encoder to construct a generating network, which can restore user-rating data to solve the problem of poor robustness and over-fitting caused by large-scale data. Meanwhile, for the existing KL-vanishing problem in the variational inference deep learning model, this paper optimizes the model by the KL annealing and Free Bits methods.

The effect of the basic model is considerably improved after using the KL annealing or Free Bits method to solve KL vanishing. The proposed models evidently perform worse than competitors on small data sets, such as MovieLens 1 M. By contrast, they have better effects on large data sets such as MovieLens 10 M and MovieLens 20 M.

This paper presents the usage of the variational inference model for collaborative filtering recommendation and introduces the KL annealing and Free Bits methods to improve the basic model effect. Because the variational inference training denotes the probability distribution of the hidden vector, the problem of poor robustness and overfitting is alleviated. When the amount of data is relatively large in the actual application scenario, the probability distribution of the fitted actual data can better represent the user and the item. Therefore, using variational inference for collaborative filtering recommendation is of practical value.

This study aims to assess the use of variational quantum imaginary time evolution for solving partial differential equations using real-amplitude ansätze with full circular entangling layers. A graphical mapping technique for encoding impulse functions is also proposed.

The Smoluchowski equation, including the Derjaguin–Landau–Verwey–Overbeek potential energy, is solved to simulate colloidal deposition on a planar wall. The performance of different types of entangling layers and over-parameterization is evaluated.

Colloidal transport can be modelled adequately with variational quantum simulations. Full circular entangling layers with real-amplitude ansätze lead to higher-fidelity solutions. In most cases, the proposed graphical mapping technique requires only a single bit-flip with a parametric gate. Over-parameterization is necessary to satisfy certain physical boundary conditions, and higher-order time-stepping reduces norm errors.

Variational quantum simulation can solve partial differential equations using near-term quantum devices. The proposed graphical mapping technique could potentially aid quantum simulations for certain applications.

This study shows a concrete application of variational quantum simulation methods in solving practically relevant partial differential equations. It also provides insight into the performance of different types of entangling layers and over-parameterization. The proposed graphical mapping technique could be valuable for quantum simulation implementations. The findings contribute to the growing body of research on using variational quantum simulations for solving partial differential equations.

Our purpose of this study is to construct an algorithm for finding a zero of the sum of two maximally monotone mappings in Hilbert spaces and discus its convergence. The assumption that one of the mappings is α-inverse strongly monotone is dispensed with. In addition, we give some applications to the minimization problem. Our method of proof is of independent interest. Finally, a numerical example which supports our main result is presented. Our theorems improve and unify most of the results that have been proved for this important class of nonlinear mappings.

This paper aims to deal with the design optimization of a synchronous reluctance machine to be used in an X-ray tube, where the goal is to maximize the torque while keeping low the amount of material used, by means of gradient-based free-form shape optimization.

The presented approach is based on the mathematical concept of shape derivatives and allows to obtain new motor designs without the need to introduce a geometric parametrization. This paper presents an extension of a standard gradient-based free-form shape optimization algorithm to the case of multiple objective functions by determining updates, which represent a descent of all involved criteria. Moreover, this paper illustrates a way to obtain an approximate Pareto front.

The presented method allows to obtain optimal designs of arbitrary, non-parametric shape with very low computational cost. This paper validates the results by comparing them to a parametric geometry optimization in JMAG by means of a stochastic optimization algorithm. While the obtained designs are of similar shape, the computational time used by the gradient-based algorithm is in the order of minutes, compared to several hours taken by the stochastic optimization algorithm.

This paper applies the presented gradient-based multi-objective optimization algorithm in the context of free-form shape optimization using the mathematical concept of shape derivatives. The authors obtain a set of Pareto-optimal designs, each of which is a shape that is not represented by a fixed set of parameters. To the best of the authors’ knowledge, this approach to multi-objective free-form shape optimization is novel in the context of electric machines.

Surface roughness has a serious impact on the fatigue strength, wear resistance and life of mechanical products. Realizing the evolution of surface quality through theoretical modeling takes a lot of effort. To predict the surface roughness of milling processing, this paper aims to construct a neural network based on deep learning and data augmentation.

This study proposes a method consisting of three steps. Firstly, the machine tool multisource data acquisition platform is established, which combines sensor monitoring with machine tool communication to collect processing signals. Secondly, the feature parameters are extracted to reduce the interference and improve the model generalization ability. Thirdly, for different expectations, the parameters of the deep belief network (DBN) model are optimized by the tent-SSA algorithm to achieve more accurate roughness classification and regression prediction.

The adaptive synthetic sampling (ADASYN) algorithm can improve the classification prediction accuracy of DBN from 80.67% to 94.23%. After the DBN parameters were optimized by Tent-SSA, the roughness prediction accuracy was significantly improved. For the classification model, the prediction accuracy is improved by 5.77% based on ADASYN optimization. For regression models, different objective functions can be set according to production requirements, such as root-mean-square error (RMSE) or MaxAE, and the error is reduced by more than 40% compared to the original model.

A roughness prediction model based on multiple monitoring signals is proposed, which reduces the dependence on the acquisition of environmental variables and enhances the model's applicability. Furthermore, with the ADASYN algorithm, the Tent-SSA intelligent optimization algorithm is introduced to optimize the hyperparameters of the DBN model and improve the optimization performance.

Fault diagnosis methods based on blind source separation (BSS) for rolling element bearings are necessary tools to prevent any unexpected accidents. In the field application, the actual signal acquisition is usually hindered by certain restrictions, such as the limited number of signal channels. The purpose of this study is to fulfill the weakness of the existed BSS method.

To deal with this problem, this paper proposes a blind source extraction (BSE) method for bearing fault diagnosis based on empirical mode decomposition (EMD) and temporal correlation. First, a single-channel undetermined BSS problem is transformed into a determined BSS problem using the EMD algorithm. Then, the desired fault signal is extracted from selected intrinsic mode functions with a multi-shift correlation method.

Experimental results prove the extracted fault signal can be easily identified through the envelope spectrum. The application of the proposed method is validated using simulated signals and rolling element bearing signals of the train axle.

This paper proposes an underdetermined BSE method based on the EMD and the temporal correlation method for rolling element bearings. A simulated signal and two bearing fault signal from the train rolling element bearings show that the proposed method can well extract the bearing fault signal. Note that the proposed method can extract the periodic fault signal for bearing fault diagnosis. Thus, it should be helpful in the diagnosis of other rotating machinery, such as gears or blades.

The main motivation of this paper is to present  the Yosida approximation of a semi-linear backward stochastic differential equation in infinite dimension. Under suitable assumption and condition, an L²-convergence rate is established.

The authors establish a result concerning the L²-convergence rate of the solution of backward stochastic differential equation with jumps with respect to the Yosida approximation.

The authors carry out a convergence rate of Yosida approximation to the semi-linear backward stochastic differential equation in infinite dimension.

In this paper, the authors present the Yosida approximation of a semi-linear backward stochastic differential equation in infinite dimension. Under suitable assumption and condition, an L²-convergence rate is established.