Symmetry

In this study, nonlinear fractional Korteweg–de Vries (KdV) type equations with nonlocal operators are studied using Mittag–Leffler kernels and exponential decay. The KdV equations are well known for its use in modeling ion-acoustic waves in plasma, oceanic dynamics, and shallow-water waves. As a result, mathematicians are working to examine modified and generalized versions of the basic KdV equation. In order to find the solutions of nonlinear fractional KdV equations, an extension of this concept is described in the current paper. The solution of fractional KdV equations is carried out using the well-known natural transform decomposition method (NTDM). To evaluate the problem, we employ the fractional operator in the Caputo–Fabrizio (CF) and the Atangana–Baleanu–Caputo sense (ABC) manner. Nonlinear terms can be handled with Adomian polynomials. The main advantage of this novel approach is that it might offer an approximate solution in the form of convergent series using easy calculations. The dynamical behavior of the resulting solutions have been demonstrated using graphs. Numerical data is represented visually in the tables. The solutions at various fractional orders are found and it is proved that they all tend to an integer-order solution. Additionally, we examine our findings with those of the iterative transform method (ITM) and the residual power series transform method (RPSTM). It is evident from the comparison that our approach offers better outcomes compared to other approaches. The results of the suggested method are very accurate and give helpful details on the real dynamics of each issue. The present technique can be expanded to address other significant fractional order problems due to its straightforward implementation.

Electricity spot prices jointly encode network physics and strategic bidding outcomes. In a well-functioning market, nodal and temporal price patterns tend to remain approximately invariant under mild perturbations-exhibiting symmetry-preserving regularities in distribution shape, spatial gradients, and temporal variation. Conversely, congestion binding, net-load stress, and abnormal bidding can induce symmetry breaking, manifested as heavy tails, mean shifts, and localized price discontinuities. This study develops a symmetry-guided and explainable diagnostic framework to identify price anomalies and attribute their dominant drivers. First, representative anomaly types (spike and mean shift) are defined using statistically and operationally motivated criteria, together with robustness checks across alternative thresholds. Second, principal component analysis is applied to construct compact, anomaly-specific feature sets, filtering weakly related variables while retaining system stress, congestion proxies, and renewable-induced variability indicators. Third, leveraging the optimization structure of market clearing and the associated KKT conditions, we characterize the price–feature linkage as a piecewise mapping and quantify each feature’s contribution via a sampling-based influence scoring procedure, yielding a ranked causal attribution. Case studies on a regional day-ahead spot market dataset demonstrate that the proposed framework achieves high consistency with expert assessments, with traceability accuracy exceeding 85% overall and particularly strong performance for spike-type anomalies. The method reduces reliance on purely manual diagnosis and black-box learning, and provides symmetry-oriented, actionable evidence for market surveillance and renewable-friendly flexibility and congestion management design. The proposed framework enables transparent identification of dominant structural drivers underlying different types of electricity price anomalies, linking observed price signals to market-clearing mechanisms. The results provide actionable diagnostic insights for market monitoring and regulatory assessment in electricity markets with high renewable penetration.

In vertical mine hoisting systems, the rigid guide serves as a critical safety component whose failure may induce severe dynamic disturbances and potentially trigger cascading safety incidents. Existing data-driven diagnosis methods for rigid guides often lack robustness under variable operating conditions and require substantial labeled data. Yet in practical mine hoisting operations, variations in hoisting speed and lifting mass are inevitable, and acquiring sufficient fault samples is challenging due to safety constraints. To address these problems, this paper proposes a novel fault diagnosis framework that integrates a physics-informed feature-extraction pipeline with the zero-space observer theory. Vibration signals are processed to extract dimensionless and relative features, which are deliberately designed based on the dynamic mechanisms underlying different fault states. These features rely solely on the geometric characteristics of the waveform at the fault location, rendering them sensitive to fault types while remaining robust to variations in operating conditions. The feature set is subsequently optimized using the minimum redundancy maximum relevance (mRMR) algorithm to enhance computational efficiency, mitigate overfitting, and improve the generalization ability of the method. A set of zero-space observers is then constructed to perform efficient fault classification through geometric operations in the feature space, with each observer specifically sensitive to its corresponding health state while remaining insensitive to others. Experimental validation across multiple health states and operational variations demonstrates that the proposed method outperforms four widely used intelligent models in both classification accuracy and computational efficiency, showing strong suitability for real-world deployment in coal mining applications.