Search Results (7,561)

This study develops an effective numerical method for addressing the time-fractional gas dynamics equation formulated with the Caputo time-fractional derivative. Novel basis functions are utilized, formulated as particular generalized Fibonacci polynomials contingent on a free parameter. This family generalizes the second kind of Chebyshev family. For the proposed polynomials, we establish basic analytical properties, including closed-form series expansion, inverse relation, moment and linearization formulas, and operational matrices for both integer-order and Caputo fractional derivatives. Using these tools, the fractional model, together with its underlying conditions, can be transformed into a finite system of nonlinear algebraic equations via a collocation strategy. Using Newton’s iterative method, the resulting system can be treated. A full convergence analysis of the double generalized Chebyshev expansion is provided. We demonstrate the accuracy and reliability of the presented method through several numerical simulations. Comparisons with existing numerical methods show that this approach achieves higher accuracy and faster execution. Full article

In this study, we analyze the solution characteristics and dynamics of a variable-order fractional (V-OF) Arneodo system using the Liouville–Caputo fractional operator with variable order. The V-OF operator is used to describe the time-dependent memory effect in the system, which leads to more complex and diverse dynamics compared to integer-order systems. In this work, numerical simulations are performed to observe the effect of the order functions on the dynamic behaviors of the system. In addition, the phase portraits, time series graphs, and three-dimensional diagrams are used to analyze the dynamic behaviors and different types of oscillations present in the system. Furthermore, the bifurcations, chaotic behaviors, and stability of the system with variable orders are studied, and it is found that the system has more complex dynamics compared to the integer-order case. In this case, the Lyapunov exponents indicate that the system under investigation is sensitive to the initial conditions, and the memory effect can control the chaotic oscillation depending on the order of the functions. Full article

In this paper, we investigate the application of the relative entropy framework for safety assessments of steel elements with structural defects at the micro- and macro-scales. Mathematical theories developed by Bhattacharyya and by Kullback and Leibler (K-L) have been used for this purpose. This approach uses both expectations and variations, similar to the First-Order Reliability Method (FORM), but is extended to include 3rd- and 4th-order central probabilistic moments. It is necessary to use a hybrid computational technique that combines the Finite Element Method (FEM) software ABAQUS CAE 2017 with the implemented Gurson–Tvergaard–Needleman (GTN) damage model and the computer algebra system MAPLE. The iterative generalized stochastic perturbation technique has been used to determine the probabilistic moments of structural response, to utilize the Weighted Least Squares Method to approximate the structural response function, and to determine uncertainty in the stress, strain, and displacement state functions. This approach is based on relative entropy because of its universality. There is no need to assume a type of distribution of the state functions, in contrast to FORM, where a Gaussian distribution is required. This paper verifies whether relative entropy can serve as an alternative to FORM for determining reliability. The yield surface of the porous material with a random values of the void volume fraction f are also presented. Full article

This study addresses the need for applied sentiment analysis in engineering decision-support systems by presenting a hybrid framework for domain-specific engineering text. This study presents a hybrid sentiment classification framework by integrating transformer-based semantic embeddings with fractional-order feature modeling. The proposed BERTLR framework combines BERT and RoBERTa representations with Grünwald–Letnikov fractional derivative–enhanced TF-IDF features and logistic regression within a unified soft-voting architecture. Unlike conventional ensemble sentiment models that merely aggregate embeddings and handcrafted features, the proposed method introduces fractional-order feature transformation to capture non-local dependency patterns and memory-aware lexical variations that are often overlooked in technical review text. This design provides a structured fusion of contextual semantic information and fractional statistical representations, supported by SHAP-based explainability and ablation analysis. Experiments conducted on six real-world engineering application domains show consistent improvements over conventional TF-IDF models, LSTM baselines, and non-fractional transformer variants. The framework achieves up to 91% accuracy, together with strong precision, recall, and F1-score performance. These results demonstrate that fractional-order feature augmentation can provide a meaningful complementary signal to transformer embeddings, offering an interpretable and effective sentiment analysis solution for engineering and industrial decision-support applications. Full article

Disturbances in dynamical systems pose a major challenge for parameter identification, particularly in the presence of unknown initial conditions and uncertain external influences. To address this issue, this paper proposes an algebraic parameter estimation methodology that incorporates fractional-order calculus in the Laplace domain for controlled linear engineering systems. The proposed approach eliminates the influence of unknown initial conditions and considers external disturbances that admit a local polynomial representation through Taylor series expansions over sufficiently small time intervals, while avoiding explicit numerical differentiation in the time domain. The manuscript includes analytical, numerical, and experimental validations to highlight the benefits of incorporating fractional-order differentiation in the derivation of algebraic estimators for online parameter estimation. The method is experimentally validated on two linear differentially flat electrical circuits, whose flat representations enable the proposed algebraic formulation under distinct disturbance signals. The results demonstrate that the fractional differentiation order acts as an additional tuning parameter, and that appropriately selected fractional orders can improve estimation accuracy, yielding parameter estimates consistently closer to their true values when compared with the conventional integer-order algebraic formulation. Full article

We construct and analyze a family of support-defined Brauer-type configurations canonically associated with the Boolean geometry underlying the Grassmann algebra. The construction is governed by an x-support map on monomial labels, which identifies the vertex set with the Boolean lattice $\mathcal{P}\left(\right[n\left]\right)$. This identification yields a Boolean support quiver isomorphic to the directed Hasse diagram of $\mathcal{P}\left(\right[n\left]\right)$, equivalently, to an oriented hypercube. We then equip the family with a canonical cyclic ordering at each vertex and obtain a genuine connected reduced Brauer configuration in the standard sense, together with its associated Brauer configuration algebra and its standard Brauer quiver. A ghost-variable mechanism is introduced to obtain a connected realization without altering any support-controlled invariants. We prove that polygon membership, valencies, multiplicities, Boolean stratification, and the support quiver are invariant under support-preserving ghost relabelings. We also give an explicit description of the standard Brauer quiver and show that it is different from the Boolean support quiver. On the algebraic side, we derive closed formulas for the center dimension, the algebra dimension, and the normalization constant of the induced weighted distribution. On the probabilistic side, we distinguish the vertex entropy from the layer entropy, establish an exact decomposition of the former by Hamming layers, and show that the layer distribution is asymptotically concentrated on the middle layers, while extremal vertices and any fixed maximal path contribute a negligible fraction of the total weight. As a consequence, the layer entropy satisfies a logarithmic asymptotic law. We also investigate geometric consequences of the Boolean model transported through the support identification. Coordinate projections produce a rigidity phenomenon for antipodal pairs, providing a combinatorial analogue of Greenberger–Horne–Zeilinger (GHZ)-type fragility, whereas the first Boolean layer exhibits a persistence property analogous to W-type robustness. Together, these results exhibit a concrete bridge between Grassmann combinatorics, Brauer configuration theory, hypercube geometry, and entropy asymptotics. Full article

This study presents a combined numerical and experimental investigation of transient multiphase compressible flow inside a profiled-rotor rotary volumetric compressor. While most existing studies rely on high-fidelity CFD approaches, a low-order thermodynamic chamber-based model implemented in MATLAB Release 2023a is proposed to predict the temporal evolution of pressure, temperature, and vapor volume fraction during the compression cycle. The model is based on mass and energy conservation applied to variable-volume control chambers and incorporates a simplified cavitation criterion derived from local pressure relative to saturation vapor pressure. An open-loop experimental test bench was developed to measure air mass flow rate, suction and discharge pressures, temperatures, torque, and shaft power under controlled operating conditions. These measurements are used to validate the numerical predictions. The results show good agreement between measured and simulated pressure levels and global performance indicators, with deviations quantified using mean absolute percentage error values remaining below 5% over the investigated operating range. The numerical analysis further reveals the occurrence of localized low-pressure zones during the suction phase, indicating incipient cavitation or microbubble formation at specific rotor positions. The proposed modeling approach provides a computationally efficient alternative to full CFD simulations and enables rapid parametric analysis of rotor geometry and operating conditions. The cavitation formulation does not aim to resolve detailed bubble dynamics or erosion mechanisms, but rather to identify cavitation tendency based on thermodynamic pressure thresholds. Full article

Financial time series in turbulent markets exhibit complex long-memory dynamics and multifractal features that traditional deep learning models fail to capture due to inherent exponential forgetting mechanisms. To address this, we propose Frac-Attn-GL, a novel Fractional-order Spatiotemporal Attention-based GRU-LSTM framework. Grounded in the Fractal Market Hypothesis, the model embeds Grünwald–Letnikov fractional-order operators into a dual-channel architecture (FracLSTM and FracGRU) to characterize long-range memory with rigorous power-law decay priors. Furthermore, an extreme-aware asymmetric loss function is designed to drive a dynamic spatiotemporal routing mechanism, enabling adaptive shifts between long-term macro trends and short-term micro shocks. Empirical tests on major U.S. stock indices reveal three significant findings. First, the Frac-Attn-GL framework substantially reduces prediction errors, achieving up to a 93.1% RMSE reduction on the highly volatile NASDAQ index compared to standard baselines. Second, the adaptively learned fractional-order parameters exhibit a consistent quantitative alignment with the market’s empirical multifractal singularity spectrum, supporting the physical interpretability of the model’s endogenous memory mechanism. Finally, hybrid residual multifractal diagnostics indicate that the framework effectively captures deep long-range correlations, reducing the Hurst exponent of the prediction residuals from ~0.83 to approximately 0.50, a level consistent with the absence of significant long-range dependence. Full article

Variations in the particle size of cereal flour could influence its techno-functional properties and affect the quality of the end products. In this study, common buckwheat (Fagopyrum esculentum) seeds were milled and then sieved into five fractions (≥200, 150–200, 100–150, 80–100, and 60–80 mesh). Proximate analysis showed that the protein and ash contents of buckwheat flour decreased with decreased particle size, whereas the starch content increased. Reducing the particle size did not change the A-type crystalline structure and the short-range ordered structure of buckwheat starch, whereas the buckwheat batter flowability, foaming properties and foam stability of the batter supernatant increased. The steamed buckwheat cakes made from ≥100-mesh flour showed a desirable appearance, cross-sectional structure, color, flavor, and texture. Pearson correlation analysis revealed that the starch content and relative crystallinity of buckwheat flour were significantly positively correlated with its pasting parameters and the textural properties (springiness, cohesiveness, resilience) and overall acceptability of steamed buckwheat cake, whereas the protein, lipid, and β-sheet content of buckwheat flour showed the opposite trend. This study demonstrated that differential sieving caused a difference in particle size and chemical composition, which were key variables governing the processing performance of buckwheat flour and important to the quality of its end products. Full article

Photocatalytic hydrogen production represents a promising approach for sustainable fuel generation, particularly when driven by solar irradiation. In this study, a photocatalytic system composed of eosin Y, cobalt sulfate, triethanolamine, and graphene oxide was investigated for hydrogen evolution. The optical and structural properties of the system components were characterized using UV–Vis spectroscopy, FT-IR spectroscopy, Raman spectroscopy, and atomic force microscopy. Photocatalytic activity was evaluated under both artificial light sources (halogen lamp, xenon lamp, and LED 505 nm) and natural sunlight in order to assess system performance under realistic environmental conditions. The addition of graphene oxide significantly enhanced hydrogen production, resulting in an approximately 4-fold increase compared to the three-component system without graphene oxide. Solar-driven experiments conducted over one year demonstrated efficient hydrogen evolution under a wide range of weather and irradiance conditions. Importantly, based on combined experimental and meteorological data, it is shown that high photocatalytic performance is achievable for a substantial fraction of the year, with approximately 55% of days expected to provide at least 80% of the maximum hydrogen production efficiency under Central European climatic conditions. These findings highlight the strong potential of the investigated four-component system for efficient hydrogen generation using low amounts of catalytic material and without external electrical energy input. Overall, the system shows promising performance for solar-driven hydrogen production under real-world solar irradiation conditions. Full article

This paper investigates and characterizes Smarandache space curves, an important class of curves, using the Caputo fractional Frenet frame. The Frenet frame and fractional curvature functions have been calculated for these fractional Smarandache curves. To demonstrate the theoretical results obtained, an example of a fractional Smarandache curve derived from a helical curve is considered, and the curvatures of this curve are explicitly calculated. Finally, to show the effect of the fractional order parameter on the geometric behavior, a graphical analysis of the curvatures obtained for different fractional orders is presented. Full article

This study develops an extension of the classical Rayleigh–Schrödinger method for solving fractional-order differential equations. The primary objective is to derive the eigenvalues of a Begley–Torvik-type equation. The proposed analytical expression for the eigenvalues, obtained through this methodological advancement, shows excellent agreement with their exact values. This result is obtained by developing the Rayleigh–Schrödinger method and can be used for a wide range of applied problems. As an illustrative example, the Begley–Torvik type equation is used to describe the deformation-strength characteristics of polymer concrete and other modern granular road materials. It should be noted, however, that this represents just one of many potential applications for such fractional-order models. Full article

In this paper, an efficient and accurate framework for nonlinear spacetime fractional diffusion equations is proposed. The methods are based on the spectral deferred correction technique, which employs a compact difference scheme as the preconditioner via the Picard integral collocation formulation. The nonlinear term is incorporated into the preconditioner in a way similar to linear systems without using Newtonian methods. The preconditioner is proven to be a stable operator, and the resulting spectral deferred correction method maintains an arbitrary order of accuracy and excellent stability. Due to the dense property of the central finite difference approximation of the fractional Laplacian ${(-\Delta )}^s$, a dual accelerated algorithm for the exact computation of the matrix–vector product is presented by introducing the discrete sine transform. The numerical results demonstrate that the proposed new methods are highly efficient and precise. Full article

In this paper, we give some new Jensen, Jensen–Mercer, and Hermite–Hadamard inequalities for uniformly $\delta$-geometric convex functions. In addition, some limit bounds for Caputo–Fabrizio fractional integral operators are established as an application in the case of uniformly $\delta$-geometric convex functions. Some new examples and graphical representations are provided in order to illustrate the validity of our results. Full article

In this paper, we study a class of second-order fractional boundary value problems involving $\Theta$-Caputo derivatives of different orders. By reformulating the problem to an integral equation, we introduce an appropriate notion of a mild solution in the $\Theta$-fractional framework. Existence results are obtained via Krasnoselskii’s fixed point theorem, while uniqueness is established using the Banach contraction principle under suitable Lipschitz-type conditions. The obtained results extend several earlier works on Caputo, Hadamard–Caputo, and Riemann–Liouville fractional derivatives. Two examples are presented to illustrate the applicability of the theoretical results. Full article