Search Results (2,442)

In this study, the extended (3 + 1)-dimensional Jimbo–Miwa equation, which has not been previously studied using Lie symmetry techniques, is the focus. We derive new symmetry reductions and exact invariant solutions, including lump and rogue wave structures. Additionally, precise solitary wave solutions of the extended (3 + 1)-dimensional Jimbo–Miwa equation using the multivariate generalized exponential rational integral function technique (MGERIF) are studied. The extended (3 + 1)-dimensional Jimbo–Miwa equation is crucial for studying nonlinear processes in optical communication, fluid dynamics, materials science, geophysics, and quantum mechanics. The multivariate generalized exponential rational integral function approach offers advantages in addressing challenges involving exponential, hyperbolic, and trigonometric functions formulated based on the generalized exponential rational function method. The solutions provided by MGERIF have numerous applications in various fields, including mathematical physics, condensed matter physics, nonlinear optics, plasma physics, and other nonlinear physical equations. The graphical features of the generated solutions are examined using 3D surface graphs and contour plots, with theoretical derivations. This visual technique enhances our understanding of the identified answers and facilitates a more profound discussion of their practical applications in real-world scenarios. We employ the MGERIF approach to develop a technique for addressing integrable systems, providing a valuable framework for examining nonlinear phenomena across various physical contexts. This study’s outcomes enhance both nonlinear dynamical processes and solitary wave theory. Full article

This paper studies discontinuous quasilinear sub-elliptic systems associated with Hörmander’s vector fields under controllable and natural growth conditions. By a new $\mathcal{A}$-harmonic approximation reformulation for bilinear forms $\mathcal{A}\in \mathrm{Bil}({\mathbb{R}}^{kN},{\mathbb{R}}^{kN})$, we obtain optimal partial Hölder continuity with exact exponents for weak solutions with vanishing mean oscillation coefficients. Full article

The integration of solar generation into national energy balances is associated with a wide range of technical, economic, and organizational challenges, the solution of which requires the adoption of innovative strategies for energy system management. The inherent variability of electricity production, driven by fluctuating climatic conditions, complicates system balancing processes and necessitates the reservation of capacities from conventional energy sources to ensure reliability. Under modern market conditions, the pricing of generated electricity is commonly based on day-ahead forecasts of day energy yield, which significantly affects the economic performance of solar power plants. Consequently, achieving high accuracy in day-ahead electricity production forecasting is a critical and highly relevant task. To address this challenge, a physico-statistical model has been developed, in which the analytical approximation of daily electricity generation is represented as a function of a random variable—cloud cover—modeled by a β-distribution. Analytical expressions were derived for calculating the mathematical expectation and variance of daily electricity generation as functions of the β-distribution parameters of cloudiness. The analytical approximation of daily generation deviates from the exact value, obtained through hourly integration, by an average of 3.9%. The relative forecasting error of electricity production, when using the mathematical expectation of cloudiness compared to the analytical approximation of daily generation, reaches 15.2%. The proposed forecasting method, based on a β-parametric cloudiness model, enhances the accuracy of day-ahead production forecasts, improves the economic efficiency of solar power plants, and contributes to strengthening the stability and reliability of power systems with a substantial share of solar generation. Full article

The second virial coefficient (SVC) of the Lennard-Jones fluid is a cornerstone of molecular theory, yet its calculation has traditionally relied on the complex integration of the pair potential. This work introduces a fundamentally different approach by reformulating the problem in terms of ordinary differential equations (ODEs). For the classical component of the SVC, we generalize the confluent hypergeometric and Weber–Hermite equations. For the first quantum correction, we present entirely new ODEs and their corresponding exact-analytical solutions. The most striking result of this framework is the discovery that these ODEs can be transformed into Schrödinger-like equations. The classical term corresponds to a harmonic oscillator, while the quantum correction includes additional inverse-power potential terms. This formulation not only provides a versatile method for expressing the virial coefficient through a linear combination of functions (including Kummer, Weber, and Whittaker functions) but also reveals a profound and previously unknown mathematical structure underlying a classical thermodynamic property. Full article

The purpose of this paper is to investigate a class of novel symmetric coupled fuzzy fractional partial differential equation system involving the Caputo–Katugampola (C-K) generalized Hukuhara (gH) derivative. Within the framework of C-K gH differentiability, two types of gH weak solutions are defined, and their existence is rigorously established through explicit constructions via employing Schauder fixed point theorem, overcoming the limitations of traditional Lipschitz conditions and thereby extending applicability to non-smooth and nonlinear systems commonly encountered in practice. A typical numerical example with potential applications is proposed to verify the existence results of the solutions for the symmetric coupled system. Furthermore, we introduce Ulam–Hyers stability (U-HS) theory into the analysis of such symmetric coupled systems and establish explicit stability criteria. U-HS ensures the existence of approximate solutions close to the exact solution under small perturbations, and thereby guarantees the reliability and robustness of the systems, while it prevents significant deviations in system dynamics caused by minor disturbances. We not only enrich the theoretical framework of fuzzy fractional calculus by extending the class of solvable systems and supplementing stability analysis, but also provide a practical mathematical tool for investigating complex interconnected systems characterized by uncertainty, memory effects, and spatial dynamics. Full article

Collective superradiant decay of a tightly packed inverted quantum emitter ensemble is among the most intensely studied phenomena in quantum optics. Since the seminal work of Dicke more than half a century ago, a plethora of theoretical calculations in quantum many-body physics have followed. Widespread experimental efforts range from the microwave to the X-ray regime. Nevertheless, accurate calculations of the time dynamics of the superradiant emission pulse still remain a challenging task requiring approximate methods for large ensembles. Here, we benchmark the cumulant expansion method for describing collective superradiant decay against a newly found exact solution. The application of two variants of the cumulant expansion exhibits reliable convergence of time and magnitude of the maximum emission power with increasing truncation order. The long-term population evolution is only correctly captured for low emitter numbers, where an individual spin-based cumulant expansion proves more reliable than the collective spin-based variant. Surprisingly, odd orders show unphysical behavior. At sufficiently high spin numbers, both chosen cumulant methods agree, but still fail to reliably converge to the numerically exact result. Generally, on longer time scales the expansions substantially overestimate the remaining population. While numerically fast and efficient, cumulant expansion methods need to be treated with sufficient caution when used for long-time evolution or reliably finding steady states. Full article

On-demand urban delivery increasingly relies on electric delivery bicycles (EDBs), yet their limited battery capacity creates coupled challenges of routing efficiency and energy replenishment. We study a novel battery swapping cabinet location-routing problem (BSC-LRP) with multiple depots, which jointly optimizes routing and modular energy infrastructure deployment under time-window and battery constraints. To address the problem’s complexity, we design an improved branch-and-price algorithm enhanced with adaptive heuristic-exact labeling (IBP-HL) and a robust arc-based branching scheme. This hybrid framework accelerates column generation while preserving exactness, representing a methodological advancement over standard B&P approaches. Computational experiments on modified Solomon instances show that IBP-HL consistently outperforms Gurobi in both runtime and solution quality on small cases, and achieves substantial speedups and improved bounds over baseline B&P on medium and large cases. These results demonstrate not only the scalability of IBP-HL but also its practical relevance: the framework provides decision support for operators and planners in designing cost-efficient, reliable, and sustainable last-mile delivery systems with battery-swapping infrastructure. Full article

Through transformation and utilizing a novel extended complex method combining with the Weierstrass factorization theorem, Wiman–Valiron theory and the Painlevé test, new non-constant meromorphic solutions were constructed for the predator–prey model. These meromorphic solutions contain the rational solutions, exponential solutions, elliptic solutions, and transcendental entire function solutions of infinite order in the complex plane. The exact solutions contribute to understanding the predator–prey model from the perspective of complex differential equations. In fact, the presented synthesis method provides a new technology for studying some systems of partial differential equations. Full article

An exact analytical model based on three-dimensional (3D) thermo-elasticity theory is developed to investigate the bending behavior of fiber metal laminate (FML) plates under thermo-mechanical load. The temperature-dependent properties and the orthotropy of the component materials are considered in this model. The analytical model is based on the heat conduction theory and thermoelasticity theory, and the solutions are determined by employing the Fourier series expansion, the state space approach and the transfer matrix method. Comparison study shows that the FE results are generally in good agreement with the present analytical solutions, exhibiting relative errors of less than 2%, except in the regions near the upper and lower surfaces. The present solution is close to the experimental values for the laminated plate within the linear range, with errors less than 10%. The decoupling analysis indicates that the thermo-mechanical performance of FML plates no longer strictly adheres to the traditional superposition principle, with errors reaching 30.39%. A modified principle accounting for modulus degradation is introduced to address this discrepancy. Furthermore, parametric studies reveal that the temperature and the lamina number have significant effect on the stresses and displacements of the FML plate. Full article

In industrial equipment containing hydraulic lines for power transmission, the lines have boundary conditions defined by components such as pumps, valves, and actuators located at the ends of the lines. Sudden changes in any of the boundary conditions may result in significant pressure/flow dynamics (fluid transients) in the lines that may be detrimental or favorable to the performance of the equipment. Accurate models for line transients are defined by the exact solution to a set of simultaneous partial differential equations. In this paper, analytical solutions to the partial differential equations provide Laplace transform transfer functions applicable to any set of boundary conditions yet to be specified that satisfy the requirements of causality. Analytical solutions of these partial differential equations from previous publications are reviewed for cases of laminar and turbulent flow for Newtonian and a class of non-Newtonian fluids. This paper focuses on a method for obtaining total system analytical models and time domain solutions for cases in which the end-of-line components can be modeled with linear equations for perturbations relative to pre-transient flow conditions. Examples with pumps, valves, and actuators demonstrate the process of coupling equations for components at the ends of a line to obtain total system transfer functions and then obtain time domain solutions for outputs of interest associated with system inputs and load variations. Full article

In this paper, we develop a fully discrete finite element scheme, based on a second-order backward differentiation formula (BDF2), for numerically solving the three-dimensional incompressible Navier–Stokes equations. Under the assumption that the fully discrete solution remains bounded in a certain norm, we establish that any smooth initial data necessarily gives rise to a unique strong solution that remains smooth. Moreover, we demonstrate that the fully discrete numerical solution converges strongly to this exact solution as the temporal and spatial discretization parameters approach zero. Full article

This work investigates a new category of linear non-homogeneous discrete-matrix equations that feature multiple delays and first-order differences, under the assumption of pairwise permutable coefficient matrices. We define a novel multi-delayed discrete-matrix-exponential function that extends previous concepts. Using this function and specific commutativity properties, we construct an explicit matrix solution. The study highlights the advantages of our approach by contrasting it with prior research and identifying new open questions. Finally, a numerical example illustrates the application and relevance of the derived solution. Full article

Background: Electrochemical impedance spectroscopy (EIS) is indispensable for disentangling charge-transfer, capacitive, and diffusive phenomena, yet reproducible parameter estimation and objective model selection remain unsettled. Methods: We derive closed-form impedances and analytical Jacobians for seven equivalent-circuit models (Randles, constant-phase element (CPE), and Warburg impedance (ZW) variants), enforce physical bounds, and fit synthetic spectra with 2.5% and 5.0% Gaussian noise using hybrid optimization (Differential Evolution (DE) → Levenberg–Marquardt (LM)). Uncertainty is quantified via non-parametric bootstrap; parsimony is assessed with root-mean-square error (RMSE), Akaike Information Criterion (AIC), and Bayesian Information Criterion (BIC); physical consistency is checked by Kramers–Kronig (KK) diagnostics. Results: Solution resistance ($R_s$) and charge-transfer resistance ($R_{ct}$) are consistently identifiable across noise levels. CPE parameters $(Q,n)$ and diffusion amplitude $\left(\sigma \right)$ exhibit expected collinearity unless the frequency window excites both processes. Randles suffices for ideal interfaces; Randles+CPE lowers AIC when non-ideality and/or higher noise dominate; adding Warburg reproduces the ${45}^{\circ }$ tail and improves likelihood when diffusion is present. The $(R_{ct}+Z_W)\Vert \mathrm{CPE}$ architecture offers the best trade-off when heterogeneity and diffusion coexist. Conclusions: The framework unifies analytical derivations, hybrid optimization, and rigorous statistics to deliver traceable, reproducible EIS analysis and clear applicability domains, reducing subjective model choice. All code, data, and settings are released to enable exact reproduction. Full article

Mathematical optimization, in both continuous and discrete forms, is well established and widely applied. This work addresses a gap in the literature by focusing on large-number optimization, where integers or fractions with hundreds of digits occur in decision variables, objective functions, or constraints. Such problems challenge standard optimization tools, particularly when exact solutions are required. The suitability of computer algebra systems and high-precision arithmetic software for large-number optimization problems is discussed. Our first contribution is the development of Python implementations of an exact Simplex algorithm and a Branch-and-Bound algorithm for integer linear programming, capable of handling arbitrarily large integers. To test these implementations for correctness, analytic optimal solutions for nine specifically constructed linear, integer linear, and quadratic mixed-integer programming problems are derived. These examples are used to test and verify the developed software and can also serve as benchmarks for future research in large-number optimization. The second contribution concerns constructing partially increasing subsequences of the Collatz sequence. Motivated by this example, we quickly encountered the limits of commercial mixed-integer solvers and instead solved Diophantine equations or applied modular arithmetic techniques to obtain partial Collatz sequences. For any given number J, we obtain a sequence that begins at $2^J-1$ and repeats J times the pattern ud: multiply by $3x_j+1$ and then divide by 2. Further partially decreasing sequences are designed, which follow the pattern of multiplying by $3x_j+1$ and then dividing by $2^m$. The most general J-times increasing patterns (ududd, udududd, …, ududududddd) are constructed using analytic and semi-analytic methods that exploit modular arithmetic in combination with optimization techniques. Full article

Photogrammetry applied to sports provides precise data on athlete positions and time instants, especially with digital motion capture systems. However, detecting and identifying specific events in athletic movements such as discus throwing can be challenging when using only images. For example, with high-speed video, it is difficult to pinpoint the exact frame when events like foot touchdown or takeoff occur, as contact between shoe and ground may span several frames. Inertial measurement units (IMUs) can detect maxima and minima in linear accelerations and angular velocities, helping to accurately determine these specific events in throwing movements. As a result, comparing photogrammetry data with IMU data becomes challenging because of the differences in the methods used to detect events. Even if comparisons can be made with IMU data from other sports researchers, variations in methodologies can invalidate the comparison. To address this, the paper proposes a simple methodology for detecting the five phases of a discus throw using three IMUs located on the thrower’s wrist and on the instep or ankle of the feet. Experiments with three elite male discus throwers are conducted and the results are compared with existing data in the literature. The findings demonstrate that the proposed methodology is effective (100% of phases detected in the experiments without false positives) and reliable (results validated with professional coaches), offering a practical and time- and cost-effective solution for accurately detecting key moments in athletic movements. Full article