Function Method Research Articles

Wigner function method for the Gibbons–Hawking and the Unruh effect

Wigner function method for the Gibbons–Hawking and the Unruh effect

Dynamic Description of soliton solutions of nonlinear Schrödinger equation with higher order dispersion and nonlinear terms

Abstract The article presents an analytical solution for the higher-order nonlinear Schrödinger equation (NLSE), which describes the propagation of short light pulses in monomode optical fibers. Various traveling wave solutions are obtained using the generalized exponential rational function method, a technique with substantial applications in physics and mathematics. Additionally, the parameters leading to the occurrence of optical bright and multipeak solitons in this medium are provided along with their formation conditions. The derived solutions are graphically displayed to enhance the understanding of the model’s physical phenomena. This approach is credible, potent, and successful in solving a wide variety of different models of this kind that arise in the applied sciences. Its robustness, strength, and efficiency make it suitable for addressing various higher-order nonlinear problems in current research fields, extending beyond the models encountered in the applied sciences.

Azaindole: A Candidate Anchor for Regulating Charge Polarity and Inducing Resonance Transmission at the Fermi Level via Dehydrogenation.

Tuning the polarity of charge carriers is essential for designing molecular logic devices in molecular electronics. In this study, the electrical transport properties of a family of azaindole-anchored single-molecule junctions have been investigated using density functional theory combined with the nonequilibrium Green's function method. The obtained results reveal that dehydrogenation is an effective method for reversing the polarity of charge carriers. The molecular junctions based on the entire azaindole unit are n-type and contain electrons as the principal charge carriers, whereas the dehydrogenated junctions are p-type and contain holes as the main carriers. Furthermore, the azaindole anchors undergo a transition from an electron-rich to an electron-deficient state due to dehydrogenation, which is the original cause of the charge carrier polarity conversion. Dehydrogenated molecular junctions also exhibit the Fermi pinning effect and a sharp highest occupied molecular orbital (HOMO) resonance peak at the Fermi level. In addition, using Pt electrodes instead of Au electrodes is a means of producing a HOMO resonance peak a for azaindole-based molecular junctions. This work demonstrates the enormous potential of utilizing azaindole-anchored molecular junctions for the implementation of molecular logic and multifunctional molecular devices.

Quantifying Uncertainty Costs in Renewable Energy Systems Considering Probability Function Behavior and CVaR at Low-Probability Generation Extremes using Deterministic Equations

The increase and integration of renewable energy sources in electrical power systems implies an increase in uncertainty variables, both in costs and production, of economic dispatch (ED) and currently have a significant influence on wholesale electricity markets (MEM). Uncertainty costs refer to the quantification of additional expenses or economic losses associated with the variability inherent in the generation of renewable energy, such as wind, solar, or hydroelectric. Therefore, this article presents deterministic equations related to cost overestimation and underestimation, as well as CVaR, to model and evaluate the stochasticity of risks associated with the integration of renewable sources, allowing system operators and planners to make informed decisions. To mitigate or use said risks in energy systems with high penetration of elements, mainly smart networks. In this study, a mathematical analysis is carried out using the histogram spectrum formed by the power generated by the probability density function (PDF) for solar generation, although it is possible to consider other types of functions to determine energy generation. The objective of the proposed model is to provide another tool to the system operator for energy management and planning, which relieves a little of the weight of the computational load and at the same time presents more precision in the results by being able to work with a database. Historical data if these values are available. Commonly, for this type of analysis, values are estimated using probabilistic calculations by density functions when integrating these functions, or in other recent cases by estimating them by analytical methods of the same functions. A validation of the model is presented by comparing the result with the Monte Carlo simulation, developing the total cost of uncertainty only from "low probability generation extremes". Furthermore, the results are presented through analytical uncertainty cost functions (AUCF). This analysis includes the calculation of uncertainty costs for low and high-probability energy generation, determined by the Conditional Value at Risk (CVaR), using deterministic equations.

Numerical Modeling on Ocean-Bottom Seismograph P-Wave Receiver Function to Analyze Influences of Seawater and Sedimentary Layers

It is challenging to apply the receiver function method to teleseisms recorded by ocean-bottom seismographs (OBSs) due to a specific working environment that differs from land stations. Teleseismic incident waveforms reaching the area beneath stations are affected by multiple reflections generated by seawater and sediments and noise resulting from currents. Furthermore, inadequate coupling between OBSs and the seabed basement and the poor fidelity of OBSs reduce the signal-to-noise ratio (SNR) of seismograms, leading to the poor quality of extracted receiver functions or even the wrong deconvolution results. For instance, the poor results cause strong ambiguities regarding the Moho depth. This study uses numerical modeling to analyze the influences of multiple reflections generated by seawater and sediments on H-kappa stacking and the neighborhood algorithm. Numerical modeling shows that seawater multiple reflections are mixed with the coda waves of the direct P-wave and slightly impact the extracted receiver functions and can thus be ignored in subsequent inversion processing. However, synthetic seismograms have strong responses to the sediments. Compared to the waveforms of horizontal and vertical components, the sedimentary responses are too strong to identify the converted waves clearly. The extracted receiver functions correspond to the above influences, resulting in divergent results of H-kappa stacking (i.e., the Moho depth and crustal average VP/VS ratio are unstable and have great uncertainties). Fortunately, waveform inversion approaches (e.g., the neighborhood algorithm) are available and valid for obtaining the S-wave velocity structure of the crust–upper mantle beneath the station, with sediments varying in thickness and velocity.

Computational model for fractal dimension of anisotropic gear surfaces based on improved structure function method

Computational model for fractal dimension of anisotropic gear surfaces based on improved structure function method

Copula-Based Regression with Mixed Covariates

In this paper, we focused on developing copula-based modeling procedures that effectively capture the dependence between response and explanatory variables. Building upon the work of Noh et al. (J. Am. Stat. Assoc. 2013, 108, 676–688) we extended copula-based regression to accommodate both continuous and discrete covariates. Specifically, we explored the construction of copulas to estimate the conditional mean of the response variable given the covariates, elucidating the relationship between copula structures and marginal distributions. We considered various estimation methods for copulas and distribution functions, presenting a diverse array of estimators for the conditional mean function. These estimators range from non-parametric to semi-parametric and fully parametric, offering flexibility in modeling regression relationships. An adapted algorithm is applied to construct copulas and simulations are carried out to replicate datasets, estimate prediction model parameters, and compare with the OLS method. The practicality and efficacy of our proposed methodologies, grounded in the principles of copula-based regression, are substantiated through methodical simulation studies.

Computational analysis of the conformable nonlinear Schrödinger equation with Kudryashov’s refractive index model and generalized non-local nonlinearity

This paper investigates the nonlinear conformable Schrödinger equation with nonlocal nonlinearities and Kudryashov’s arbitrary refractive index. The study employs the Uniform Method (UM) and the Extended Hyperbolic Function Method (EHFM) to derive various optical soliton solutions for this present conformable equation. To demonstrate the importance of the novel solutions, the paper presents two-dimensional, three-dimensional, and contour plots, illustrating kink-type, wave, bright, bell-shaped, and mixed dark–bright soliton solutions. Further, the impact of the fractional order parameter, temporal parameter, and nonlinearity coefficient on these optical solutions is analyzed, providing valuable insights into the conformable nonlinear Schrödinger model. The behavior of these optical solutions is analyzed through illustrative graphs that account for variations in the conformable order derivative, temporal parameter, and nonlinearity coefficient. The methodologies applied in this research show potential for broader application to various nonlinear Schrödinger equations in fields such as applied mathematics and nonlinear optics. The research suggests that these methods could be applied in the future to investigate other differential equations with fractional and integer orders across different fields of applied sciences. This study expands our understanding of nonlinear optics and has potential practical implications in various fields like optical signal processing, laser technology, and telecommunications.

New analytical wave structures for generalized B-type Kadomtsev–Petviashvili equation by improved modified extended tanh function method

Abstract Recently, solving the complicated nonlinear partial differential equations has become very important demand in order to simulate their physical phenomena. This manuscript focuses on extracting the wave solutions of (3 + 1)-dimensional generalized B-type Kadomtsev-Petviashvili equation (GBKPE), which demonstrates the behavior of nonlinear waves in fluid mechanics. The improved modified extended Tanh function (IMETF) method is the suggested method to do this task as it gives different types of solutions. This method enables us to obtain many solutions, such as Jacobi elliptic, dark soliton, and singular soliton, exponential, and singular periodic wave solutions. Additionally, for more illustrations graphical visual representations of some solutions are provided.

Adaptive third-order integral sliding mode control for the mobile petroleum crane-form pipelines system with input bounded

The mobile petroleum crane-form pipelines (MPCFP) system is one of the most important mechanical equipment in the petrochemical industry, consisting of rigid pipelines and rotating elbows, mainly used for loading and unloading petrochemical fluids. This paper proposes a novel neural network adaptive third-order integral sliding mode tracking control strategy with bounded inputs to achieve automatic alignment between the MPCFP system and the inlet of the tank truck. The proposed strategy replaces normal system inputs with the time derivative of the control torque. Although there are discontinuous terms in the derivative of the control torque, the integrated control torque is continuous. This unique feature ensures finite-time convergence of system state errors and eliminates sliding mode chattering. To address the issues of bounded input and modeling error, a state compensation method is introduced, and a neural network adaptive algorithm is utilized to estimate the modeling error. The stability proof process employs the Lyapunov function method, demonstrating that the system state error can converge to zero. Numerous simulation experiments validate the effectiveness of this control strategy.

Nonlinear Dynamics of an Electromagnetically Actuated Cantilever Beam Under Harmonic External Excitation

The present work is devoted to the study of nonlinear vibrations of an electromagnetically actuated cantilever beam subject to harmonic external excitation. The soft actuator that controls the vibratory motion of such components of a robotic structure led to a strongly nonlinear governing differential equation, which was solved in this work by using a highly accurate technique, namely the Optimal Auxiliary Functions Method. Comparisons between the results obtained using our original approach with those of numerical integration show the efficiency and reliability of our procedure, which can be applied to give an explicit analytical approximate solution in two cases: the nonresonant case and the nearly primary resonance. Our technique is effective, simple, easy to use, and very accurate by means of only the first iteration. On the other hand, we present an analysis of the local stability of the model using Routh–Hurwitz criteria and the eigenvalues of the Jacobian matrix. Global stability is analyzed by means of Lyapunov’s direct method and LaSalle’s invariance principle. For the first time, the Lyapunov function depends on the approximate solution obtained using OAFM. Also, Pontryagin’s principle with respect to the control variable is applied in the construction of the Lyapunov function.

Penalty 4-Node Quadrilateral Element Formulation for Axisymmetric Couple Stress Problems

To address the issue of size effects in axisymmetric deformation of small-scale solids, this work proposes a 4-node 12-DOF element for axisymmetric problems based on the consistent couple stress theory (CCST), following the framework of the unsymmetric finite element method. With the use of the penalty function method, an independently assumed rotational field is introduced into the virtual work principle to approximate the physical rotation, ensuring the satisfaction of the C1 continuity requirement of the CCST in a weak form. As a benefit, the enriched C0 isoparametric-based interpolation is employed to construct the test functions for displacement and rotation. Furthermore, the force-stress field that satisfies the equilibrium equations related to axisymmetric deformation is employed as the element’s force-stress trial function. In order to circumvent locking issues, reduced integration is employed in the penalty stiffness integration process. The numerical results demonstrate that the new element exhibits high computational accuracy and convergence rate in both static and modal analysis problems, effectively capturing size-dependent phenomena.

Riemannian trust-region methods for strict saddle functions with complexity guarantees

Riemannian trust-region methods for strict saddle functions with complexity guarantees

Lag-WALS approach incorporating ENSO-related quantities for altimetric interannual SLA forecasts in the South China Sea

A novel approach using lag weighted-average least squares (Lag-WALS) is proposed to forecast the interannual sea level anomaly (SLA) in the South China Sea (SCS) using lagged equatorial Pacific El Niño–Southern Oscillation (ENSO)-related quantities. Through empirical orthogonal function (EOF) and wavelet coherence method, we first investigated the relationships between sea surface temperature (SST) and SLA (both steric sea level (SSL) and non-steric sea level (NSSL)) in the equatorial Pacific, and then explored their cross-correlations with the interannual SCS SLA. A robust alignment was found between the first spatiotemporal mode of EOF (i.e. EOF1 and first principal component (PC1)) from SLA/SSL and SST across the equatorial Pacific, both of which exhibited a typical ENSO horseshoe spatial pattern in EOF1. Good consistency between the SCS SLA and the SST/SLA/SSL PC1 was revealed, with the SCS SLA lagging behind the SST, SLA, and SSL by several months at most grid locations. In contrast, the NSSL exhibited large disparities with the SST PC1 or the interannual SCS SLA. The lag-WALS model performed better at the SCS boundaries than in the central region, with an average STD/MAE/Bias (RMSE/MAE/Bias) for internal (external) accuracies of 1.01/0.80/–0.002 cm (1.39/1.13/–0.08 cm), respectively. The altimetric-observed SLA seasonal patterns agreed with the Lag-WALS model-forecasted SLA. A similar situation applies to regionally-averaged SLA time series. These results underscore the ability of the Lag-WALS model to accurately forecast the SCS SLA at the interannual scale, which is crucial for early warning of abnormal sea level changes in the SCS.

Nonlinear mechanical response of finite-length soft composites with random dislocations

The effects of random dislocations on the vibrational properties of finite-length RNA and DNA macro-structures have been investigated by means of a harmonic Hamiltonian and the Green’s function method. The RNA molecule has been modeled using a half ladder model, and three models (a fishbone model and two different strand models) have been employed to model the structure of DNA. The lengths of the finite and cyclic systems are gradually increased to more accurately approximate the structures of RNA and DNA. Springs whose behaviors are governed by Hooke’s constitutive law have been used to represent the bonds between the masses, with the stiffness of the vertical springs randomly changing along the length of each model. This results in a more realistic representation of the inherent randomness of the studied structures. To investigate the effect of random dislocations on the mechanical response of the studied systems, it has been assumed that a random mass-spring ensemble has been knocked out of place by an external force. It has been found that increasing the number of building blocks along the length of the models suppresses the influence of dislocations on the vibration spectra of RNA and DNA. It was also observed that at low frequencies, the influence of dislocations becomes more pronounced. Besides, taking into account the collective mass of the sugar-phosphate backbone results in the appearance of gaps in the vibration spectra. By introducing dislocations into the models, additional dislocation-induced vibrational states appear in the DOS curves. What stands out from the results is that the responses of the studied systems to these changes are strictly nonlinear. The employed methodology can be applied to investigate the mechanical response of damaged RNA and DNA.

Electron transport and quantum phase transitions in cumulene-connected C80H20 fulleryne: DFT and tight-binding studies

Molecular bridges are opening up exciting new applications in diverse fields, improving the efficiency of conductive inks, enhancing the performance of devices such as organic light-emitting diodes and low-cost solar cells, and advancing the development of highly sensitive sensors, chemical reactions, drug delivery systems, and more. In this paper, we study the electron transport properties of a C80H20 fulleryne (dodecahedryne) connected to two cumulene electrodes. Using density functional theory (DFT), we determine the optimal molecular bridge structure. Based on the IR vibration spectra, different stable phases of the molecular bridge are obtained. The corresponding tight-binding (TB) parameters of the cage are obtained by assigning appropriate values of the length and type of bonds for the fulleryne cage through matching the HOMO-LUMO gap between the DFT calculations and the TB parameters. The electron transport for the desired structures is investigated using the obtained tight-binding parameters and the non-equilibrium Green's function (NEGF) method. Finally, it is concluded that among the five possible configurations for the cumulene-dodecahedryne -cumulene molecular bridge, only one specific configuration—where the electrodes are one edge apart—exhibits metallic behavior, while other positions act as insulators. In addition, the system exhibits quantum phase transitions from metal to semiconductor and from insulator to metal in the presence of critical electric fields. The ability to control quantum phase transitions in these molecular systems can be leveraged to develop qubits for quantum computing. The unique properties can be utilized to design advanced molecular electronic devices.

Time-fractional of cubic-quartic Schrödinger and cubic-quartic resonant Schrödinger equations with parabolic law: various optical solutions

Abstract Schrödinger's nonlinear equation is a fundamental model in fiber optics and many other areas of science. Using the Jacobi elliptic expansion function method, the time-fractional cubic-quartic nonlinear Schrödinger equation and cubic-quartic resonant nonlinear Schrödinger equation are investigated. By applying the effective Jacobi elliptic expansion function method, optical soliton solutions such as bright, dark, singular, periodic singular, exponential, and Jacobi elliptic function solutions have been obtained. The effect of the time-fractional derivative on the solutions is also revealed. Graphical representations are illustrated to showcase the physical properties of raised solutions, providing a comprehensive understanding of the solutions’ functionality.

Anti-bump control for switched systems with decentralised adaptive event-triggering

A control system may exist bumps during mode transitions and discontinuous control input would seriously affect performance. In this paper, the anti-bump (or bumpless) switching control for triggered and switched systems under injection attacks is studied. This control method combines decentralised adaptive event-triggering mechanism (DAETM) and anti-bump condition. The main work is as follows: First, a DAETM is considered to allocate network resources properly and resist data injection attacks. Then, a data buffer is adopted to consolidate data from decentralised channels for ensuring timely utilisation. In addition, a controller gain condition is presented to achieve the bumpless switching between subsystems. Moreover, by adopting piecewise Lyapunov function method, sufficient conditions are obtained for triggered switched systems to be exponentially stable and anti-bump under injection attacks. Finally, the design method effectiveness is verified by actual circuit simulation.

Photometric detection of internal gravity waves in upper main-sequence stars. IV. Comparable stochastic low-frequency variability in SMC, LMC, and Galactic massive stars

Massive main-sequence stars have convective cores and radiative envelopes, but can also have sub-surface convection zones caused by partial ionisation zones. However, the convective properties of such regions strongly depend on opacity and therefore a star's metallicity. Non-rotating 1D evolution models of main-sequence stars between $7 M odot $ and metallicity of the Small Magellanic Cloud (SMC) galaxy suggest tenuous (if any) sub-surface convection zones when using the Rayleigh number as a criterion for convection owing to their substantially lower metallicity compared to Galactic massive stars. We test if massive stars of different metallicities both inside and outside of asteroseismically calibrated stability windows for sub-surface convection exhibit different properties in stochastic low-frequency (SLF) variability. Thus we aim to constrain the metallicity dependence of the physical mechanism responsible for SLF variability commonly found in light curves of massive stars. We extract customised light curves from the ongoing NASA Transiting Exoplanet Survey Satellite (TESS) mission for a sample of massive stars using an effective point spread function (ePSF) method, and compare their morphologies in terms of characteristic frequency, $ char $, and amplitude using a Gaussian process (GP) regression methodology. We demonstrate that the properties of SLF variability observed in time-series photometry of massive stars are generally consistent across the metallicity range from the Milky Way down to the SMC galaxy, for stars both inside and outside of the sub-surface stability windows based on the Rayleigh number as a criterion for convection. We conclude that non-rotating 1D stellar structure models of sub-surface convection cannot alone be used to explain the mechanism giving rise to SLF variability in light curves of massive stars. Additionally, the similar properties of SLF variability across a large range in metallicity, which follow the same trends in mass and age in the Hertzsprung--Russell (HR) diagram at both high and low metallicity, support a transition in the dominant mechanism causing SLF variability from younger to more evolved stars. Specifically, core-excited internal gravity waves (IGWs) are favoured for younger stars lacking sub-surface convection zones, especially at low-metallicity, and sub-surface convection zones are favoured for more evolved massive stars.

Optimized design of energy absorption of aluminum foam filled CFRP thin-walled square tube based on agent model

PurposeAluminum foam-filled thin-walled unit structures have received much attention for their excellent energy absorption properties. To improve the energy absorption effect of car energy absorption box under axial compression, this paper optimizes the fiber lay-up sequence, fiber angle and aluminum foam density of aluminum foam filled carbon fiber reinforced plastic (CFRP) thin-walled square tubes.Design/methodology/approachDesign of sample points required to construct the proxy model using design of experiments (DOE) method, and the data sample points of different models are obtained through Abaqus simulation and test. A double high-precision proxy model with the maximum specific energy absorption (SEA) and the minimum initial peak crash force (PCF) as the evaluation index is constructed based on the response surface function method. The NSGA-II multi-objective genetic algorithm was used to optimize the design parameters and obtain the optimal solution for the Pareto front, and the results were verified by using the multi-objective optimization toolbox in design-expert.FindingsThe results show that the optimal solution to the multi-objective optimization problem with the inclusion of the lay-up sequence is ρ = 0.5g/cm3 for a fiber lay-up angle varying in the range ±15–90° and an aluminum foam density varying in the range 0.2g/cm3-0.5g/cm3, with a lay-up method of [±87°/±16°/±15°/±89°]. The two optimization methods correspond to SEA and PCF errors of 2.109% and 4.1828%, respectively. The optimized SEA value is 18.2 J/g and the PCF value is 18,230 N. The optimized design reduces the peak impact force and increases the specific energy absorption, which improves the energy absorption effect of thin-walled energy-absorbing boxes for automobiles.Originality/valueIn this paper, the impact resistance of CFRP thin-walled square tubes filled with aluminum foam is optimized. Based on numerical simulations and experiments to obtain the sample point data for constructing the dual-agent model, we investigate the effect of filling with different densities of aluminum foam under the simultaneous change of fiber lay-up angle and order on its mechanical properties in this process, and carry out the multi-objective optimization design with NSGA-II algorithm.