Search Results (218)

In this paper, we study direct and inverse problems for a spatial-fractional Black–Scholes equation with space-dependent volatility. For the direct problem, we provide CN-WSGD (Crank–Nicholson and the weighted and shifted Grünwald difference) scheme to solve the initial boundary value problem. The latter aims to recover the implied volatility via observable option prices. Using a linearization technique, we rigorously derive a mathematical formulation of the inverse problem in terms of a Fredholm integral equation of the first kind. Based on an integral equation, an efficient numerical reconstruction algorithm is proposed to recover the coefficient. Numerical results for both problems are provided to illustrate the validity and effectiveness of proposed methods. Full article

This article investigates modeling issues of fractional-order singular systems. The state estimation can be solved by using the particle filter. An improved Adaptive Moment Estimation (Adam) method—the Amsgrad algorithm can handle the optimization problem caused by parameter estimation. Thus, a hybrid approach that combines the particle filter and Amsgrad is proposed to estimate both parameters and states in fractional-order singular systems. This method leverages the strengths of the particle filter in handling nonlinear and high-dimensional problems, as well as the stability of the Amsgrad algorithm in optimizing parameters for dynamic systems. Then, the identification process is concluded to achieve a more accurate joint estimation. To validate the feasibility of the proposed hybrid algorithm, simulations involving three-order and four-order fractional-order singular systems are conducted. A comparative analysis with other algorithms demonstrates that the proposed method behaves better than the standard particle filter, Amsgrad and Gravitational search algorithm-Kalman filter algorithms. Full article

The chiral nonlinear Schrödinger equation (CNLSE) serves as a simplified model for characterizing edge states in the fractional quantum Hall effect. In this paper, we leverage the generalization and parameter inversion capabilities of physics-informed neural networks (PINNs) to investigate both forward and inverse problems of 1D and 2D CNLSEs. Specifically, a hybrid optimization strategy incorporating exponential learning rate decay is proposed to reconstruct data-driven solutions, including bright soliton for the 1D case and bright, dark soliton as well as periodic solutions for the 2D case. Moreover, we conduct a comprehensive discussion on varying parameter configurations derived from the equations and their corresponding solutions to evaluate the adaptability of the PINNs framework. The effects of residual points, network architectures, and weight settings are additionally examined. For the inverse problems, the coefficients of 1D and 2D CNLSEs are successfully identified using soliton solution data, and several factors that can impact the robustness of the proposed model, such as noise interference, time range, and observation moment are explored as well. Numerical experiments highlight the remarkable efficacy of PINNs in solution reconstruction and coefficient identification while revealing that observational noise exerts a more pronounced influence on accuracy compared to boundary perturbations. Our research offers new insights into simulating dynamics and discovering parameters of nonlinear chiral systems with deep learning. Full article

This article discusses the transformation of a continuous-time model of the fractional system into a discrete-time model of the fractional system. For the continuous-time model, the exact solution of the nonlinear equation with fractional derivatives (FDs) that has the form of the damped rotator type with power non-locality in time is obtained.This equation with two FDs and periodic kicks is solved in the general case for the arbitrary orders of FDs without any approximations. A three-stage method for solving a nonlinear equation with two FDs and deriving discrete maps with memory (DMMs) is proposed. The exact solutions of the nonlinear equation with two FDs are obtained for arbitrary values of the orders of these derivatives. In this article, the orders of two FDs are not related to each other, unlike in previous works. The exact solution of nonlinear equation with two FDs of different orders and periodic kicks are proposed. Using this exact solution, we derive DMMs that describe a kicked damped rotator with power-law non-localities in time. For the discrete-time model, these damped DMMs are described by the exact solution of nonlinear equations with FDs at discrete time points as the functions of all past discrete moments of time. An example of the application, the exact solution and DMMs are proposed for the economic growth model with two-parameter power-law memory and price kicks. It should be emphasized that the manuscript proposes exact analytical solutions to nonlinear equations with FDs, which are derived without any approximations. Therefore, it does not require any numerical proofs, justifications, or numerical validation. The proposed method gives exact analytical solutions, where approximations are not used at all. Full article

Advancements in Internet of Things (IoT) technologies have had a profound impact on interconnected devices, leading to exponentially growing networks of billions of intelligent devices. However, this growth has exposed Internet of Things (IoT) systems to cybersecurity vulnerabilities. These vulnerabilities are primarily caused by the inherent limitations of these devices, such as finite battery resources and the requirement for ubiquitous connectivity. The rapid evolution of deep learning (DL) technologies has led to their widespread use in critical application domains, thereby highlighting the need to integrate DL methodologies to improve IoT security systems beyond the basic secure communication protocols. This is essential for creating intelligent security frameworks that can effectively address the increasingly complex cybersecurity threats faced by IoT networks. This study proposes a hybrid methodology that combines fractional discrete Tchebichef moment analysis with deep learning for the prevention of IoT attacks. The effectiveness of our proposed technique for detecting IoT threats was evaluated using the UNSW-NB15 and Bot-IoT datasets, featuring illustrative cases of common IoT attack scenarios, such as DDoS, identity spoofing, network reconnaissance, and unauthorized data access. The empirical results validate the superior classification capabilities of the proposed methodology in IoT cybersecurity threat assessments compared with existing solutions. This study leveraged the synergistic integration of discrete Tchebichef moments and deep convolutional networks to facilitate comprehensive attack detection and prevention in IoT ecosystems. Full article

The stochastic correlation for Brownian motions is the integrand in the formula of their quadratic covariation. The estimation of this stochastic process becomes available from the temporally localized correlation of latent price driving Brownian motions in stochastic volatility models for asset prices. By analyzing this process for Apple and Microsoft stock prices traded minute-wise, we give statistical evidence for the roughness of its paths. Moment scaling indicates fractal behavior, and both fractal dimensions (approx. 1.95) and Hurst exponent estimates (around 0.05) point to rough paths. We model this rough stochastic correlation by a suitably transformed fractional Ornstein–Uhlenbeck process and simulate artificial stock prices, which allows computing tail dependence and the Herding Behavior Index (HIX) as functions in time. The computed HIX is hardly variable in time (e.g., standard deviation of 0.003–0.006); on the contrary, tail dependence fluctuates more heavily (e.g., standard deviation approx. 0.04). This results in a higher correlation risk, i.e., more frequent sudden coincident appearance of extreme prices than a steady HIX value indicates. Full article

The issue of multi-switch generalized projective anti-synchronization of fractional-order chaotic systems is investigated in this work. The model is constructed using Caputo–Fabrizio derivatives, which have been rarely addressed in previous research. In order to expand the symmetric and asymmetric synchronization modes of chaotic systems, we consider modeling chaotic systems under such fractional calculus definitions. Firstly, a new fractional-order differential inequality is proven, which facilitates the rapid confirmation of a suitable Lyapunov function. Secondly, an effective multi-switching controller is designed to confirm the convergence of the error system within a short moment to achieve synchronization asymptotically. Simultaneously, a multi-switching parameter adaptive principle is developed to appraise the uncertain parameters in the system. Finally, two simulation examples are presented to affirm the correctness and superiority of the introduced approach. It can be said that the symmetric properties of Caputo–Fabrizio fractional derivative are making outstanding contributions to the research on chaos synchronization. Full article

Natural gas stands out as a promising alternative fuel, and utilizing late intake valve close (LIVC) can further enhance its potential by improving internal combustion engine performance. The present study investigated the effect of LIVC on the performance of a Nissan Qashqai J10 four-cylinder internal combustion ignition engine (ICE) operating on gasoline (G) and natural gas (NG), with a focus on both energy and ecological aspects at stoichiometric points. Experimental tests were performed under the usual engine operating conditions, with engine speeds of 2000 and 3000 rpm and brake mean effective pressures (BMEPs) of 0.31, 0.55, and 0.79 MPa, while the intake valve closing moment was delayed at 24°, 31°, 38°, 45°, 52°, and 59° after bottom dead center (aBDC). The software AVL BOOST™ (version R2021.2) and its utility BURN were used to calculate the rate of heat release (ROHR), mass fraction burned (MFB), in-cylinder temperature, and the rate of temperature rise. The substitution of natural gas for gasoline substantially decreases CO₂ and NO_x emissions while enhancing the engine’s energy efficiency. Implementing a LIVC strategy can further boost brake thermal efficiency and reduce CO₂, though it negatively impacts CO, HC, and NO_x emissions. Optimal performance necessitates balancing efficiency improvements and CO₂ reduction against the control of other pollutants, potentially through combining LIVC with alternative engine control methodologies. Full article

To address the critical challenge of controlling tiltrotor aircraft during transition mode, this paper proposes a fractional-order model reference adaptive control (FO-MRAC) method based on the unique modeling of the tiltrotor aircraft. A nonlinear model capturing the dynamic characteristics of the tiltrotor aircraft during the transition mode is developed based on an accurate analysis of the forces and moments acting on key components. This model is subsequently linearized to obtain a stable flight envelope. Considering the complexity of transition, the FO-MRAC method is designed based on the shift of the equilibrium point for superior parameter tuning and disturbance rejection. Then, the stability of the closed-loop system is analyzed using the Lyapunov stability theory. Finally, an experimental platform is constructed to verify the validity of the aerodynamic modeling and the designed control method. Full article

A fragment of mortar from the pedestal ruin belonging to the central statue in Forum Vetus, Ulpia Traiana archaeological site, Romania, was investigated. The ruin is well-documented and unrestored, and radiocarbon dating was deemed suitable to determine its moment of construction. Preliminary analyses were used to establish the composition of the material and the sources of carbon-14, selecting the most reliable fraction for radiocarbon dating by the AMS method. Although sampling was carried out according to the recommendations, a younger apparent age was obtained than that expected. This is in fact a concrete-like mortar according to the analyses, and the phenomenon of delayed hardening of mortar in masonry was detected. The difference between the real and apparent ages quantifies this phenomenon. X-ray diffraction, scanning electron microscopy with energy-dispersive X-ray spectroscopy, Fourier-transform infrared spectroscopy, differential scanning calorimetry with thermogravimetric analysis, and gamma spectrometry were used. Pyrogenic calcium carbonate and carbonates from calcium silicate/calcium aluminate hydrates were the only forms present in mini-nodules/lumps. The reactivation of binder calcite or geogenic calcite, the other problems encountered when dating mortars, were not spotted. This case study highlights the limitations of the radiocarbon dating method, and we introduce gamma spectrometry as a technique for additional investigations into direct exposure to the environment or the origins of raw materials. Full article

Structures composed of classical dipoles in higher-dimensional space present a unique opportunity to venture beyond the conventional paradigm of few-body or cluster physics. In this work, we consider the six convex regular polychora that exist in an Euclidean four-dimensional space as a theoretical benchmark for hte investigation of dipolar systems in higher dimensions. The structures under consideration represent the four-dimensional counterparts of the well-known Platonic solids in three-dimensions. A dipole is placed in each vertex of the structure and is allowed to interact with the rest of the system via the usual dipole–dipole interaction generalized to the higher dimension. We use numerical tools to minimize the total interaction energy of the systems and observe that all six structures represent dipole clusters with a zero net dipole moment. The minimum energy is achieved for dipoles arranging themselves with orientations whose angles are commensurate or irrational fractions of the number $\pi$. Full article

This study investigates the computation of fractional and higher integer-order moments for a stochastic process governed by a one-dimensional, non-homogeneous linear stochastic differential equation (SDE) driven by fractional Brownian motion (fBm). Unlike conventional approaches relying on moment-generating functions or Fokker–Planck equations, which often yield intractable expressions, we derive explicit closed-form formulas for these moments. Our methodology leverages the Wick–Itô calculus (fractional Itô formula) and the properties of Hermite polynomials to express moments efficiently. Additionally, we establish a recurrence relation for moment computation and propose an alternative approach based on generalized binomial expansions. To validate our findings, Monte Carlo simulations are performed, demonstrating a high degree of accuracy between theoretical and empirical results. The proposed framework provides novel insights into stochastic processes with long-memory properties, with potential applications in statistical inference, mathematical finance, and physical modeling of anomalous diffusion. Full article

This paper presents a fast Crank–Nicolson L1 finite difference scheme for the two-dimensional time fractional mobile/immobile diffusion equation with weakly singular solution at the initial moment. First, the time fractional derivative is discretized using the Crank–Nicolson formula on uniform meshes, and a local truncation error estimate is provided. The spatial derivative is discretized using the central difference quotient on uniform meshes. Then, energy analysis methods are utilized to provide an optimal error estimates. On the other hand, the numerical scheme is optimized based on the sum-of-exponentials approximation, effectively reducing computation and memory requirements. Finally, numerical examples are simulated to verify the effectiveness of the algorithm. Full article

We introduce a novel class of coherent states, termed ${\mathcal{W}}_{\alpha ,\beta ,\nu }$-coherent states, constructed using a deformed boson algebra based on the generalised factorial ${\left[n\right]}_{\alpha ,\beta ,\nu }!$. This algebra extends conventional factorials, incorporating advanced special functions such as the Mittag-Leffler and Wright functions, enabling the exploration of a broader class of quantum states. The mathematical properties of these states, including their continuity, completeness, and quantum fluctuations, are analysed. A key aspect of this work is the resolution of the Stieltjes moment problem associated with these states, achieved through the inverse Mellin transformation method. The framework provides insights into the interplay between the classical and quantum regimes, with potential applications in quantum optics and fractional quantum mechanics. By extending the theoretical landscape of coherent states, this study opens avenues for further exploration in mathematical physics and quantum technologies. Full article

To analyze the impact of capacity uncertainty on the seismic collapse fragility of reinforced concrete (RC) frame structures, a fragility analysis framework based on seismic reliability methods is proposed. First, incremental dynamic analysis (IDA) curves are plotted by IDA under a group of natural seismic waves. Subsequently, collapse points are identified based on recommendations from relevant standards, yielding the probability distribution of the maximum inter-story drift ratios (MIDRs) at collapse points. Then, the distribution of the MIDRs under various intensity measures (IMs) of artificial seismic waves is calculated by using the fractional exponential moments-based maximum entropy method (FEM-MEM). Next, the structural failure probability is determined based on the combined performance index (CPI), and a seismic collapse fragility curve is plotted using the four-parameter shifted generalized lognormal distribution (SGLD) model. The results indicate that the collapse probability is lower considering the capacity uncertainty. Compared to deterministic MIDR limits of 1/25 and 1/50, the median values of the structure’s collapse resistance increased by 13.2% and 87.3%, respectively. Additionally, the failure probability obtained by considering the capacity uncertainty is lower than the results based on deterministic limits alone. These findings highlight the importance of considering capacity uncertainty in seismic risk assessments of RC frame structures. Full article