Accurate Solutions Research Articles

A novel multilevel finite element method for a generalized nonlinear Schrödinger equation

In this article, we focus on an efficient multilevel finite element method to solve the time-dependent nonlinear Schrödinger equation which is one of the most important equations of mathematical physics. For the time derivative, we adopt implicit schemes including the backward Euler method and the Crank–Nicolson method. Based on these stable implicit schemes, the proposed method requires solving a nonlinear elliptic problem at each time step. For these nonlinear elliptic equations, a multilevel mesh sequence is constructed. At each mesh level, we first derive a rough approximation by correcting the approximation of the previous mesh level in a special correction subspace. The correction subspace is composed of a coarse finite element space and an additional approximate solution derived from the previous mesh level. Next, we only need to solve a linearized elliptic equation by inserting the rough approximation into the nonlinear term. Then, we derive an accurate approximate solution by performing the aforementioned solving process on the multilevel mesh sequence until we reach the final mesh level. Owing to the special construct of the correction subspace, we derive a multilevel finite element method to solve the nonlinear Schrödinger equation for the first time, and meanwhile we also derive an optimal error estimate with linear computational complexity. Additionally, unlike the existing multilevel methods for nonlinear problems, that typically require bounded second-order derivatives of the nonlinear terms, the nonlinear term in our study requires only one-order derivatives. Numerical results are provided to support our theoretical analysis and demonstrate the efficiency of the presented method.

A novel hybrid BWO-BiLSTM-ATT framework for accurate offshore wind power prediction

Accurate wind power forecasting is pivotal in facilitating the efficient integration of renewable energy sources into existing power grids. This study proposes a novel hybrid framework, termed BWO-BiLSTM-ATT, which synergistically combines the strengths of a bidirectional long short-term memory (BiLSTM) network, a self-attention mechanism, and the Beluga Whale Optimization (BWO) algorithm. The BiLSTM architecture, with its unique ability to capture bidirectional temporal dependencies, effectively models the intricate dynamics present in wind power time series data. Integrating the self-attention mechanism further enhances the model's performance by discerning and emphasizing the most salient features within the input data. Furthermore, employing the BWO algorithm optimizes the model's hyperparameters, ensuring optimal configuration and enhancing its predictive accuracy and generalization capabilities. The proposed BWO-BiLSTM-ATT framework was rigorously evaluated using real-world data from an offshore wind farm in Yangjiang City, China. Comparative analyses were conducted against several baseline algorithms, including persistence, ARIMA, SVR, and vanilla LSTM models. The results demonstrate the superior performance of the BWO-BiLSTM-ATT framework, achieving higher R-squared and explained variance scores, coupled with lower error metrics such as MSE, RMSE, MedAE, and MAE. Statistical significance tests further corroborated the substantial performance improvements offered by the proposed framework. The combination of advanced deep learning techniques and metaheuristic optimization algorithms embedded in the BWO-BiLSTM-ATT framework provides a robust and accurate solution for wind power forecasting, enabling efficient management and seamless integration of renewable energy resources into existing power grids.

Optimization of Power Control for Autonomous Hybrid Electric Vehicles with Flexible Power Demand

Abstract Technology advancement for on-road vehicles has gained significant momentum in the past decades, particularly in the field of vehicle automation and powertrain electrification. The optimization of powertrain controls for autonomous vehicles typically involves a separated consideration of the vehicle's external dynamics and powertrain dynamics, with one key aspect often overlooked. This aspect, known as flexible power demand, recognizes that the powertrain control system does not necessarily have to precisely match the power requested by the vehicle motion controller at all times. Leveraging this feature can lead to control designs achieving improved fuel economy by adding an extra degree of freedom to the powertrain control while maintaining safety and drive comfort. The present research investigates the use of an Approximate Dynamic Programming (ADP) approach to develop a powertrain controller, which takes into account the flexibility in power demand within the ADP framework. The concept of reachable sets is incorporated into the ADP framework to ensure safety, improve ride comfort, and enhance the accuracy of the optimization solution. The formulation is based on an autonomous hybrid electric vehicle (HEV), while the methodology can also be applied to other types of vehicles. It is also found that necessary customization of the ADP algorithm is needed for this particular control problem to prevent convergence issues. Finally, a case study is presented to evaluate the effectiveness of flexible power demand, as addressed by the ADP method. The experiment demonstrates a 14.1% improvement in fuel economy compared to a scenario without flexible power demand.

Enhancing photoacoustic imaging for lung diagnostics and BCI communication: simulation of cavity structures artifact generation and evaluation of noise reduction techniques

Pandemics like COVID-19 have highlighted the potential of Photoacoustic imaging (PAI) for Brain-Computer Interface (BCI) communication and lung diagnostics. However, PAI struggles with the clear imaging of blood vessels in areas like the lungs and brain due to their cavity structures. This paper presents a simulation model to analyze the generation and propagation mechanism within phantom tissues of PAI artifacts, focusing on the evaluation of both Anisotropic diffusion filtering (ADF) and Non-local mean (NLM) filtering, which significantly reduce noise and eliminate artifacts and signify a pivotal point for selecting artifact-removal algorithms under varying conditions of light distribution. Experimental validation demonstrated the efficacy of our technique, elucidating the effect of light source uniformity on artifact-removal performance. The NLM filtering simulation and ADF experimental validation increased the peak signal-to-noise ratio by 11.33% and 18.1%, respectively. The proposed technique adds a promising dimension for BCI and is an accurate imaging solution for diagnosing lung diseases.

Novel interpretable and robust web-based AI platform for phishing email detection

Phishing emails continue to pose a significant threat, causing financial losses and security breaches. This study addresses limitations in existing research, such as reliance on proprietary datasets and lack of real-world application, by proposing a high-performance machine learning model for email classification. Utilizing a comprehensive and largest available public dataset, the model achieves a f1 score of 0.99 and is designed for deployment within relevant applications. Additionally, Explainable AI (XAI) is integrated to enhance user trust. This research offers a practical and highly accurate solution, contributing to the fight against phishing by empowering users with a real-time web-based application for phishing email detection.

Multi-version interactive assessment through the integration of GeoGebra with Moodle

AI systems are now capable of providing accurate solutions to questions presented in text format, causing a major problem in assessment integrity. To address this issue, interactive material can be integrated with the questions, preventing current AI systems from processing the requirements. This study proposes a novel approach that combines two important tools: GeoGebra and Moodle. GeoGebra is a widely used tool in schools and universities for creating dynamic and interactive material in the STEM field. On the other hand, Moodle is a popular learning management system with integrated tools capable of generating multiple versions of the same question to enhance academic integrity. We combine these two tools to automatically create unique interactive questions for each student in a computer-based assessment. Detailed implementation steps that do not require prior coding experience or the installation of additional plugins are presented, making the technique accessible to a wider range of instructors. The proposed approach was tested on a group of students and showed enhanced performance in animation-based questions compared to traditional question formats. Moreover, a survey exploring the students’ opinions on the proposed approach reported strong student endorsement of animated questions.

Enhancing topology optimization with colored body-fitted mesh using adaptive filter, dual re-meshing strategy, and OOP programming paradigm

This study introduces a novel topology optimization approach by employing power law-based material interpolation and adaptive filtering in the framework of the unstructured grids. As an extension of the established Solid Isotropic Material with Penalization (SIMP) method that utilizes the fixed structured mesh, the proposed Colored Body-Fitted Optimization (CBFO) method adopts the body-fitted grids to enhance efficiency, accuracy, and adaptability for diverse engineering applications. Notably, incorporating body-fitted meshes with intermediate density profiles enables improved flexibility in the numerical simulations and eliminates the need for re-meshing in each iteration. The dual re-meshing strategy drastically reduces computational costs, with only two re-meshing procedures required throughout the optimization process. This approach facilitates the generation of dense mesh regions around critical boundaries to augment solution accuracy while enabling sparse mesh configurations in the low-sensitivity regions, thereby boosting computational efficiency without compromising performance. The effectiveness, robustness, and efficiency of the CBFO method are validated through testing on multiple standard minimum compliance and compliant mechanism problems. The proposed optimization method can converge in dozens of iterations, obtain better objective function values, and save computational costs by up to 69 % compared to the previous method using the body-fitted mesh. Additionally, a concise MATLAB script implementing the proposed method using an Object-Oriented Programming (OOP) paradigm is provided in the appendix and the supplementary material, complete with annotations.

A Non-Standard Finite Difference Schemes for the Solution of Stiff Initial Value Problems

In this study, we introduce a novel non-standard finite difference (NSFD) scheme designed to address the challenges posed by stiff initial value problems. Stiffness in differential equations often leads to numerical instability and requires specialized methods for stable and accurate solutions. A novel set of numerical schemes for solving stiff ordinary differential equations caused by the decay of radioactive substances developed. This paper demonstrates the power of normalization in the discretization function. We employed non-local approximation and renormalization of the denominator function to create qualitatively stable schemes for a stiff ordinary differential equation. The schemes' stability properties were verified using numerical experiments. The schemes' performance is evaluated in comparison to other typical finite difference schemes

An improved solution of the probability density evolution equation for analyzing stochastic structural responses with adaptive time steps and initial conditions

The probability density evolution method (PDEM) offers an innovative approach for analyzing the stochastic vibration responses of structures. Given the complexities inherent in real-world challenges, the probability density evolution equations (PDEE) often require numerical methods for effective resolution. Consequently, improving computational efficiency and solution accuracy is of paramount importance. Currently, the impulse function discretization method (IFDM) faces several challenges, including limited differentiability in the spatial direction for initial value conditions. Furthermore, the probability density function produced by this method frequently deviates significantly from the true value, undermining the accuracy of the PDEE. Additionally, when addressing vibration responses characterized by rapid evolutionary velocities, the use of a globally fixed uniform time step size (FUTSS) results in a marked decrease in computational efficiency. To tackle these challenges, this study introduces an improved probability density evolution method (IPDEM), which integrates the Gaussian kernel density estimation discretization method (GKDEDM) and an adaptive non-uniform time step size (ANUTSS). The advantages of the IPDEM are illustrated through two numerical simulation examples. The findings indicate that the GKDEDM not only enhances spatial differentiability but also provides a more accurate estimation of the probability density at the initial moment, thereby improving the accuracy of the PDEE. Furthermore, the ANUTSS significantly reduces the number of iterations required while preserving precision, leading to a substantial optimization of computation time.

Second-Order Modified Nonstandard Explicit Euler and Explicit Runge–Kutta Methods for n-Dimensional Autonomous Differential Equations

Nonstandard finite-difference (NSFD) methods, pioneered by R. E. Mickens, offer accurate and efficient solutions to various differential equation models in science and engineering. NSFD methods avoid numerical instabilities for large time steps, while numerically preserving important properties of exact solutions. However, most NSFD methods are only first-order accurate. This paper introduces two new classes of explicit second-order modified NSFD methods for solving n-dimensional autonomous dynamical systems. These explicit methods extend previous work by incorporating novel denominator functions to ensure both elementary stability and second-order accuracy. This paper also provides a detailed mathematical analysis and validates the methods through numerical simulations on various biological systems.

Investigation of the dynamics of transverse oscillations of a vertical rod under gravity, friction, and thermal expansion

The paper considers the mathematical formulation of the problem of transverse oscillations of a vertical rod under gravity, friction and external pulse effect, leading to thermal expansion of the rod. The dynamics of the system under consideration corresponds to the behavior of a fuel element (FE) in a pulsed reactor and is related to the dynamic stability of the processes occurring in it.The FE dynamics is described by inhomogeneous linear differential equation of the fourth order with non-constant coefficients. Initial boundary conditions are not smooth since they correspond to the instant heating of the part of the FE surface exposed to neutron pulse radiation. The general solution of the homogeneous linear equation can be found using the concept of generalized functions expressed as a series expansion in terms of coordinate eigenfunctions dependent on p parameter. The p parameter is related with the FE mass and at some certain values results in bifurcation points of the boundary value problem for eigenfunctions. The partial solution can be found in several ways: using the Fourier transform, the method of Green's function, and in terms of series expansion by eigenfunctions.Eigenfunction expansion coefficients in an explicit form have been obtained and the numerical solution accuracy has been estimated for some important particular cases. The results of the obtained exact solutions appear to be very close to the results of the ANSYS numerical estimations. In the future the obtained exact solutions will be used as input data to simulate self-consistent system dynamics of more than 400 FE. Therefore, the advantage of the exact solution in practical use is that its calculation is much faster as compared with the finite element method done with ANSYS.

BO-GRNN: Machine Learning for Underwater Acoustic Source Localization

This article proposes a Bayesian optimization-based generalized regression neural network (BO-GRNN) framework for underwater source localization in shallow water environment. Specially, by leveraging kernel regression analysis, BO-GRNN fits a conditional probability density function using training and test samples to infer the source location of unseen test samples. Furthermore, BO-GRNN models the objective function as a Gaussian process (GP) based on observed samples, utilizing an acquisition function to sample around the best objective function values and high variance regions. By evaluating these samples and updating the GP iteratively, the optimal hyperparameters are identified, ultimately improving the training efficiency and localization performance of the GRNN. Simulation experiments and error analysis validate this framework’s effectiveness. Further validation using SWellEx-96 experimental data demonstrates the superior localization performance of this framework, even with limited data snapshots. This framework estimates the range and depth of underwater sources within a 10 km range, achieving mean absolute errors of less than 0.15 km for range and 0.2 m for depth. Compared to GRNN, this framework demonstrates a 5 times improvement in training efficiency. These advantages make BO-GRNN an efficient, accurate, and robust solution for underwater source localization.

Viscoelasticity of maxell fluid in a permeable porous channel

This study examines the flow dynamics and heat transfer characteristics of Maxwell fluid in a channel influenced by magnetohydrodynamics (MHD), Joule heating, thermal radiation, and boundary layer suction/blowing effects. The governing partial differential equations (PDEs) for momentum, energy, and concentration are transformed into ordinary differential equations (ODEs) using similarity transformations. The boundary value problem (BVP) is solved numerically using the bvp4c solver in MATLAB, yielding accurate solutions for velocity, temperature, and concentration profiles under various parameters. Key findings reveal that the porous parameter decreases the velocity profile but increases the temperature profile for both suction and blowing effects. Additionally, the MHD, Deborah, and Eckert numbers significantly influence the velocity and temperature profiles differently under these conditions. This study highlights the crucial role of integrating MHD, thermal, and boundary control effects to optimize performance and efficiency in engineering systems involving Maxwell fluids, with applications in polymer processing, biomedical engineering, electronics cooling, oil recovery, and chemical processing.

Characterizing magnetohydrodynamic effects on developed nanofluid flow in an obstructed vertical duct under constant pressure gradient

Abstract This research endeavors to conduct an examination of the thermal characteristics within the duct filled with the copper nanoparticles and water as base fluid. In exhaust systems, like car exhausts, chimneys, and kitchen hoods, duct flows are crucial. These systems safely discharge odors, smoke, and contaminants into the atmosphere after removing them from enclosed places. The study focuses on a laminar flow regime that is both hydrodynamically and thermally developed, with a specified constraints at any cross-sectional plane. To address this, we employ the finite volume method as it stands as a judicious choice, offering a balance between computational efficiency and solution accuracy. Notably, we have observed that the deceleration of flow induced by elevated Rayleigh numbers can be effectively regulated by the application of an appropriately calibrated external magnetic field. The prime parameters of the problem with ranges are: pressure gradient ( 1 ≤ p 0 ≤ 100 ) (1\le {p}_{0}\le 100) , Hartmann number ( 0 ≤ Ha ≤ 50 ) (0\le \text{Ha}\le 50) , Rayleigh number ( 1 , 000 ≤ Ra ≤ 40 , 000 ) (1,000\le \text{Ra}\le 40,000) , and magnetic parameter ( 0 ≤ M ≤ 50 ) (0\le M\le 50) . Furthermore, our analysis reveals that the Nusselt number exhibits a nearly linear correlation with the nanoparticle volume fraction parameter, a trend observed across a range of Rayleigh numbers and magnetic parameter values. We have noted that a mere 20% nanoparticle volume fraction can result in up to 62% rise in the Nusselt number while causing an almost 50% decrease in the factor f Re. This research framework serves as a robust foundation for understanding the intricate interplay between magnetic influences and thermal-hydraulic behavior within the delineated system.

Analytical solutions for static longitudinal displacements of suspension bridges under a moving vertical concentrated load

The longitudinal displacement at the girder ends of long-span suspension bridges is a crucial issue that affects structural safety, durability, as well as traffic safety and ride comfort. The primary cause of these reciprocating displacements is the action of moving live loads, which induces quasi-static displacements in the longitudinal direction. This paper theoretically investigates the static longitudinal displacements of suspension bridges under a moving vertical concentrated load and elucidates the underlying deformation mechanisms. Considering geometrical nonlinearity, analytical equations are established for the longitudinal deformations of the main cable and stiffening girder under a vertical concentrated load. By analyzing the geometrical deformation conditions of the suspenders, the relationship between the longitudinal displacements of the stiffening girder and the main cable is derived. Due to the coupling between longitudinal and vertical displacements, the vertical displacement of the stiffening girder must be solved to obtain its longitudinal displacement. Based on deflection theory and considering the bending stiffness of the stiffening girder, this paper derives the vertical deformations of the stiffening girder for single-span suspension bridges and those with short continuous side spans under a vertical concentrated load, which is then used to calculate the longitudinal displacement. Finite element models are used to verify the accuracy of the analytical solutions. Differences between the proposed solutions and previous solutions neglecting the bending stiffness of the stiffening girder are compared and analyzed. The effects of neglecting the longitudinal displacement at the bridge tower top and the presence of short continuous side spans on the longitudinal displacement of stiffening girder are also investigated. The results demonstrate the high accuracy of the proposed analytical solution for the longitudinal displacement at the girder ends of suspension bridges. Importantly, the bending stiffness of the stiffening girder cannot be neglected in calculating the longitudinal displacement. The longitudinal displacement at the girder ends is primarily caused by the longitudinal deformation of the main cable, while the contribution from the flexural deformation of the girder is negligible. Furthermore, the longitudinal displacement at the bridge tower top has a negligible effect on the longitudinal displacement of the girder, and the presence of short continuous side spans is beneficial for reducing the longitudinal displacement of the stiffening girder. The proposed analytical solution method provides a rapid approach for calculating the static longitudinal displacement at the girder ends of suspension bridges under passing live loads and reveals the deformation mechanism. This is beneficial for preliminary design and structural optimization, as it avoids the complex finite element modeling and solution process.

Horseshoes and spiral waves: capturing the 3D flow induced by a low-mass planet analytically

Abstract The key difficulty faced by 2D models for planet-disc interaction is in appropriately accounting for the impact of the disc’s vertical structure on the dynamics. 3D effects are often mimicked via softening of the planet’s potential; however, the planet-induced flow and torques often depend strongly on the choice of softening length. We show that for a linear adiabatic flow perturbing a vertically isothermal disc, there is a particular vertical average of the 3D equations of motion which exactly reproduces 2D fluid equations for arbitrary adiabatic index. There is a strong connection here with the Lubow-Pringle 2D mode of the disc. Correspondingly, we find a simple, general prescription for the consistent treatment of planetary potentials embedded within ‘2D’ discs. The flow induced by a low-mass planet involves large-scale excited spiral density waves which transport angular momentum radially away from the planet, and ‘horseshoe streamlines’ within the co-orbital region. We derive simple linear equations governing the flow which locally capture both effects faithfully simultaneously. We present an accurate co-orbital flow solution allowing for inexpensive future study of corotation torques, and predict the vertical structure of the co-orbital flow and horseshoe region width for different values of adiabatic index, as well as the vertical dependence of the initial shock location. We find strong agreement with the flow computed in 3D numerical simulations, and with 3D one-sided Lindblad torque estimates, which are a factor of 2 to 3 times lower than values from previous 2D simulations.

Nonlinear interactions of an n-layer X-shape low-frequency vibration isolator equipped with a nonlinear vibration absorber at 1:1 internal resonance: analytical and numerical investigations

This work addresses, for the first time, the nonlinear dynamics of an n− layers bioinspired X− shape low-frequency vibration isolation system equipped with a nonlinear vibration absorber. The comprehensive mathematical model governing the two-degree-of-freedom system has been derived using Lagrangian mechanics. The method of averaging was applied to obtain an accurate analytical solution for the derived mathematical model. Utilizing the established analytical solution, the dynamical characteristics of the X− shape vibration isolator have been explored both as a single-degree-of-freedom system and after coupling with the proposed vibration absorber, as a two-degree-of-freedom system. The influence of different system parameters, including the number of structure layers, the initial inclination angle, the X− shape rod length, and the absorber stiffness and damping coefficients, on the vibration isolation performance has been investigated. The main findings indicate that while the single-degree-of-freedom X− shape low-frequency vibration isolator performs well in eliminating low-frequency vibration excitations, it exhibits strong vibration levels when the base excitation frequency exceeds the structure's natural frequency. Additionally, it was found that coupling a highly damped linear vibration absorber to the X− shape isolator not only eliminates the strong oscillations of the structure when subjected to high-frequency base excitations but also enhances the structure's low-frequency vibration isolation performance. Moreover, it was reported that integrating a nonlinear vibration absorber with either soft or hard spring characteristics instead of a linear one may degrade the vibration isolation performance of the structure rather than enhance it. Finally, numerical validation of all the analytical results was performed, which showed excellent correspondence with the analytical findings.

Estimation of corn crop damage caused by wildlife in UAV images

Abstract Purpose This paper proposes a low-cost and low-effort solution for determining the area of corn crops damaged by the wildlife facility utilising field images collected by an unmanned aerial vehicle (UAV). The proposed solution allows for the determination of the percentage of the damaged crops and their location. Methods The method utilises image segmentation models based on deep convolutional neural networks (e.g., UNet family) and transformers (SegFormer) trained on over 300 hectares of diverse corn fields in western Poland. A range of neural network architectures was tested to select the most accurate final solution. Results The tests show that despite using only easily accessible RGB data available from inexpensive, consumer-grade UAVs, the method achieves sufficient accuracy to be applied in practical solutions for agriculture-related tasks, as the IoU (Intersection over Union) metric for segmentation of healthy and damaged crop reaches 0.88. Conclusion The proposed method allows for easy calculation of the total percentage and visualisation of the corn crop damages. The processing code and trained model are shared publicly.

Simulating the Growth of TAF15 Inclusions in Neuron Soma.

To the best of the author's knowledge, this paper presents the first attempt to develop a mathematical model of the formation and growth of inclusions containing misfolded TATA-box binding protein associated factor 15 (TAF15). It has recently been shown that TAF15 inclusions are involved in approximately 10% of cases of frontotemporal lobar degeneration (FTLD). FTLD is the second most common neurodegenerative disease after Alzheimer's disease (AD). It is characterized by a progressive loss of personality, behavioral changes, and a decline in language skills due to the degeneration of the frontal and anterior temporal lobes. The model simulates TAF15 monomer production, nucleation and autocatalytic growth of free TAF15 aggregates, and their deposition into TAF15 inclusions. The accuracy of the numerical solution of the model equations is validated by comparing it with analytical solutions available for limiting cases. Physiologically relevant parameter values were used to predict TAF15 inclusion growth. It is shown that the growth of TAF15 inclusions is influenced by two opposing mechanisms: the rate at which free TAF15 aggregates are deposited into inclusions and the rate of autocatalytic production of free TAF15 aggregates from monomers. A low deposition rate slows inclusion growth, while a high deposition rate hinders the autocatalytic production of new aggregates, thus also slowing inclusion growth. Consequently, the rate of inclusion growth is maximized at an intermediate deposition rate of free TAF15 aggregates into TAF15 inclusions.

Innovative Solutions to the Fractional Diffusion Equation Using the Elzaki Transform

This study explores the application of advanced mathematical techniques to solve fractional differential equations, focusing particularly on the fractional diffusion equation. The fractional diffusion equation, used to simulate a range of physical and engineering phenomena, poses considerable difficulties when applied to fractional orders. Thus, by utilizing the mighty powers of fractional calculus, we employ the variational iteration method (VIM) with the Elzaki transform to produce highly accurate approximations for these specific differential equations. The VIM provides an iterative framework for refining solutions progressively, while the Elzaki transform simplifies the complex integral transforms involved. By integrating these methodologies, we achieve accurate and efficient solutions to the fractional diffusion equation. Our findings demonstrate the robustness and effectiveness of combining the VIM and the Elzaki transform in handling fractional differential equations, offering explicit functional expressions that are beneficial for theoretical analysis and practical applications. This research contributes to the expanding field of fractional calculus, providing valuable insights and useful tools for solving complex, nonlinear fractional differential equations across various scientific and engineering disciplines.