Second-order Differentiation Research Articles

Prediction of soil heavy metal content around mine tailings using multiple methods combined with transformed hyperspectral reflectance data

The extensive accumulation of tailings can potentially cause heavy metal contamination in the surrounding farmland soil. Accurately predicting the spatial distribution of heavy metals in farmland soil is crucial for assessing the potential environmental hazards of tailings.This study focuses on the spatial distribution and the quantitative prediction of heavy metals (chromium (Cr), vanadium (V), and copper (Cu)) in soils surrounding mine tailings using advanced spectral data analysis and multiple prediction models. The original hyperspectral reflectance data were processed using first-order differential (FD), second-order differential (SD), reciprocal logarithmic (LR), and continuum removal (CR) transformations to highlight the positions of characteristic bands. Multiple linear regression (MLR), stepwise linear regression (SLR), partial least squares regression (PLSR), random forest (RF), and back propagation artificial neural network (BP-ANN) models were used to establish inversion models for Cr, V, and Cu based on bands with high correlation coefficients. The performance of the inversion models was evaluated using the coefficient of determination (R2), root mean square error (RMSE), mean absolute error (MAE), and residual predictive deviation (RPD). The results indicate that the raw hyperspectral data from the measured soil exhibit a weak response to heavy metal content in the study area. However, applying FD, SD, and CR transformations significantly enhances the sensitivity of soil spectral data to heavy metal concentrations, facilitating subsequent modeling. Among these, the SD transformation is particularly beneficial for modeling the Cr and Cu elements in the soil. For the V element, the FD transformation yields data that are more suitable for modeling. Regarding the inversion models based on the measured spectral data, the BP-ANN model exhibited the best predictive performance. Specifically, when combined with SD spectral data, the BP-ANN achieved the highest predictive accuracy for Cu content (R² = 0.85, RPD = 2.12). The RF model demonstrated the next best performance, with its optimal inversion model also utilizing SD spectral data for predicting Cu content (R² = 0.76, RPD = 1.90). On the other hand, the MLR model exhibited the poorest performance and is unsuitable for predicting heavy metal content in the region using the measured spectral data. This study highlights the potential of spectral data in environmental monitoring and provides a technical reference for the inversion assessment and regulation of heavy metals in farmlands surrounding tailing sites.

Implicit EXP-RBF techniques for modeling unsaturated flow through soils with water uptake by plant roots

Modeling unsaturated flow through soils with water uptake by plan root has many applications in agriculture and water resources management. In this study, our aim is to develop efficient numerical techniques for solving the Richards equation with a sink term due to plant root water uptake. The Feddes model is used for water absorption by plant roots, and the van-Genuchten model is employed for capillary pressure. We introduce a numerical approach that combines the localized exponential radial basis function (EXP-RBF) method for space and the second-order backward differentiation formula (BDF2) for temporal discretization. The localized RBF methods eliminate the need for mesh generation and avoid ill-conditioning problems. This approach yields a sparse matrix for the global system, optimizing memory usage and computational time. The proposed implicit EXP-RBF techniques have advantages in terms of accuracy and computational efficiency thanks to the use of BDF2 and the localized RBF method. Modified Picards iteration method for the mixed form of the Richards equation is employed to linearize the system. Various numerical experiments are conducted to validate the proposed numerical model of infiltration with plant root water absorption. The obtained results conclusively demonstrate the effectiveness of the proposed numerical model in accurately predicting soil moisture dynamics under water uptake by plant roots. The proposed numerical techniques can be incorporated in the numerical models where unsaturated flows and water uptake by plant roots are involved such as in hydrology, agriculture, and water management.

A linear second order unconditionally maximum bound principle-preserving scheme for the Allen-Cahn equation with general mobility

In this work, we investigate a linear second-order numerical method for the Allen-Cahn equation with general mobility. The proposed scheme is a combination of the two-step first- and second-order backward differentiation formulas for time approximation and the central finite difference for spatial discretization, two additional stabilizing terms are also included. The discrete maximum bound principle of the numerical scheme is rigorously proved under mild constraints on the adjacent time-step ratio and the two stabilization parameters. Furthermore, the error estimates in H1-norm for the case of constant mobility and L∞-norm for the general mobility case, as well as the energy stability for both cases are obtained. Finally, we present extensive numerical experiments to validate the theoretical results, and develop an adaptive time-stepping strategy to demonstrate the performance of the proposed method.

Hybrid-tracked finite-time path following control of underactuated underwater vehicles with 6-DOF

This paper introduces a novel hybrid-tracked finite-time path following control approach for controlling the surge speed and attitude angles of underactuated underwater vehicles with six degrees of freedom (6-DOF). This method integrates 6-DOF line-of-sight (LOS) guidance with nonzero roll dynamics. In kinematics layer, the Rodrigues rotation matrix ensures a unique solution and a slight desired roll angle in LOS guidance calculations. A second-order finite-time differentiator bridges the kinematics and dynamics layers, processing LOS guidance commands and ensuring differential errors converge to zero in finite time. In dynamics layer, a hybrid thrust calculation strategy achieves finite-time convergence of attitude angles in the inertial frame and surge speed in body-fixed frame. Extensive numerical simulations comparing various initial conditions and control strategies demonstrate the superior performance and efficiency of the proposed hybrid-tracked finite-time controller (HTFTTC). The HTFTTC significantly outperforms existing hybrid-tracked backstepping and direct finite-time controllers, reducing the root mean square error of attitude angles or speed by at least 67.2% in steady-state.

Конечно-элементное моделирование нелинейных задач упругости в абсолютных узловых координатах на неструктурированных шестигранных сетках

We consider a finite element approach to solving the problem of elasticity theory in terms of absolute nodal coordinate formulation (ANCF), in which large body displacements are described in a global reference frame without using any local coordinate system. The main feature of the method is the absence of gyroscopic effects and, as a result, constancy of the mass matrix and the vector of the generalized force of gravity. In contrast to the traditional ANCF approach, the sets of nodal degrees of freedom of the finite element are formed only on the basis of absolute coordinates of nodes, which allows us to solve the problem using, in particular, unstructured hexahedral meshes. To construct the stiffness matrix, we apply a second-order automatic differentiation algorithm, which ensures its symmetrical form (Hessian matrix) and is analytically accurate in calculating the derivative. This approach also makes it possible to carry out calculations for models of hyperelastic materials without the corresponding Piola-Kirchhoff tensor. It has been shown that within the framework of discretization of the equation of motion, along with the well-known Newmark numerical integration scheme, it is possible to use the HTT-α scheme, which is unconditionally stable, second-order accurate and dissipative for high frequencies. We present several examples of solving static and dynamic elasticity problems for compressible and incompressible models of hyperelastic materials, in which the functions of internal energy density of the body are specified in terms of the deformation gradient.

Spin–orbit optical broadband achromatic spatial differentiation imaging

Spatial optical analog differentiation allows ultrahigh-speed and low-power-consumption of image processing, as well as label-free imaging of transparent biological objects. Optical analog differentiation with broadband and incoherent sources is appealing for its multi-channels and multi-task information processing, as well as the high-quality differentiation imaging. Currently, broadband and incoherent optical differentiation is still challenging. Here, a compact and broadband achromatic optical spatial differentiator is demonstrated based on the intrinsic spin–orbit coupling in a natural thin crystal. By inserting a uniaxial crystal just before the camera of a conventional microscope, the spin to orbit conversion will embed an optical vortex to the image field and make a second-order topological spatial differentiation to the field, thus an isotropic differential image will be captured by the camera. The wavelength-independent property of the intrinsic spin–orbit coupling effect allows us to achieve broadband analog computing and achromatic spatial differentiation imaging. With this differentiation imaging method, both amplitude and pure phase objects are detected with high contrast. Transparent living cells and biological tissues are imaged with their edge contours and intracellular details protruded in the edge detection mode and edge enhancement mode, respectively. These findings pave the way for optical analog computing with broadband incoherent light sources and concurrently drive the advancement of high-performance and cost-effective phase contrast imaging.

Disturbance-Observer-Based Adaptive Prescribed Performance Formation Tracking Control for Multiple Underactuated Surface Vehicles

This study proposes a new disturbance-observer-based adaptive distributed formation control scheme for multiple underactuated surface vehicles (USVs) subject to unknown synthesized disturbances under prescribed performance constraints. A modified sliding mode differentiator (MSMD) is applied as a nonlinear disturbance observer to estimate unknown synthesized disturbances, which contain unknown environmental disturbances and system modelling uncertainties, thus enhancing the robustness of the system. Based on this, we impose the time-varying performance constraints on the position tracking error between the neighboring USVs. A novel differentiable error transformation equation is embedded in the prescribed performance control, and an adaptive prescribed performance controller is constructed by employing the backstepping method to ensure that the position tracking error remains within the prescribed transient and steady performance, and each USV realizes collision-free formation motion. Furthermore, a novel second-order nonlinear differentiator is introduced to extract the derivative information of the virtual control law. Finally, the numerical simulation results verify the effectiveness of the proposed control scheme.

Inverse design of ultrawideband one-dimensional metamaterial photonic filters using Riccati equation constrained gradient descent

One-dimensional photonic wave devices exhibit a pivotal role in wave engineering. Despite their relative simplicity, designing 1D wave devices that implement complex functionalities over a broad frequency range is challenging and requires careful sculpting and multiple optimizations. This paper theoretically and experimentally demonstrates a new inverse design paradigm to achieve a desired broadband frequency response efficiently. Specifically, we calculate the required dielectric profile along the device using constrained gradient descent optimization to minimize the L2 norm between the desired and actual responses. In each optimization step, we avoid the need to solve the complete set of Maxwell equations by using Riccati’s equation or its discrete ancestor as the optimization constraint for calculating the local reflection coefficient. Using this approach, we design several unorthodox filters, such as dual-band narrowband bandpass filters located within a wideband bandstop and ultrawideband first and second-order differentiators. The technique produces excellent results for ultrawideband frequency ranges, with very low computational complexity and, remarkably, with a single trivial guess for the optimization starting point. We experimentally implemented the two differentiator designs in radio frequencies using electronic circuit elements that comprise a metamaterial transmission line structure.

Distributed synchronization method of multi-motor driving system’s accelerated backstepping tracking control

This paper proposes a distributed synchronization control method and an accelerated backstepping tracking control scheme for the multi-motor driving system (MMDS). In the first step, we create a dynamic model of the MMDS with complex nonlinear dynamics, encompassing elements such as the dead zone, frictions, and disturbances. Next, in order to tackle the challenge of load tracking, we fuse a speed function, a cosine barrier function, a second-order tracking differentiator (TD), and a disturbance compensator into the backstepping approach. Lastly, to address potential issues related to diverse torque inputs, which could result in the overload occurrences, we put forward a novel distributed synchronization control scheme. This scheme aims to achieve torque synchronization for the MMDS while simultaneously ensuring superior load tracking performance. In the distributed synchronization control, a communication network is built to achieve the local coupling and improve the synchronization efficiency, and a corresponding mean deviation coupling synchronization control scheme is designed. Lyapunov theory is utilized to demonstrate the stability of the introduced control scheme. The simulation experimental results for the MMDS show the effectiveness of the proposed scheme.

Estimating the chromium concentration of farmland soils in an arid zone from hyperspectral reflectance by using partial least squares regression methods

Heavy metal contamination of farmland soils is a severe concern worldwide.Chromium (Cr), a major environmental carcinogen, is a crucial indicator of farmland soil pollution. Compared with conventional chemical analysis, the hyperspectral analysis is superior in accurately and rapidly surveilling heavy metals in soils. However, various heavy metals, and types or geochemical traits of soils often affect the accuracy of hyperspectral estimation models, which makes guaranteeing the concentration estimation precision for a designated heavy metal in different regions considerably challenging. We here propose an optimal hyperspectral model for determining the Cr concentration of farmland soils first time in the Yanqi Basin of China's northwest arid zone. In total, 171 soil samples were gathered, and their Cr concentrations and relevant spectral reflectance data were determined. The best spectral transformation for measuring the Cr concentration was determined by using 12 transformation types of soil spectral curves, and characteristic wavebands were obtained. Then, by using random forest (RF), support vector machine (SVM), and partial least squares regression (PLSR) models, we constructed hyperspectral estimation models for Cr concentration in farmland soils. The estimated results of these models were compared to identify the best estimation model. Results show that the spectral transformations allow an improvement in the association between the soil Cr concentration and spectra. The accuracy of PLSR was significantly higher than that of SVM and RF. Finally, the second-order differentiation (SD) combined with the PLSR model (R2 = 0.903, RMSE = 40.363, MAE = 30.631) was identified as the optimal model for estimating the Cr concentration of farmland soils in the investigated area. The study findings offer a technical reference for the hyperspectral assessment of the Cr concentration in arid zone farmland soils.

Collision-free automatic berthing of maritime autonomous surface ships via safety-certified active disturbance rejection control

This paper addresses the automatic berthing of a maritime autonomous surface ship operating in a confined water environment subject to static obstacles, dynamic obstacles, thruster constraints, and space constraints due to shorelines. A safety-certified active disturbance rejection control (ADRC) method is proposed for achieving the automatic berthing task of an MASS in the presence of model uncertainties and ocean disturbances. An extended state observer (ESO) based on a second-order robust exact differentiator (RED) is employed to estimate an extended state vector consisting of internal model uncertainties and external ocean disturbances. With the aid of the RED-based ESO, a nominal ADRC law is designed to achieve the position and heading stabilization. To avoid collisions with static obstacles, dynamic obstacles, and shorelines, input-to-state safe high-order control barrier functions are used to guarantee safety. Optimized control signals are obtained based on a constrained quadratic programming (QP) problem within safety constraints. In order to translate the control signals into the individual thruster command, a constrained QP problem is further used to search for optimized commands in real time. It is proven that the closed-loop automatic berthing system is input-to-state stable. By using the proposed method, the MASS is able to reach the desired position and heading with collision avoidance. Simulation results verify the effectiveness of the proposed safety-certified ADRC method for automatic berthing.

Unconditional error analysis of linearized BDF2 mixed virtual element method for semilinear parabolic problems on polygonal meshes

In this paper, we construct, analyze, and numerically validate a class of H(div)-mixed virtual element method for the semilinear parabolic problem in mixed form, in which the parabolic problem is reformulated in terms of the velocity and the pressure of the time-dependent Darcy flow. The Newton linearized method for the nonlinear term is designed to cooperate with the second-order backward differentiation formula of the temporal discretization scheme. This allows each time step to only require the solution of a small and well-structured linear system rather than the solution of a nonlinear system. The linearization improves computational efficiency without decreasing convergence rates. Moreover, the “Fortin” operator and its approximation properties are applied to derive an optimal error estimates O(hk+1+τ2) for the virtual element solution of the velocity and the pressure. Finally, its remarkable performance is illustrated by several numerical examples that also validate the theoretical rates of convergence.

An SMC-Based Accurate and Robust Load Speed Control Method for Elastic Servo System

In this paper, a modified sliding mode speed controller is designed in order to improve the load speed control performance of an elastic motor drive system with internal and external disturbances. Firstly, based on the load speed deviation and its first-order and second-order differentials, a two-dimensional sliding surface function is presented. The parameters of the sliding surface are determined by a novel particle swarm optimization (PSO) method. The method has improved inertia weights and evaluation functions, which can improve the accuracy and efficiency of the parameter search and enhance the immunity and robustness of the system. Secondly, on the basis of the integral of time-weighted absolute error (ITAE) criterion, the parameters of the exponential approach law are determined, which improves the tracking accuracy. Finally, the designed sliding mode control (SMC) with high tracking accuracy, fast dynamics, and strong robustness is verified by both simulations and experiments.

On the Solvability of a Periodic Problem for a System of Second-Order Nonlinear Ordinary Differential Equations

On the Solvability of a Periodic Problem for a System of Second-Order Nonlinear Ordinary Differential Equations

An efficient two-grid high-order compact difference scheme with variable-step BDF2 method for the semilinear parabolic equation

Due to the lack of corresponding analysis on appropriate mapping operator between two grids, high-order two-grid difference algorithms are rarely studied. In this paper, we firstly discuss the boundedness of a local bi-cubic Lagrange interpolation operator. And then, taking the semilinear parabolic equation as an example, we first construct a variable-step high-order nonlinear difference algorithm using compact difference technique in space and the second-order backward differentiation formula with variable temporal stepsize in time. With the help of discrete orthogonal convolution kernels, temporal-spatial error splitting idea and a cut-off numerical technique, the unique solvability, maximum-norm stability and corresponding error estimate of the high-order nonlinear difference scheme are established under assumption that the temporal stepsize ratio satisfies rk := τk/τk−1 < 4.8645. Then, an efficient two-grid high-order difference algorithm is developed by combining a small-scale variable-step high-order nonlinear difference algorithm on the coarse grid and a large-scale variable-step high-order linearized difference algorithm on the fine grid, in which the constructed piecewise bi-cubic Lagrange interpolation mapping operator is adopted to project the coarse-grid solution to the fine grid. Under the same temporal stepsize ratio restriction rk < 4.8645 on the variable temporal stepsize, unconditional and optimal fourth-order in space and second-order in time maximum-norm error estimates of the two-grid difference scheme is established. Finally, several numerical experiments are carried out to demonstrate the effectiveness and efficiency of the proposed scheme.

The Second-Order Numerical Approximation for a Modified Ericksen–Leslie Model

In this study, two numerical schemes with second-order accuracy in time for a modified Ericksen–Leslie model are constructed. The highlight is based on a novel convex splitting method for dealing with the nonlinear potentials, which is integrated with the second-order backward differentiation formula (BDF2) and leap frog method for temporal discretization and the finite element method for spatial discretization. The unconditional energy stability of both schemes is further demonstrated. Finally, several numerical examples are presented to demonstrate the efficiency and accuracy of the proposed schemes.

Magnetotactic bacteria and Fe3O4–water in a wavy walled cavity

PurposeThe purpose of this study is to investigate the interaction between magnetotactic bacteria and Fe3O4–water nanofluid (NF) in a wavy enclosure in the presence of 2D natural convection flow.Design/methodology/approachUniform magnetic field (MF), Brownian and thermophoresis effects are also contemplated. The dimensionless, time-dependent equations are governed by stream function, vorticity, energy, nanoparticle concentration and number of bacteria. Radial basis function-based finite difference method for the space derivatives and the second-order backward differentiation formula for the time derivatives are performed. Numerical outputs in view of isolines as well as average Nusselt number, average Sherwood number and flux density of microorganisms are presented.FindingsConvective mass transfer rises if any of Lewis number, Peclet number, Rayleigh number, bioconvection Rayleigh number and Brownian motion parameter increases, and the flux density of microorganisms is an increasing function of Rayleigh number, bioconvection Rayleigh number, Peclet number, Brownian and thermophoresis parameters. The rise in buoyancy ratio parameter between 0.1 and 1 and the rise in Hartmann number between 0 and 50 reduce all outputs average Nusselt, average Sherwood numbers and flux density of microorganisms.Research limitations/implicationsThis study implies the importance of the presence of magnetotactic bacteria and magnetite nanoparticles inside a host fluid in view of heat transfer and fluid flow. The limitation is to check the efficiency on numerical aspect. Experimental observations would be more effective.Practical implicationsIn practical point of view, in a heat transfer and fluid flow system involving magnetite nanoparticles, the inclusion of magnetotactic bacteria and MF effect provide control over fluid flow and heat transfer.Social implicationsThis is a scientific study. However, this idea may be extended to sustainable energy or biofuel studies, too. This means that a better world may create better social environment between people.Originality/valueThe presence of magnetotactic bacteria inside a Fe3O4–water NF under the effect of a MF is a good controller on fluid flow and heat transfer. Since the magnetotactic bacteria is fed by nanoparticles Fe3O4 which has strong magnetic property, varying nanoparticle concentration and Brownian and thermophoresis effects are first considered.

Differentiation of Optical Signals Using an Integrated Metal–Dielectric–Metal Structure

The authors consider the optical properties of integrated metal–dielectric–metal structures embedded in a dielectric slab waveguide and their application to the problems of first- and second-order optical differentiation of the profile of the incident optical beam in reflection. Numerical simulation results show the investigated structures allow the spatial differentiation operation to be performed with high quality. The presented results could find application in new systems for optical information processing and analog optical computing based on integrated nanophotonic structures.

Power law in Sandwiched Volterra Volatility model

The paper presents an analytical proof demonstrating that the Sandwiched Volterra Volatility (SVV) model is able to reproduce the power-law behavior of the at-the-money implied volatility skew, provided the correct choice of the Volterra kernel. To obtain this result, the second-order Malliavin differentiability of the volatility process is assessed and the conditions that lead to explosive behavior in the Malliavin derivative are investigated. As a supplementary result, a general Malliavin product rule is proved.

Dynamic Analysis and Neural-Adaptive Prescribed-Time Control of the FO Memristive Magnetic-Field Electromechanical Transducer.

This article is concerned with dynamic analysis and neural-adaptive prescribed-time control of the magnetic-field electromechanical transducer incorporating a memristor. First, a fractional-order (FO) mathematical model is developed, which comprehensively characterizes fractional properties of various dielectrics and establishes the relationship between magnetic flux and electric charge. The dynamical analysis explores internal evolution and complexity performance concerning a single factor or double factors among the FO, system parameter, and memristor configuration by the Bifurcation diagram, sample entropy, and C0 complexity from multiple perspectives. Subsequently, a neural-adaptive prescribed-time control scheme is proposed to transform detrimental chaotic oscillations into orderly motions, achieve the pregiven tracking precision and accommodating both actuator fault and system uncertainty. The controller design consists of three key steps: 1) a deferred constraint function is imposed on the tracking error starting from anywhere to get assignable tracking precision within a specified time, ensuring collision avoidance; 2) a type-2 fuzzy wavelet neural network (FWNN) is utilized effectively to handle parameter perturbations and system uncertainties; and 3) a second-order FO tracking differentiator (TD) is utilized to address the "explosion of complexity" of traditional backstepping under actuator fault model. It is shown that the proposed scheme is able to ensure the boundness of all signals of the closed-loop system. Finally, extensive simulation experiments are conducted to validate the effectiveness and robustness of the rendered scheme.