Second-order Partial Derivatives Research Articles

A G-Modified Helmholtz Equation with New Expansions for the Earth’s Disturbing Gravitational Potential, Its Functionals and the Study of Isogravitational Surfaces

The G-modified Helmholtz equation is a partial differential equation that enables us to express gravity intensity g as a series of spherical harmonics having radial distance r in irrational powers. The Laplace equation in three-dimensional space (in Cartesian coordinates, is the sum of the second-order partial derivatives of the unknown quantity equal to zero) is used to express the Earth’s gravity potential (disturbing and normal potential) in order to represent other useful quantities—which are also known as functionals of the disturbing potential—such as gravity disturbance, gravity anomaly, and geoid undulation as a series of spherical harmonics. We demonstrate that by using the G-modified Helmholtz equation, not only gravity intensity but also disturbing potential and its functionals can be expressed as a series of spherical harmonics. Having gravity intensity represented as a series of spherical harmonics allows us to create new Global Gravity Models. Furthermore, a more detailed examination of the Earth’s isogravitational surfaces is conducted. Finally, we tabulate our results, which makes it clear that new Global Gravity Models for gravity intensity g will be very useful for many geophysical and geodetic applications.

Port berth allocation and microgrid cluster joint optimization scheduling based on master-slave game

The port microgrid cluster, which integrates berth allocation and energy scheduling for joint optimization, is a highly interconnected system of logistics and energy closely coupled microgrids. It can reduce port load fluctuations and ensure the stable and reliable operation of the port energy system. A port berth allocation and microgrid cluster joint optimization scheduling method based on master-slave game is proposed to address issues such as large fluctuations in current port loads, vulnerability of port power supply system to the impact of vessel loading and unloading operations, and conflicts of interest among various participants in the power market. To ensure power quality while achieving peak shaving and valley filling, a two-stage optimization model for microgrid cluster was constructed, enabling joint optimization of berth allocation and energy scheduling. In the first stage, considering the impact of random events during vessel navigation, berth allocation is conducted with the objectives of maximizing peak shaving and valley filling rates and minimizing overall costs, thereby achieving preliminary load leveling while ensuring power supply system quality. In the second stage, dispatching is performed for power generation equipment and adjustable loads, aiming to further achieve peak shaving and valley filling while maintaining the economic viability of the microgrids. Addressing conflicting interests among stakeholders in the port's electricity market, a master-slave game strategy has been established. This strategy aims to balance the interests between the electricity trading agent and the microgrid aggregator composed of multiple microgrid models. By solving the second-order partial derivatives of each optimization variable with respect to the objective function, we demonstrate the existence and uniqueness of the equilibrium solution in the master-slave game. In response to the challenge of difficulties in solving the multi-objective optimization model due to conflicting objective functions, a Dung Beetle Optimization (DBO) algorithm incorporating non-dominated sorting strategy and crowding distance was introduced. This resulted in a novel multi-objective optimization algorithm termed Non-Dominated Sorting Dung Beetle Optimizer (NSDBO). The NSDBO algorithm was demonstrated to possess favorable characteristics in terms of distribution, convergence, and uniformity. Using a northern Chinese port as an example, berth allocation and energy scheduling for three port areas were jointly optimized, and various scenarios were compared. The results indicated that, compared to Scenario 1, which did not consider random events, the proposed joint optimization method increased the peak shaving and valley filling rate by 8.46 %, reduced network losses by 42 kW, and increased the revenue of the microgrid aggregation merchant by 2.11 %. Compared to Scenario 2, which only considered energy scheduling, the joint optimization increased the peak shaving and valley filling rate by 36.89 %, reduced network losses by 18 kW, and increased the revenue of the microgrid aggregation merchant by 2.29 %. Compared to Scenario 3, which was individually optimized, the joint optimization increased the peak shaving and valley filling rate by 48.42 %, reduced network losses by 24 kW, and increased the revenue of the microgrid aggregation merchant by 2.73 %. This validates the superiority of the proposed joint optimization method in mitigating the impact of random events, reducing network losses, stabilizing port load fluctuations, and enhancing the revenue of the microgrid cluster.

Numerical assessment of hyperbolic type double interface problems via Haar wavelets

In this manuscript, we have addressed wave propagation challenges within heterogeneous media and hyperbolic interface model. A hybrid numerical approach is introduced for solving these problems, combining the finite difference method and Haar wavelet collocation method. The method entails approximating the second-order space partial derivative through a truncated Haar wavelet series, while the temporal derivative is approximated using finite difference method. For linear hyperbolic interface model, the resulting algebraic systems are solved using the Gauss elimination technique. In the case of non-linear problems, the nonlinearity is addressed through the quasi-Newton linearization formula. To assess the accuracy of the proposed technique, we compute the computational rate of convergence, root mean square errors and maximum absolute errors employing various collocation points. The proposed method perform very well and produces a stable solution if sharp transitions exists in the solution space or if there is a discontinuity between initial and boundary conditions, whereas, the other existing method loses its accuracy in such cases. The numerical experiments, stability and rate of convergence both theoretical and computational, confirm the accuracy and diverse applicability of the method.

Estimates for stress concentration between two adjacent rigid inclusions in Stokes flow

In this paper, we establish the estimates for the gradient and the second-order partial derivatives for the Stokes flow in the presence of two closely located strictly convex inclusions in dimension three. Moreover, the blow-up rate of the gradient is showed to be optimal by a pointwise upper bound and a lower bound in the narrowest region. We also show the optimal blow-up rate of Cauchy stress tensor. In dimensions greater than three, the upper bounds of the gradient are established. These results answer the questions raised in [25].

New geometry-inspired numerical convex analysis method for yield functions under isotropic and anisotropic hardenings

Convexity of a yield function must be guaranteed to ensure unique relationship between plastic strain increments and stress components. A geometry-inspired numerical convex analysis (GINCA) approach is developed to analyze the convexity of a yield function. The numerical approach is verified by the computation of convex domains for several typical yield functions of Drucker, Cazacu-Barlat2004, Hu2017, Cazacu2018 and a newly proposed function for differential-anisotropic hardening. The verified GINCA is applied to analyze the convexity of the Gao2011 yield function since it was not determined yet. The numerical approach is also used to determine the convex domain evolution with respect to plastic deformation in an anisotropic hardening function. It is also applied to investigate the convexity of polynomial yield functions for strong anisotropic metals. The result shows that the GINCA can effectively and correctly compute the convex domain for different yield functions. Besides, the numerical approach only needs to compute the effective stress from a yield function without any computation of the first- and second-order partial derivatives, compared to the complicated convex analysis by the Hessian matrix. Therefore, the proposed GINCA is user-friendly, effective and accurate to analyze the convexity of a yield function both for isotropic hardening and anisotropic/differential hardening. The Matlab codes are shared as electric attachments for scientists and engineers to check the convexity of a yield surface under biaxial loading, plane stress with shear stress and triaxial loading conditions.

Convergent finite element methods for the perfect conductivity problem with close-to-touching inclusions

Abstract In the perfect conductivity problem (i.e., the conductivity problem with perfectly conducting inclusions), the gradient of the electric field is often very large in a narrow region between two inclusions and blows up as the distance between the inclusions tends to zero. The rigorous error analysis for the computation of such perfect conductivity problems with close-to-touching inclusions of general geometry still remains open in three dimensions. We address this problem by establishing new asymptotic estimates for the second-order partial derivatives of the solution with explicit dependence on the distance $\varepsilon $ between the inclusions, and use the asymptotic estimates to design a class of graded meshes and finite element spaces to solve the perfect conductivity problem with possibly close-to-touching inclusions. In particular, we propose a special finite element basis function that resolves the asymptotic singularity of the solution by making the interpolation error bounded in $W^{1,\infty }$ in a neighborhood of the close-to-touching point, even though the solution itself is blowing up in $W^{1,\infty }$. This is crucial in the error analysis for the numerical approximations. We prove that the proposed method yields optimal-order convergence in the $H^1$ norm, uniformly with respect to the distance $\varepsilon $ between the inclusions, in both two and three dimensions for general convex smooth inclusions, which are possibly close-to-touching. Numerical experiments are presented to support the theoretical analysis and to illustrate the convergence of the proposed method for different shapes of inclusions in both two- and three-dimensional domains.

Legendre-finite difference method for solving fractional nonlinear Sobolev equation with Caputo derivative

The Sobolev equation is known as a partial differential equation with second-order partial derivatives. Initially, the Sobolev equation was used to solve the problem of turbulence in physics. However, due to its extensive use in mathematics and geometry, it soon became one of the most important mathematical equations. Despite the significance of this equation, its solution in fractional and nonlinear cases has received less attention in research. This paper presents a numerical algorithm for solving fractional nonlinear Sobolev model using Legendre polynomials and a finite difference scheme. Firstly, the finite difference plan is employed to discretize the time variable, where stability and convergence of the method are demonstrated. The stability of the method is dependent on the nonlinear part of the problem. The two-dimensional Legendre polynomials are utilized at collocation points to discretize the spatial variables. Finally, several illustrative examples are conducted to evaluate the accuracy of the proposed approach.

New classes of interval-valued variational problems and inequalities

The paper discusses two distinct classes of variational problems. The first class focuses on optimization problems governed by interval-valued functionals, which are influenced by second-order partial derivatives. The second class pertains to variational inequalities associated with the first class. The paper explores relationships between solutions within these variational models and introduces the concept of convexity assumption for interval-valued functionals. The significance of these findings lies in the practical applications, particularly in the context of objective functionals with physical implications, such as the computation of mechanical work.

Controlled nonlinear dynamics generated by isoperimetric constrained optimization problems involving second-order partial derivatives

Controlled nonlinear dynamics generated by isoperimetric constrained optimization problems involving second-order partial derivatives

Application of Polling Scheduling in Mobile Edge Computing

With the Internet of Things (IoT) development, there is an increasing demand for multi-service scheduling for Mobile Edge Computing (MEC). We propose using polling for scheduling in edge computing to accommodate multi-service scheduling methods better. Given the complexity of asymmetric polling systems, we have used an information-theoretic approach to analyse the model. Firstly, we propose an asymmetric two-level scheduling approach with priority based on a polling scheduling approach. Secondly, the mathematical model of the system in the continuous time state is established by using the embedded Markov chain theory and the probability-generating function. By solving for the probability-generating function’s first-order partial and second-order partial derivatives, we calculate the exact expressions of the average queue length, the average polling period, and the average delay with an approximate analysis of periodic query way. Finally, we design a simulation experiment to verify that our derived parameters are correct. Our proposed model can better differentiate priorities in MEC scheduling and meet the needs of IoT multi-service scheduling.

Students’ geometric understanding of partial derivatives and the locally linear approach

We use Action-Process-Object-Schema (APOS) theory to study students’ geometric understanding of partial derivatives of functions of two variables. This study contributes to research on the teaching and learning of differential multivariable calculus and its didactics. This is an important area due to its multiple applications in science, mathematics, engineering, and technology (STEM). The study tests a previously proposed model of mental constructions students may use to understand partial derivatives through a set of activities designed to help students make the conjectured constructions. The model is based on the local linearity of differentiable two-variable functions, and the model-based activities explore the relationship between partial derivatives and tangent plane in different representations. We used semi-structured interviews with eleven students whose teacher used the three-part cycle—Activities designed with the genetic decomposition; collaborative work in small groups and Class discussion; and Exercises for home (ACE)—as pedagogical strategy. The model-based activity set based on local linearity and the ACE strategy helped students construct a geometric understanding of partial derivatives. Results led to reconsider and further refine the model. This study also resulted in improving activity sets and obtaining information on students’ construction of second-order and mixed partial derivatives.

Структурный анализ функции управления динамической системой в частных производных разного порядка

For a dynamic system described by a partial differential equation of different order, the problem of constructing software control in an analytical form is solved. The primary method of research is the cascade decomposition method, the improved algorithm of which includes three main stages: the forward move, the central stage and the reverse move. The method is based on the properties of the matrix coefficient with a second-order partial derivative of the control function. The noetherianness of the coefficient causes the splitting of the original space into direct sums of subspaces. A scheme for structuring subspaces in accordance with the properties of the matrix coefficient is given. The direct move of decomposition is realized, which consists in a step-by-step transition to equivalent hierarchically structured systems of two levels in subspaces. The step-by-step structuring of the components of the state function into functions from subspaces while using matrix coefficients - projectors is performed. The resulting functions from subspaces are called pseudo-state and pseudo-control functions. Graphical visualization of the hierarchical structure of the source space in the form of a diagram is performed. This scheme reflects the essential connections between the components of the subspaces of each decomposition level. Finite-dimensional spaces are considered, which causes the complete completion of the first stage of the algorithm in a finite number of steps not exceeding the dimension of the original space. During the decomposition, the conditions at the start and end points are reduced, so that at the end of each step of the forward stroke, one additional condition appears at each point for each second-level system. In the process of implementing the first stage of the algorithm, the properties of matrix coefficients entailing full controllability or unmanageability of the initial system are established; the properties of functions in the initial conditions necessary for the implementation of the controlled process are also revealed. The criterion of complete controllability of the initial system is deduced. For a fully controlled system, a transition is made to the central stage of the algorithm – the construction of a defining basis function that satisfies all additional conditions for partial derivatives in time at each point resulting from the reduction of the original ones. It is the presence of this defining basic function that lays the prerequisites for constructing the state and control functions of the initial system at the final stage of the algorithm. A diagram visualizing the procedure of step-by-step restoration of the components of the state function in the process of implementing the reverse course is given. The reverse is completed by explicitly constructing first the state function, then the control function. A qualitative analysis of the management structure of the system under study is carried out.

The gravitational potential and its first- and second-order partial derivatives of an ellipsoidal tesseroid based on the Cartesian integral kernel

The gravitational potential and its first- and second-order partial derivatives of an ellipsoidal tesseroid based on the Cartesian integral kernel

Noncentral Wishart matrices, asymptotic normality of vec and smooth statistics

Wishart matrices play an important role in normal multivariate statistical analysis. In this work, we present an approach that has been already used for normal vectors and is now applied to noncentral Wishart matrices. We show that, under general conditions, the vec of the Wishart matrix and a large class of its statistics have asymptotic normal distributions when the norm of the noncentrality parameter diverges ∞. These statistics are called smooth and are given by functions whose component functions have continuous second-order partial derivatives in a neighbourhood of a ‘pivot’ point. Moreover, we derive the application domain of the asymptotic normal distributions for the vec of the Wishart matrix and its smooth statistics. Thus we have an attraction to the normal model, for the increasing predominance of noncentrality and not for increasing sample sizes. A simulation study shows that the threshold for the use of asymptotic normal distributions is quite acceptable.

Covariance Analysis of the Optimal Orbital Interception With Navigation Errors

This paper derives the analytical linear covariances for the optimal single-impulse orbital interception problems with navigation errors. The initial position and velocity errors are considered for both the interceptor and the non-maneuvering target. Firstly, for free impulse time and free interception time, the necessary conditions for the minimum-fuel and the minimum-time interceptions are derived, and the optimal solutions are obtained by two-dimensional numerical search methods. Then, the linear covariance method is used to derive the error covariances for the minimum-fuel solution and the minimum-time solution, respectively. Finally, based on the first-order and second-order partial derivatives, the error covariances of the optimal impulse time, the optimal interception time, the required impulse vector and the interception position vector are analytically obtained. Numerical results indicate that the proposed analytical linear covariance method is accurate for both the minimum-fuel and the minimum-time interception problems.

Recurrent Neural Network-Based Nonlinear Optimization for Braking Control of Electric Vehicles

In this paper, electro-hydraulic braking (EHB) force allocation for electric vehicles (EVs) is modeled as a constrained nonlinear optimization problem (NOP). Recurrent neural networks (RNNs) are advantageous in many folds for solving NOPs, yet existing RNNs’ convergence usually requires convexity with calculation of second-order partial derivatives. In this paper, a recurrent neural network-based NOP solver (RNN-NOPS) is developed. It is seen that the RNN-NOPS is designed to drive all state variables to asymptotically converge to the feasible region, with loose requirement on the NOP’s first-order partial derivative. In addition, the RNN-NOPS’s equilibria are proved to meet Karush–Kuhn–Tucker (KKT) conditions, and the RNN-NOPS behaves with a strong robustness against the violation of the constraints. The comparative studies are conducted to show RNN-NOPS’s advantages for solving the EHB force allocation problem, it is reported that the overall regenerative energy of RNN-NOPS is 15.39% more than that of the method for comparison under SC03 cycle.

Adaptive window rotated second-order synchroextracting transform and its application in fault diagnosis of wind turbine gearbox

Time frequency (TF) analysis (TFA) based on vibration signals is a vital method for the health monitoring of wind turbine gearboxes. Based on classical synchrosqueezing transform and short-time Fourier transform (STFT), synchroextracting transform (SET) abstractly retains primary TF energy around the interested frequency components by reassigning the TF transformation coefficient. However, as the fixed window of these traditional TFA methods, they have poor matching effects on multi-component signals with fast-varying frequencies. To address this problem, this paper first optimizes the width of the short-term window based on STFT, and then proposes adaptive short-time Fourier transform (ASTFT). That is, the direction of the window function in the STFT is changed by automatically matching a series of rotation operators. Then, based on ASTFT and SET, the second-order partial derivatives of time and frequency are used to modify the formula of instantaneous frequency estimation. Therefore, adaptive window rotated second-order synchroextracting transform (AWRSSET) is put forward to obtain and sharpen TF representations of multi-component vibration signals of wind turbines in this paper. In order to accurately diagnose faults in wind turbines, this paper extracts the rotation speed curve from the TF plane, which is generated through AWRSSET, and then makes subsequent order-frequency analyses without a tachometer. AWRSSET can be applied to diagnosing faulty wind turbine gearboxes and gears with broken teeth under time-varying speed, both of which testify to the advantages of this method.

Autofocusing of Maneuvering Targets in Terahertz Inverse Synthetic Aperture Radar Imaging Based on Damped Newton Method.

Maneuvering target imaging based on inverse synthetic aperture radar (ISAR) imaging has recently drawn significant attention. Among the many autofocusing technologies which are crucial in ISAR imaging, minimum-entropy-based autofocusing (MEA) is highly robust. However, traditional MEA is not suitable for terahertz (THz) ISAR imaging. For one thing, the iterative process in traditional MEA is too complicated to be utilized for THz-ISAR imaging with tremendous data. For another, THz wavelengths are very short and extremely sensitive to phase errors, so the compensation accuracy of the traditional MEA method can hardly meet the requirements of THz radar high-resolution imaging. Therefore, in this paper, the MEA algorithm based on the damped Newton method is proposed, which improves computational efficiency by approximating the first- and second-order partial derivatives of the image entropy function with respect to the phase errors, as well as by the fast Fourier transform (FFT). The search step size factor is introduced to ensure that the algorithm can converge along the declination direction of the entropy function and obtain the globally optimal ISAR image. The experimental results validated the efficiency of the proposed algorithm, which is promising in THz-ISAR imaging of maneuvering targets.

Recent Advances of Constrained Variational Problems Involving Second-Order Partial Derivatives: A Review

This paper comprehensively reviews the nonlinear dynamics given by some classes of constrained control problems which involve second-order partial derivatives. Specifically, necessary optimality conditions are formulated and proved for the considered variational control problems governed by integral functionals. In addition, the well-posedness and the associated variational inequalities are considered in the present review paper.

Temporal resolution and temporal extent of orientation repulsion

A vertical target is perceived as tilted against a slightly tilted inducer surrounding it. To identify the temporal resolution and temporal extent of this phenomenon of orientation repulsion in the same paradigm, we used an alternating pair of inducer stimuli having complementary orientation distributions and quantified repulsion at various alternation frequencies. The duration of each inducer stimulus was inversely proportional to the frequency. When an orthogonal pair of D2 patterns, a type of grating whose luminance modulation in a particular orientation was the second-order partial derivative of an isotropic 2D-Gaussian, was used as the inducer, repulsion occurred when the duration exceeded 20 ms and leveled off at 30 ms and beyond. When a custom-made texture with a narrowband orientation distribution and another texture with a complementary orientation distribution were alternated as the inducer, repulsion gradually increased until the inducer duration reached 200 ms. The gradual increase in repulsion was observed regardless of whether the orientation of the inducer that appeared simultaneously with the target was discernible. These findings reveal that contextual modulation in orientation occurs at a high temporal resolution and continues to a long temporal extent under optimal conditions.