Higher Order Derivatives Research Articles

Fuzzy adaptive asynchronous sampled-data consensus control for nonlinear multi-agent systems under DoS attacks

This article investigates the problem of fuzzy adaptive resilient asynchronous sampled-data consensus control (SDCC) for a class of uncertain nonlinear strict-feedback multi-agent systems (MASs) under denial-of-service (DoS) attacks and with a directed graph. Firstly, the upper bounds on the sampling intervals and a distributed sampled-data observer are proposed to estimate the leader's output and its high-order derivatives under DoS attacks. Then, the sufficient conditions to ensure the asymptotic convergence of the designed distributed sampled-data observer are established. Based on the distributed sampled-data observer and backstepping control design technique, a distributed fuzzy adaptive resilient control algorithm is developed. It is proved that the proposed control scheme can guarantee that the controlled MASs are stable, and the consensus errors converge to a small neighbourhood of zero. Finally, a simulation example on heterogeneous vehicular platoons verifies the feasibility of the proposed control scheme.

Fuzzy Adaptive Event-Triggered Consensus Control for Nonlinear Multiagent Systems Under Jointly Connected Switching Networks.

This article studies the fuzzy adaptive event-triggered (ET) consensus control issue of nonlinear multiagent systems (NMASs) under jointly connected switching networks. Since the leader and its high-order derivatives are unknown under jointly connected switching networks, a novel distributed ET reference generator equipped with an ET mechanism is constructed to estimate them. Meanwhile, the continuous information transmission among agents is avoided and the network channel utilization is optimized. Subsequently, fuzzy logic systems (FLSs) are employed to approximate unknown dynamics, and a fuzzy adaptive ET consensus control algorithm only using intermittent communication is designed by backstepping control methodology. It is demonstrated that all the closed-loop signals are semi-globally uniformly ultimately bounded (SGUUB), with the tracking errors converging to a small neighborhood around zero. Finally, we apply the developed fuzzy adaptive ET consensus control algorithm to unmanned surface vehicles (USVs), and the simulation results verify the effectiveness of the proposed ET consensus control algorithm.

Fermionic Casimir energy in Horava–Lifshitz scenario

In this work, we investigate the violation of Lorentz symmetry through the Casimir effect, one of the most intriguing phenomena in modern physics. The Casimir effect, which represents a macroscopic quantum force between two neutral conducting surfaces, is widely regarded as a triumph of Quantum Field Theory. In this study, we present new results for the Casimir effect, focusing on the contribution of mass associated with fermionic quantum fields confined between two large parallel plates, in the context of Lorentz symmetry violation within the Horava–Lifshitz formalism. To calculate the Casimir energy and pressure, we impose a MIT bag boundary condition on the two plates, compatible with the higher-order derivative term in the modified Dirac equation. Our results reveal a strong influence of Lorentz violation on the Casimir effect. We observe that the Casimir energy is significantly affected, both in intensity and sign, potentially resulting in a repulsive or attractive force between the plates, depending on the critical exponent associated with the Horava–Lifshitz formalism.

An open-source time-domain wave-based room acoustic software in Python based on the nodal discontinuous Galerkin method

In this work, we introduce an open-source implementation of a time-domain wave-based room acoustic modeling software package, named DG_RoomAcoustics. In this software, the linear acoustic equations are spatially discretized by the nodal discontinuous Galerkin method, and are integrated in time by either the explicit Runge-Kutta or the arbitrary high-order derivatives (ADER) integration schemes. Following the principles of object-oriented programming paradigm, the software is structured to ensure generic applicability and to facilitate future extensions with additional functionalities (e.g., different time integration schemes, boundary conditions). A comprehensive presentation of the physical and numerical aspects of the problem is provided. A detailed exposition of the code structure and components is presented, all of which are released under an open-source license, fostering community feedback and collaborative contributions for ongoing improvements. A brief overview of the current capabilities of the software is introduced. Future work regarding possible functionality extensions and performance optimizing are discussed.

A generalized refined Moore–Gibson–Thompson thermoelastic model based on the concept of memory-dependent higher-order derivatives

The inclusion of memory-dependent derivatives (MDD) in constitutive models improves the ability to predict and analyze time-dependent responses of materials, providing a more detailed depiction of their mechanical properties and structural changes. In this paper, a new thermoelasticity model is created that combines the Moore-Gibson-Thompson (MGT) equation with higher-order memory-dependent derivatives (MDD), any optional kernel function, and time delay. The objective of this model is to provide a more accurate mathematical depiction of the thermal and mechanical reactions of materials, particularly those that exhibit complex behaviors over time. A theoretical study was conducted to provide additional clarification of the proposed concept. For this purpose, thermal-mechanical waves were studied in a semi-infinite region, surrounded by a magnetic field, and exposed to a direct heat source uniformly distributed on its outer surface. To solve the coupled partial differential equations governing the system, the Laplace transform methodology was used. The effects of different kernel functions, time delays, and higher-order (HO) derivatives on the behavior of thermoelastic materials are discussed and illustrated using figures and tables.

A New Composite Dissimilarity Measure for Planar Curves Based on Higher-Order Derivatives

With the rapid development of information technology, the problem of curve matching has appeared in many application domains, including sequence analysis, signals processing, speech recognition, etc. Many similarity measures have been studied for matching curves based on Euclidean distance, which shows fragility in portraying the morphological information of curve data. In this paper, we propose a novel weighted composite curve dissimilarity metric (WCDM). First, the WCDM measures the dissimilarity based on the higher-order semantic difference between curve shapes and location difference. These two differences are calculated using the curvature difference and Euclidean distance between the curves, respectively. Second, a new dynamic weighting function is defined by employing the relationship between the trends of the curves. This function aims at adjusting the contributions of the curvature difference and the Euclidean distance to compose the dissimilarity measure WCDM. Finally, to ascertain the rationality of the WCDM, its metric properties are studied and proved theoretically. Comparison experiments on clustering and classification tasks are carried out on curve sets transformed from UCR time series datasets, and an application analysis of the WCDM is conducted on spectral data. The experimental results indicate the effectiveness of the WCDM. Specifically, clustering and classification based on the WCDM are superior to those based on ED, DTW, Hausdorff, Fréchet, and LCSS on at least 8 out of 14 datasets across all evaluation indices. In particular, the Purity and ARI on the Beetlefly dataset are improved by more than 7.5%, while accuracy on the Beef, Chinatown, and OliveOil datasets increases by 13.32%, 10.08%, and 12.83%, respectively.

Double Laplace Transform Method for Solving Fractional Fourth-Order Partial Integro-Differential Equations with Weakly Singular Kernel

Objectives: To investigates the solutions of fourth order partial integro-differential equations with high-order non-integer derivatives and weakly singular kernels. Methods: Weakly singular kernels present challenges in both analytical and numerical treatments due to their intricate behaviour near singular points. In this article, we introduce a novel approach utilizing the double Laplace transform method to effectively address these challenges. Findings: By solving a series of precise and understandable examples, the double Laplace transform clearly transforms the fractional partial integro-differential equation into an algebraic equation that can be solved easily. Novelty: The double Laplace transform is an effective instrument for solving fractional partial integro-differential equations, which are commonly used in fields such as physics, fluid mechanics, gas dynamics, and signal processing. Mathematics Subject Classifications: 35R09, 35R11, 44A10, 44A30. Keywords: Fractional PIDE, Weakly singular kernel, Double Laplace transform, Inverse double Laplace transform, Exact solution

A Combined Quasi-Newton-Type Method Using 4th-Order Directional Derivatives for Solving Degenerate Problems of Unconstrained Optimization

Introduction. Methods of unconstrained optimization play a significant role in machine learning [1–6]. When solving practical problems in machine learning, such as tuning nonlinear regression models, the extremum point of the chosen optimality criterion is often degenerate, which significantly complicates its search. Therefore, degenerate problems are the most challenging in optimization. Known numerical methods for solving the general unconstrained optimization problem, up to the second order, have very low convergence rates when solving degenerate problems [7, 8]. This is explained by the fact that for a significant improvement in convergence rate in this case, it is necessary to use higher-order derivatives in the method than the second order [10]. The purpose of the paper is to develop an efficient quasi-Newton method for solving degenerate unconstrained optimization problems, the idea of which (unlike regularization) involves dividing the entire space into the sum of two orthogonal subspaces. This idea was introduced in [23]. The space division at each iteration of the method is based on the spectral decomposition of the matrix approximating the Hessian of the objective function using the BFGS formula [3]. Each subspace exhibits its own behavior of the objective function, and therefore, an appropriate minimization method is applied on it. Results. A combined quasi-Newton method is presented for solving degenerate unconstrained optimization problems, based on orthogonal decomposition of the Hessian approximation matrix and division of the entire space into the sum of two orthogonal subspaces. On one subspace (the kernel of the Hessian approximation matrix), a method is applied where derivatives in the direction of the 4th order are computed, while on the orthogonal complement to it, a quasi-Newtonian method is applied. A separate one-dimensional search is applied on each of these subspaces to determine the step multiplier in the respective direction. The effectiveness of the presented combined method is confirmed by numerical experiments conducted on widely accepted test functions for unconstrained optimization problems. The proposed method allows obtaining fairly accurate solutions to test tasks in case of degeneracy of the minimum point with significantly lower computational costs for gradient calculations compared to optimization procedures of well-known mathematical packages. Conclusions. The idea of dividing the entire space into the sum of two (or possibly more) orthogonal subspaces when solving complex optimization problems is quite promising in terms of applying a combination of different numerical methods on separate subspaces. In the future, it is planned to conduct theoretical research on the convergence rate of the presented combined method in solving degenerate unconstrained optimization problems. Keywords: unconstrained optimization, quasi-Newton methods, degenerate minimum point, spectral matrix decomposition, Machine Learning.

A space-time generalized finite difference scheme for long wave propagation based on high-order Korteweg-de Vries type equations

In this paper, the space-time generalized finite difference scheme is applied to solve the nonlinear high-order Korteweg-de Vries equations in multiple dimensions. The proposed numerical scheme combines the space-time generalized finite difference method, the Levenberg-Marquardt algorithm, and a time-marching approach. The space-time generalized finite difference method treats the temporal axis as a spatial axis, enabling the proposed scheme to discretize all derivatives in the governing equation. This is accomplished through Taylor series expansion and the moving least squares method. Due to the expandability of the Taylor series to any order, the proposed numerical scheme excels in efficiently handling mixed and higher-order derivatives. These capabilities are distinct advantages of the proposed scheme. The resulting system of algebraic equations is sparse but overdetermined. Therefore, the Levenberg-Marquardt algorithm is directly applied to solve this nonlinear algebraic system. During the calculation process, the time-marching approach reduces computational effort and improves efficiency by dividing the space-time domain.

High-Order Models for Gas Transport in Multiscale Coal Rock.

The coal reservoirs exhibit great heterogeneity and strong anisotropy in multiscale pore/fracture structures. Developing highly accurate multiscale models for real-time prediction of microgaseous flow in complicated porous media with pronounced contrast in transport coefficients is crucial but not yet available, which is time-consuming, expensive, and even computationally impossible. In this study, a multiscale approximate solution of the gas flow and pressure field is derived in this paper for predicting coalbed methane (CBM) transport in macro-microscopic two-scale porous media of typical coal rocks in Guizhou Province, China, and the detailed finite element algorithm is established. The multiscale approximate solutions are constructed from the first- and second-order auxiliary cell functions and low- or high-order derivatives of the homogenized pressure field, which can accurately approximate the solution of the original problem to a certain extent. The numerical accuracy and validity of the multiscale approximate solutions under various working conditions are analyzed in detail by comparing and verifying the results with the direct numerical simulation results of fine-mesh high-precision finite elements. The results show that the homogenized approximate solution can only roughly capture the characteristics of the macroscopic pressure evolution, the first-order approximate solution can partially reproduce the macro-microscopic pressure oscillation phenomenon, and the second-order approximate solution can accurately predict the pressure profile and its evolution law with time in porous media with macro-micro configuration under various complex conditions. The second- and high-order two-scale approximate solutions constructed in this paper exhibit high numerical accuracy and excellent computational efficiency. We also develop multiscale approximate solutions for predicting microgaseous flow at a low reservoir pressure where the slippage effects are very pronounced. Potential applications of our high-order models to the coal reservoirs with a hierarchical matrix and fractures are discussed. The developed multiscale models exhibit excellent applicability to investigating the crossflow effects and coupling mechanisms between the matrix and cleats or fractures with multiple configurations in the CBM and shale gas reservoirs, which is crucial for the effective implementation of hydraulic fracturing technology. This study provides a fundamental understanding of gas transport in multiscale matrix-fracture networks, which helps to improve the optimization design of hydraulic fracturing in tight reservoirs.

Efficient finite element strategy using enhanced high-order and second-derivative-free variants of Newton's method

In this work, we propose a stable finite element approximation by extending higher-order Newton's method to the multidimensional case for solving nonlinear systems of partial differential equations. This approach relies solely on the evaluation of Jacobian matrices and residuals, eliminating the need for computing higher-order derivatives. Achieving third and fifth-order convergence, it ensures stability and allows for significantly larger time steps compared to explicit methods. We thoroughly address accuracy and convergence, focusing on the singular p-Laplacian problem and the time-dependent lid-driven cavity benchmark. A globalized variant incorporating a continuation technique is employed to effectively handle high Reynolds number regimes. Through two-dimensional and three-dimensional numerical experiments, we demonstrate that the improved cubically convergent variant outperforms others, leading to substantial computational savings, notably halving the computational cost for the lid-driven cavity test at large Reynolds numbers.

Is Quantum Above-Barrier Reflection Important for Molecular Barrier Crossing Rates?

Understanding quantum tunneling and above-barrier reflection effects on unimolecular and bimolecular reaction rate constants remains challenging to this very day. In many applications, especially when considering moderate-to-high temperatures, the "standard" procedure is to use the parabolic barrier approximation. Recent work has shown though that this may be insufficient, and one cannot ignore anharmonicity. In this work, we study the analytic theory, including anharmonicity obtained when expanding the thermal rate up to order ℏ4. Such theories need high-order derivatives of the potential at the barrier top. We show that such derivatives are computed straightforwardly for six different reactions. We suggest a straightforward methodology for assessing whether the parabolic barrier approximation is valid and show that when the reaction asymmetry is large, this may lead to significant quantum above-barrier reflection and transmission coefficients, which are less than unity.

Chien-physics-informed neural networks for solving singularly perturbed boundary-layer problems

A physics-informed neural network (PINN) is a powerful tool for solving differential equations in solid and fluid mechanics. However, it suffers from singularly perturbed boundary-layer problems in which there exist sharp changes caused by a small perturbation parameter multiplying the highest-order derivatives. In this paper, we introduce Chien’s composite expansion method into PINNs, and propose a novel architecture for the PINNs, namely, the Chien-PINN (C-PINN) method. This novel PINN method is validated by singularly perturbed differential equations, and successfully solves the well-known thin plate bending problems. In particular, no cumbersome matching conditions are needed for the C-PINN method, compared with the previous studies based on matched asymptotic expansions.

A new shifted generalized Chebyshev approach for multi-dimensional sinh-Gordon equation

The numerical treatment of multi-dimensional non-linear sinh-Gordon equations is the focus of this paper. We numerically solve the (1 + 1) and (2 + 1) sinh-Gordon equations using two collocation algorithms. We select the set of basis functions as a set of generalized Chebyshev polynomials (CPs), which we express as orthogonal combinations of CPs. We develop and utilize some formulas related to these polynomials to propose our numerical algorithms. Specific values for the high-order derivatives of the basis functions serve in the derivation of the two presented algorithms. Additionally, we provide an estimation of the basis functions used in the convergence analysis study. We follow the two collocation algorithms to transform the sinh-Gordon equations into non-linear equation systems, which any suitable solver can handle. We provide some examples and comparisons to confirm the effectiveness of our presented algorithms.

Second-Order Terminal Sliding Mode Control for Trajectory Tracking of a Differential Drive Robot

This paper proposes a second-order terminal sliding mode (2TSM) approach to the trajectory tracking of the differential drive mobile robot (DDMR). Within this cascaded control scheme, the 2TSM dynamic controller, at the innermost loop, tracks the robot’s velocity quantities while a kinematic controller, at the outermost loop, regulates the robot’s positions. In this manner, chattering is greatly attenuated, and finite-time convergence is guaranteed by the second-order TSM manifold, which involves higher-order derivatives of the state variables, resulting in an inherently robust as well as fast and better tracking precision. The simulation results demonstrate the merit of the proposed control methods.

Exploration of Multivalent Harmonic Functions: Investigating Essential Properties

Within this manuscript, we introduce an innovative subclass of multivalent harmonic functions, encompassing higher-order derivatives within the confines of an open unit disk. Our investigation extends to the analysis of coefficient bounds, growth estimates, starlikeness, and convexity radii uniquely associated with this particular class. Furthermore, we scrutinize the property of closure under convolution operations for this subclass.

Variational consistent one-point integration with Taylor's expansion-based stabilization in the second-order meshfree Galerkin method for strain gradient elasticity

A generalized variational principle with five independent variables is proposed for strain gradient elasticity, including displacement, strain, strain gradient, stress, and double stress. Based on the principle, a one-point integration scheme is designed for the second order meshfree Galerkin method through nodal smoothed derivatives and their high order derivatives by Taylor's expansion. Since the proposed integration scheme meets the orthogonality conditions, it is variational consistent. The weak form expanded with Taylor's polynomials can be well evaluated by nodal smoothed derivatives and their high order derivatives on one quadrature point. Numerical one- and two-dimensional case studies show that the proposed integration scheme performs better than the standard Gaussian integration method in terms of accuracy, convergence, efficiency, and stability.

Two-Step Quadratic Programming for Physically Meaningful Smoothing of Longitudinal Vehicle Trajectories

Longitudinal vehicle trajectories suffer from errors and noise because of detection and extraction techniques, challenging their applications. Existing smoothing methods either lack physical meaning or cannot ensure solution existence and uniqueness. To address this, we propose a two-step quadratic programming method that aligns smoothed speeds and higher-order derivatives with physical laws, drivers’ behaviors, and vehicle characteristics. Unlike the well-known smoothing splines method, which minimizes a weighted sum of discrepancy and roughness in a single quadratic programming problem, our method incorporates prior knowledge of position errors into two sequential quadratic programming problems. Step 1 solves half-smoothed positions by minimizing the discrepancy between them and raw positions, subject to physically meaningful bounds on speeds and higher-order derivatives of half-smoothed positions. Step 2 solves smoothed positions by minimizing the roughness while maintaining physically meaningful bounds and allowing the deviations from raw data of smoothed positions by at most those of the half-smoothed positions and prior position errors. The second step’s coefficient matrix is not positive definite, necessitating the matching of the first few smoothed positions with corresponding half-smoothed ones, with equality constraints equaling the highest order of derivatives. We establish the solution existence and uniqueness for both problems, ensuring their well-defined nature. Numerical experiments using Next Generation Simulation (NGSIM) data demonstrate that a third-order derivative constraint yields an efficient method and produces smoothed trajectories comparable with manually re-extracted ones, consistent with the minimum jerk principle for human movements. Comparisons with an existing approach and application to the Highway Drone data set further validate our method’s efficacy. Notably, our method is a postprocessing smoothing technique based on trajectory data and is not intended for systematic errors. Future work will extend this method to lateral vehicle trajectories and trajectory prediction and planning for both human-driven and automated vehicles. This approach also holds potential for broader smoothing problems with known average error in raw data. History: This paper has been accepted for the Transportation Science Special Issue on ISTTT25 Conference. Funding: The authors extend their gratitude to the Pacific Southwest Region University Transportation Center and the University of California Institute of Transportation Studies (UC ITS) Statewide Transportation Research Program (STRP) for their valuable financial support.

Contributions to the characterization of the Schema using APOS theory: Graphing with derivative

This study contributes to Action, Process, Object, Schema (APOS) theory research by showing two approaches used by advanced mathematics students to construct relations between higher-order derivatives to solve complex problems. We show evidence of students’ ability to perform Actions on their graphing derivative Schema, that is, of its thematization. It also contributes to the literature on the learning of differential calculus by showing how advanced students use their knowledge to construct relations between concepts when facing complex situations. The work of three graduate students on transforming complex graphs and determining their properties and their relation to the domain structure is analyzed to determine their solution approaches. Their graphing derivative Schema is analyzed in depth in terms of the construction of relations among the Schema structures and assimilation and accommodation mechanisms involved in thematization in APOS theory. These findings are important in informing and developing didactic strategies to foster university students’ understanding of derivatives, which can smoothe the transition to the study of advanced mathematics courses.

A new multiscale imaging method for estimating the depth and structural index of a magnetic source

The fast automatic technique for determining the source parameter is commonly used to interpret magnetic data. A new method is proposed to estimate the magnetic source parameter based on the any-order analytic signals of magnetic anomalies at different altitudes. The new method is based on the relationship between the location, depth, and structural index of the source and the expressions of analytic signals and uses the altitude z and a depth scaling factor [Formula: see text] to establish a new multiscale imaging method called variable depth mirror imaging (VDMI), whose extreme points are related to the source parameters. Two equations are given to calculate the source depth and structural index on the basis of the vertical positions of the peaks of the VDMI sections with two different [Formula: see text]. Moreover, a series of solutions of the source parameters will be obtained when a number of [Formula: see text] is selected, which can make the results more reasonable. The method is stable and can be directly applied to noisy anomalies or high-order derivatives because it is based on magnetic anomalies of upward continuation. In addition, the method is flexible, as we can select different [Formula: see text] as desired. Moreover, the method can be applied to multisource cases and can simultaneously estimate the depth and structural index for each source. The method is tested on noise-free and noise-corrupted synthetic magnetic anomalies. In all cases, the VDMI method effectively estimates the depths and structural indices of the sources. The VDMI method is also applied to real aeromagnetic data from the Hamrawein area, Egypt, and ground magnetic data over the Neibei Farm of Heilongjiang Province, China.