Higher-order Equations Research Articles

Reinforcement Learning for Efficient Identification of Soliton System Parameters Across Expansive Domains

Abstract Optical solitons in mode-locked fiber lasers and optical communication links have a wide range of applications. A crucial step in studying the transmission modes of optical solitons is to investigate the relationship between equation parameters and soliton evolution using deep learning techniques. However, current identification models have a limited parameter domain search range and are greatly influenced by initialization, often leading to divergence towards incorrect parameter values. This research harnesses reinforcement learning to revamp the iterative process of the parameter identification model. Through the development of a two-stage optimization strategy, the model is capable of conducting an accurate parameter search across arbitrary domains. The investigation encompasses a series of experiments on various standard and higher-order equations, illustrating that the innovative model overcomes the impact of initialization on the parameter search, and the identified parameters are guided towards their correct values. The enhanced model markedly improves experimental efficiency and holds significant promise for advancing the research of soliton propagation dynamics and addressing intricate scenarios.

Higher-order fractional equations and related time-changed pseudo-processes

We study Cauchy problems of fractional differential equations in both space and time variables by expressing the solution in terms of “stochastic composition” of the solutions to two simpler problems. These Cauchy sub-problems respectively concern the space and the time differential operator involved in the main equation. We provide some probabilistic and pseudo-probabilistic applications, where the solution can be interpreted as the pseudo-transition density of a time-changed pseudo-process. To extend our results to higher order time-fractional problems, we introduce stable pseudo-subordinators as well as their pseudo-inverses. Finally, we present our results in the case of more general differential operators and we interpret the results by means of a linear combination of pseudo-subordinators and their inverse processes.

APOS Framework Didactize Equivalence Linear Simultaneous Equations in Senior High School Mathematics

Many students followed rote-learned procedural rules without thinking about the meaning of quadratic equations, and the Chief examiners’ reports attributed the trend to poor teaching methods. In certain instances, candidates cannot use even the conventional methods to factorize quadratics equations due to a lack of understanding of the zero-product property in graphing, factorizing, and completing the squares or quadratic formula. In the worst circumstances, the candidates failed to scaffold higher-order quadratic equations using the Conjugale and Equivalence Linear Simultaneous Equation methods. However, the action, process, object, and schema theory which came out of the constructivist learning theory and created by Dubinsky can be applied to teach mathematics. With the aid of the APOS framework, this study sought to digitize the Equivalence Linear Simultaneous Equations in senior high schools. In this mixed-method embedded design, there were 286 first-year students selected from one school. All the students received four phases of the APOS framework. The four phases were collected based on the Actions for Factorization, Processes for Quadratic Formula, Object for Conjugale, and Schema for ELSE method. The data, which was both qualitatively and quantitatively, was analyzed using IBM SPSS Statistics (Version 26) deterministic analytics software. The data collection covered four contact periods of a total duration of four hours. The results showed that students in the Actions and Processes were not statistically significant. However, the results were statistically significant at the Object and Schema phases. It was concluded that students’ learning through the APOS framework improved their academic performance. The positive effect was the triumphant mental constructions involving encapsulation and interiorization in the conjugale and ELSE methods. It was, therefore, recommended that the framework be promoted to find solutions to many other complex mathematics problems.

Data-driven solutions and parameter estimations of a family of higher-order KdV equations based on physics informed neural networks

Physics informed neural network (PINN) demonstrates powerful capabilities in solving forward and inverse problems of nonlinear partial differential equations (NLPDEs) through combining data-driven and physical constraints. In this paper, two PINN methods that adopt tanh and sine as activation functions, respectively, are used to study data-driven solutions and parameter estimations of a family of high order KdV equations. Compared to the standard PINN with the tanh activation function, the PINN framework using the sine activation function can effectively learn the single soliton solution, double soliton solution, periodic traveling wave solution, and kink solution of the proposed equations with higher precision. The PINN framework using the sine activation function shows better performance in parameter estimation. In addition, the experiments show that the complexity of the equation influences the accuracy and efficiency of the PINN method. The outcomes of this study are poised to enhance the application of deep learning techniques in solving solutions and modeling of higher-order NLPDEs.

Well-posedness for the NLS hierarchy

We prove well-posedness for higher-order equations in the so-called NLS hierarchy (also known as part of the AKNS hierarchy) in almost critical Fourier–Lebesgue spaces and in modulation spaces. We show the jth equation in the hierarchy is locally well-posed for initial data in H^rs(R)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\hat{H}}}^s_r({\\mathbb {R}})$$\\end{document} for s≥j-1r′\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$s \\ge \\frac{j-1}{r'}$$\\end{document} and 1<r≤2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$1 < r \\le 2$$\\end{document} and also in M2,ps(R)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$M^s_{2, p}({\\mathbb {R}})$$\\end{document} for s=j-12\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$s = \\frac{j-1}{2}$$\\end{document} and 2≤p<∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$2 \\le p < \\infty $$\\end{document}. Supplementing our results with corresponding ill-posedness results in Fourier–Lebesgue and modulation spaces shows optimality. Using the conserved quantities derived in Koch and Tataru (Duke Math J 167(17), 3207–3313, 2018), we argue that the hierarchy equations are globally well-posed for data in Hs(R)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H^s({\\mathbb {R}})$$\\end{document} for s≥j-12\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$s \\ge \\frac{j-1}{2}$$\\end{document}. Our arguments are based on the Fourier restriction norm method in Bourgain spaces adapted to our data spaces and bi- and trilinear refinements of Strichartz estimates.

Long-time dynamics of a random higher-order Kirchhoff model with variable coefficient rotational inertia

This paper delves into the stochastic asymptotic behavior of a non-autonomous stochastic higher-order Kirchhoff equation with variable coefficient rotational inertia. The equation is solved using the Galerkin method, and a stochastic dynamical system is established on this basis. Uniform estimation demonstrates a family of Dk−absorbing sets in the stochastic dynamical system Φk, and the asymptotic compactness of Φk is proved via decomposition. Finally, the family of Dk−random attractors is acquired for the stochastic dynamical system Φk in Vm+k(Ω)×Vm+kb0(Ω). These results improve and extend those in recent literature (Lv et al., 2021). The findings promote the relevant conclusions of the non-autonomous stochastic higher-order Kirchhoff model and provide a theoretical basis for its subsequent application and research.

The Riemann–Hilbert Approach to the Higher-Order Gerdjikov–Ivanov Equation on the Half Line

The Fokas method exhibits remarkable versatility in solving boundary value problems associated with both linear and nonlinear partial differential equations, particularly when conventional approaches encounter challenges in handling intricate boundary conditions. The existing literature often lacks the incorporation of unconventional boundary conditions, and this study addresses this issue by extending the application of the Fokas method to the higher-order Gerdjikov-Ivanov equation on the half line (−∞,0]. We have demonstrated the exclusive representation of the potential function u(z,t) in the higher-order Gerdjikov–Ivanov equation through the solution of a Riemann–Hilbert problem. The characteristic function is partitioned on the complex plane, and we obtain the jump matrix between each partition based on the positive and negative values of the partition as well as the spectral matrix determined by the initial data and boundary value data. The findings suggest that the spectral functions are not mutually independent; instead, they conform to a global relationship. The novel aspect of this study is the application of the Fokas method to a previously unexplored case, contributing to the theoretical and practical understanding of complex partial differential equations and filling a gap in the treatment of boundary conditions.

Chaotic systems based on higher-order oscillatory equations

This paper discusses the design process toward new lumped chaotic systems that originates in higher-order ordinary differential equations commonly used as description of ideal oscillators. In investigated third-order case, two chaotic oscillators were constructed. These systems are dual in the sense of vector field geometry local to fixed points. The existence of robust chaos was proved by both standard routines of numerical analysis and practical measurement. For the fourth-order oscillatory equation, the concept based on interaction between superinductor and supercapacitor was examined in detail. Since both “superelements” are active, the nonlinearity essential to the evolution of chaos is fully passive. It is demonstrated that complex motion is robust and does not represent long transient behavior or numerical artefact. The existence of chaos was verified using standard quantifiers of the flow, such as the largest Lyapunov exponents, recurrence plots, approximate entropy and sensitivity calculation. A good final agreement between theoretical assumptions and practical results will be concluded, on a visual comparison basis.

On Accuracy of the Lattice Boltzmann Equations of Low and High Orders as Applied to Slow Isothermal Microflows

On Accuracy of the Lattice Boltzmann Equations of Low and High Orders as Applied to Slow Isothermal Microflows

Assessment of Health Values, Beliefs, Norms, and Behavior towards Consumption Intention of 3D-Bioprinted Meat.

Continuous innovation in product development further enhances consumer appeal and contributes to a more sustainable and ethical food system. This study used the health belief model (HBM) and value-belief-norm (VBN) theory to investigate the customer perceptions of and intentions towards 3D-bioprinted meat. Specifically, this study examined consumer behavior factors using higher-order partial least squares structural equation modeling (PLS-SEM). Data were collected from 738 meat consumers through online survey questions, distributed among social groups and face-to-face distribution-limiting only to respondents who are familiar with 3D-bioprinted meats. Using a filtering question, only those who are familiar with and have knowledge of the topic were considered valid respondents. Based on the results, all variables under the integrated theories were deemed significant. Consumers' perceptions of 3D-bioprinted meat are also shaped by altruism, egoism, biospheric concern, and willingness to change. The findings revealed that buyers rationally choose benefits over social or personal values. The study emphasized educating consumers, being transparent about production, and constantly innovating for higher acceptance of 3D-bioprinted meat. In order to foster consumer confidence, it is essential to prioritize transparency in the production process, encompassing information regarding sourcing and manufacturing methods. Certifications that validate safety and quality standards serve to reinforce this notion. In addition, the implementation of competitive pricing strategies has the potential to enhance the accessibility of 3D-bioprinted meat, whereas industry partnerships can aid in distribution operations and improve market visibility-all of which extend the practical implications developed for this study. Moreover, the foundation of the integrated framework promotes its extension and application outside technology-based meat production. This could also be considered and utilized among other studies on developed food and food consumption.

One-sweep moment-based semi-implicit-explicit integration for gray thermal radiation transport

Thermal radiation transport (TRT) is a time dependent, high dimensional partial integro-differential equation. In practical applications such as inertial confinement fusion, TRT is coupled to other physics such as hydrodynamics, plasmas, etc., and the timescales one is interested in capturing are often much slower than the radiation timescale. As a result, TRT is treated implicitly, and due to its stiffness and high dimensionality, is often a dominant computational cost in multiphysics simulations. Here we develop a new approach for implicit-explicit (IMEX) integration of gray TRT in the deterministic SN setting, which requires only one sweep per stage, with the simplest first-order method requiring only one sweep per time step. The partitioning of equations is done via a moment-based high-order low-order formulation of TRT, where the streaming operator and first two moments are used to capture the asymptotic stiff regimes of the streaming limit and diffusion limit. Absorption-reemission is treated explicitly, and although stiff, is sufficiently damped by the implicit solve that we achieve stable accurate time integration without incorporating the coupling of the high order and low order equations implicitly. Due to nonlinear coupling of the high-order and low-order equations through temperature-dependent opacities, to facilitate IMEX partitioning and higher-order methods, we use a semi-implicit integration approach amenable to nonlinear partitions. Results are demonstrated on thick Marshak and crooked pipe benchmark problems, demonstrating orders of magnitude improvement in accuracy and wallclock compared with the standard first-order implicit integration typically used.

A multi-orbital Hund´s rules-based Ionic Hamiltonian for transition metal atoms: High-order Equation of Motion Method approach and Kondo resonances.

A multi-orbital ionic Hamiltonian is presented to analyze the many-body properties of the d-transition metal atoms. This Hamiltonian considers all the atomic states obeying the first Hund´s rule and also includes all orbital degeneracy, as well as the interaction of the atom with a metal. We analyze the solution of this ionic Hamiltonian by means of the Equation of Motion (EOM) method up to the fourth order, V4, in the atom-metal interaction. Equations for the appropriate Green-functions for analyzing the chemical and transport properties of the system are given for different atom occupancies. In particular, we introduce a full analysis of the multi-orbital Hamiltonian including atomic configurations with N, N+1 and N-1 electrons, and discuss its Kondo properties. The shells d1, d2 and d3 are analyzed in detail and Kondo energies are deduced in all these cases showing good agreement with the conventional known results.&#xD.

Superelastic electromechanical behaviors in ferroelectric PbTiO[formula omitted] ceramics under coupled mechanical-electric fields: Higher-order piezoelectric constitutive equations from first-principles

With the recent discovery of superelastic behaviors of ferroelectric (piezoelectric) ceramics, understanding their non-linear mechanical and electromechanical responses under high stress and electric field conditions is crucial for engineering and designing advanced piezoelectric devices, such as ultrasmall actuator systems. In this study, we carry out first-principles finite electric field calculations to clarify the mechanical and electromechanical properties under the electric field for typical ferroelectric ceramics of PbTiO3. Superelastic-like non-linear deformation behavior is enhanced or suppressed depending on the electric field direction, suggesting the possible design of superelasticity. We clarified that positive and negative electric fields to spontaneous polarization promote and inhibit deformation due to strain loading, thereby providing tunable superelasticity. We also investigate the mechanical loading and electric field dependence of the piezoelectric coefficient of PbTiO3. The tunable piezoelectric response can be provided with a coupling of mechanical loading and electric field. Finally, we present higher-order piezoelectric constitutive equations applicable for superelastic-like non-linear deformations under an electric field. The present results will open promising paradigms for functional piezoelectric devices, and the proposed piezoelectric constitutive equations broaden the design scope through finite element method analysis.

Spatial positioning method based on range-only measurement of multi-station radar

Aiming at the characteristics that affect the high ranging accuracy of shipborne radar, a multi station radar spatial positioning method based on ranging is proposed. This method is based on the ECEF (Earth Centered Earth Fixed) coordinate system, and adopts the method of intersecting the detection distances of multiple radars to the target, and directly calculates the spatial position of the target. This method only needs the geodetic coordinates of each radar station and the detection range of the target to calculate the geodetic coordinates of the target, which can effectively solve the coordinate conversion problem caused by the curvature of the earth, and does not involve higher-order equations and multi-value solutions, and it does not involve iterative initial value calculation. The positioning system can not only obtain high positioning accuracy, but also has a long operating distance, which can better solve the multi-radar range measurement intersection positioning problem.

Asymptotical convergence of solutions of boundary value problems for singularly perturbed higher-order integro-differential equations

The article considers a two-point boundary value problem for a linear integro-differential equation of the n + m order with small parameters for m higher derivatives, provided that the roots of the additional characteristic equation are negative. The aim of the study is to obtain asymptotic estimates of the solution, to find out the asymptotic behavior of solutions in the vicinity of points where additional conditions are set, as well as to construct a degenerate problem, the solution of which tend to the solution of the initial perturbed boundary value problem. The Cauchy function and boundary functions of a boundary value problem for a singularly perturbed homogeneous differential equation are constructed, and their asymptotic estimates are obtained. Using the Cauchy functions and boundary functions, an analytical formula for solutions to the boundary value problem is obtained. A theorem on an asymptotic estimate for the solution of the considered boundary value problem is proved. The asymptotic behavior of the solution with respect to a small parameter and the order of growth of its derivatives are established. It is shown that the solution of the boundary value problem under consideration at the left end of this segment has the phenomenon of an initial jump and the order of this jump is determined. A modified degenerate boundary value problem containing initial jumps of the solution and the integral term is constructed.

On Higher-Order Nonlinear Fractional Elastic Equations with Dependence on Lower Order Derivatives in Nonlinearity

The paper studied high-order nonlinear fractional elastic equations that depend on low-order derivatives in nonlinearity and established the existence and uniqueness results by using the Leray–Schauder alternative theorem and Perov’s fixed point theorem on an appropriate space under mild assumptions. Examples are given to illustrate the key results.

Existence results for the higher-order Q-curvature equation

Existence results for the higher-order Q-curvature equation

Dirichlet Problem for Nonlinear Higher-Order Equations in Upper Half Plane

In this article, we consider the Dirichlet problem for nonlinear higher-order equations in upper half plane. Firstly we introduce the solutions of inhomogeneous polyanalytic equation in upper half plane. Then we investigate the properties of relevant integral operators. Lastly we transform the Dirichlet problem for nonlinear higher-order equations in upper half plane into the system of integro-differential equations and we obtain the existence of unique solution using Banach fixed point theorem.

Measurement of phase velocity in the functionally graded concrete systems via improved higher-order shear deformation theory

The measurement of phase velocity (PV) in concrete systems is critically important for civil engineers as it provides essential insights into the material properties and structural integrity of concrete. PV, which refers to the speed at which a wave phase propagates through a medium, is directly related to the stiffness and density of the material. By accurately measuring PV, civil engineers can assess the quality and homogeneity of concrete, detect potential flaws, and predict the material’s behavior under various load conditions. This information is crucial for ensuring the safety and durability of concrete structures, from bridges and buildings to pavements and dams. Moreover, understanding PV helps in the optimization of mix designs and the development of new, advanced concrete materials, ultimately leading to more resilient and cost-effective construction practices. For this issue, in this work, for the first time, the measurement of PV in the concrete doubly curved panel reinforced by graphene oxide powders (GOPs) via improved higher-order shear deformation theory is presented. Hamilton’s principle and analytical method are used for extracting and solving the higher-order equations. Finally, some recommendations for improving the efficiency and stability of the concrete structures are presented for related engineering industries. The findings underscore the potential of combining state-of-the-art theoretical models with cutting-edge material reinforcements to push the boundaries of structural engineering and material performance.

The effectiveness of voluntary collective action in China's rural land development

China is gradually and directly making rural collective land available on the land market. It is important to understand how stakeholders collectively treat and manage this land, and if this collective action leads to effective and sustainable results. This paper, however, revises Ostrom's framework of collective action by constructing the joint analysis of social capital, trust, and cooperation performance, in order to better capture the distinction between voluntary action and enforced action of stakeholders towards a sustainable goal. We analyze the role of voluntary collective action through higher-order structural equation modeling (HSEM), applied to a database consisting of 324 households in Jiangsu Province, China, collected in 2022 and 2023. The results suggest that social capital effectively promotes villagers' trust in policy implementers by establishing new institutions, increasing actors' trustworthiness, and improving channels of information exchange, and the first two means are more effective than the latter. We also demonstrate the significant positive effect of trust on cooperation performance, with trusting relationships with familiar groups and the credible behavior of policy implementers affecting performance outcomes of collective action. Moreover, social capital drives cooperation performance through the mediating effect of mutual trust that it breeds. With this revised framework and verified hypotheses one can thus create more effective policies to avoid inequality and informality in the land marketization process in China.