Equations In Spaces Research Articles

Solutions to the Landau–Lifshitz–Gilbert equation in the frequency space: Discretization schemes for the dynamic-matrix approach

The dynamic matrix method addresses the Landau–Lifshitz–Gilbert (LLG) equation in the frequency domain by transforming it into an eigenproblem. Subsequent numerical solutions are derived from the eigenvalues and eigenvectors of the dynamic matrix. In this work we explore discretization methods needed to obtain a matrix representation of the dynamic operator, a fundamental counterpart of the dynamic matrix. Our approach opens a new set of linear algebra tools for the dynamic matrix method and expose the approximations and limitations intrinsic to it. Moreover, our discretization algorithms can be applied to various discretization schemes, extending beyond micromagnetism problems. We present some application examples, including a technique to obtain the dynamic matrix directly from the magnetic free energy function of an ensemble of macrospins, and an algorithmic method to calculate numerical micromagnetic kernels, including plane wave kernels. We also show how to exploit symmetries and reduce the numerical size of micromagnetic dynamic-matrix problems by a change of basis. This procedure significantly reduces the size of the dynamic matrix by several orders of magnitude while maintaining high numerical precision. Additionally, we calculate analytical approximations for the dispersion relations in magnonic crystals. This work contributes to the understanding of the current magnetization dynamics methods, and could help the development and formulations of novel analytical and numerical methods for solving the LLG equation within the frequency domain.

Fredholm integral equation in composed-cone metric spaces

AbstractThe current paper introduces a novel generalization of cone metric spaces called type I and type II composed cone metric spaces. Therefore, examples of a type I and type II composed cone metric space, which is not a cone metric space, are given. We establish some results of fixed point precisely about Hardy–Rogers type contraction on C2CMS and provide examples. Finally, we present an application of our results and how our results solve the Fredholm integral equation of generalizing several existing and unique fixed point theorems.

Bifurcation of solutions of Ornstein–Zernike equation in the gas–liquid coexistence region of simple fluids

One objective of this paper is to investigate bifurcation of solutions of Ornstein–Zernike equation in liquid state theory for the gas–liquid coexistence region of simple fluids. The analysis uses a discrete form of the Ornstein–Zernike equation in real space, together with a closure relation which provides nearly self consistent thermodynamic properties of fluids governed by interaction potentials like the Lennard-Jones potential. Accurate asymptotic forms of correlation functions are incorporated in the real space algorithm so as to mitigate the influence of cutoff length used in the discrete approximations. The global error reduction of this algorithm is similar to that of Simpson’s rule. It is found that there is no spinodal on the vapor side on a sub-critical isotherm. However, there is a density at which compressibility vanishes, but this lies on a nonphysical branch. There are fold bifurcation points on the vapor and liquid sides together with a ‘no-real-solution-region’ in between. This is followed by a region of negative compressibility extending up to the spinodal on the liquid side. Importantly, it is found that in regions where compressibility is non-positive, there are infinite number of solutions to OZE which violate an integral constraint that is needed for applications to thermodynamics. The emerging picture is somewhat similar to that in the case of the hypernetted-chain-closure. It is also found that complex solutions connect the physical solutions that exist outside the ‘no-real-solution-region’ as well as in region of negative compressibility. The second objective of the paper, in the light of bifurcation scenario, is to evolve an interpretation of the coupling-parameter expansion of correlation functions pertaining to the region of negative compressibility. It is shown that the expansion provides an accurate description of results of the restricted ensembles, implied in mean field theories and simulation methods. Bifurcation of singlet density to an in-homogeneous distribution at a spinodal is also established using a simple equation, which uses the direct correlation function. Thus the paper re-establishes the fact that restriction to a homogeneous singlet density is the root cause of all inconsistencies found in the gas–liquid coexistence region.

On the Stability of Hyers Orthogonality Functional Equations in Non-Archimedean Spaces

On the Stability of Hyers Orthogonality Functional Equations in Non-Archimedean Spaces

Kinetic equation for stochastic vector bundles

The kinetic equation is crucial for understanding the statistical properties of stochastic processes, yet current equations, such as the classical Fokker–Planck, are limited to local analysis. This paper derives a new kinetic equation for stochastic systems on vector bundles, addressing global scale randomness. The kinetic equation was derived by cumulant expansion of the ensemble-averaged local probability density function, which is a functional of state transition trajectories. The kinetic equation is the geodesic equation for the probability space. It captures global and historical influences, accounts for non-Markovianity, and can be reduced to the classical Fokker–Planck equation for Markovian processes. This paper also discusses relative issues concerning the kinetic equation, including non-Markovianity, Markov approximation, macroscopic conservation equations, gauge transformation, and truncation of the infinite-order kinetic equation, as well as limitations that require further attention.

Physics-informed neural network for lithium-ion battery degradation stable modeling and prognosis

Accurate state-of-health (SOH) estimation is critical for reliable and safe operation of lithium-ion batteries. However, reliable and stable battery SOH estimation remains challenging due to diverse battery types and operating conditions. In this paper, we propose a physics-informed neural network (PINN) for accurate and stable estimation of battery SOH. Specifically, we model the attributes that affect the battery degradation from the perspective of empirical degradation and state space equations, and utilize neural networks to capture battery degradation dynamics. A general feature extraction method is designed to extract statistical features from a short period of data before the battery is fully charged, enabling our method applicable to different battery types and charge/discharge protocols. Additionally, we generate a comprehensive dataset consisting of 55 lithium-nickel-cobalt-manganese-oxide (NCM) batteries. Combined with three other datasets from different manufacturers, we use a total of 387 batteries with 310,705 samples to validate our method. The mean absolute percentage error (MAPE) is 0.87%. Our proposed PINN has demonstrated remarkable performance in regular experiments, small sample experiments, and transfer experiments when compared to alternative neural networks. This study highlights the promise of physics-informed machine learning for battery degradation modeling and SOH estimation.

Some Aspects of Maxwell’s Equations, Klein-Gordon Equations, and Heat and Mass Transfer Equations in an n-Dimensional Maximally Symmetric Space-Time from the Classical and Quantum Mechanical Standpoints

This manuscript examines Maxwell’s equations, Klein-Gordon equations, and heat and mass transfer equations in n-dimensional maximally symmetric space-time. It investigates these equations in spherical and hyperbolic spaces embedded in higher-dimensional Euclidean and Minkowski spaces. The study focuses on the implications of these geometries and symmetries on the behaviour of the equations, highlighting how specific transformations and parametrizations impact their solutions. The findings reveal the underlying connections between geometric symmetries and physical laws, providing insights into their possible applications in theoretical physics. We touch upon both classical and quantum mechanical aspects of density and velocity evolutions with time in the universe. Quantum mechanical aspects of single and two-particle state evolution and statistical moments of the matter four-current are derived from the quantum Boltzmann equation and Feynman’s path integral method for fields applied to gravity interacting with electrons and positrons.

Trajectory tracking control of the pushing ore truck manipulator based on adaptive sliding mode control with interference observer

An improved sliding mode controller was designed to solve the problem of low tracking accuracy caused by various disturbances and errors existing in the hydraulic-driven manipulator for pushing ore truck. The state space equation of the hydraulic system of the pushing ore truck manipulator was modeled and derived, and a sliding mode function was designed. For interference signals, use interference observers for observation. For the unobserved parts, adaptive laws are used for compensation to reduce system chattering, and system’s stability is verified using Lyapunov functions. The simulation results show that, compared with the traditional Sliding mode control and adaptive sliding mode control, the adaptive sliding mode control strategy based on interference observer proposed in this paper can effectively weaken the system chattering, reduce the uncertainty caused by external interference and modeling error, and improve the trajectory tracking accuracy of the pushing ore truck manipulator.

Some results on a nonlinear fractional equation with nonlocal boundary condition

The aim of this paper is to derive sufficient conditions for the existence, uniqueness, and Hyers–Ulam stability of solutions to a new nonlinear fractional integro‐differential equation with functional boundary conditions, using several fixed‐point theorems, the multivariate Mittag‐Leffler function and Babenko's approach. A few examples are also presented to illustrate the applications of our results based on approximate values of a couple of Mittag‐Leffler functions calculated by Python codes. Furthermore, the approaches used have a wide range of applications to various fractional differential equations with initial or boundary conditions or integral equations in complete spaces.

Interior solution of azimuthally symmetric case of Laplace equation in orthogonal similar oblate spheroidal coordinates

Curvilinear coordinate systems distinct from the rectangular Cartesian coordinate system are particularly valuable in the field calculations as they facilitate the expression of boundary conditions of differential equations in a reasonably simple way when the coordinate surfaces fit the physical boundaries of the problem. The recently finalized orthogonal similar oblate spheroidal (SOS) coordinate system can be particularly useful for a physical processes description inside or in the vicinity of the bodies or particles with the geometry of an oblate spheroid. The solution of the azimuthally symmetric case of the Laplace equation was found for the interior space in the orthogonal SOS coordinates. In the frame of the derivation of the harmonic functions, the Laplace equation was separated by a special separation procedure. A generalized Legendre equation was introduced as the equation for the angular part of the separated Laplace equation. The harmonic functions were determined as relations involving generalized Legendre functions of the first and of the second kind. Several lower-degree functions are reported. Recursion formula facilitating determination of the higher-degree harmonic functions was found. The general solution of the azimuthally symmetric Laplace equation for the interior space in the SOS coordinates is reported.

Multi-USV cooperative oil spill source seeking via extremum seeking combined with information fusion

This paper addresses the issue of cooperative source seeking for oil spill pollution by multiple unmanned surface vehicles (USVs), where the oil spill plume propagation is modeled by a linear advection–diffusion equation in two-dimensional space. The main objective is to construct a cooperative source seeking scheme for a group of USVs in the framework of extremum seeking combined with information fusion to solve such cooperative source seeking problem. To do this, both a gradient-based control law and a formation control law are included in the cooperative source seeking scheme. The gradient-based control law that is constructed by extremum-seeking control combined with information fusion drives the multiple USVs to approach the oil spill source, where the gradient is estimated online via the extremum-seeking control and the fused concentration value that is derived via a linear H∞ filter from the local concentration measurements at the USV’s current position. In the source seeking process, the USVs driven by the formation control law based on the virtual structure method maintain a preset circular formation. The convergence performance of the cooperative source seeking scheme is analyzed rigorously by the Lyapunov method. Finally, extensive numerical simulations are included to validate the theoretical algorithm.

Fractional stochastic Schrödinger evolution system with complex potential and poisson jumps: Qualitative behavior and T-controllability

Establishing the fractional stochastic Schrödinger equations (FSSEs) in Hilbert space with complex potential and Poisson jumps is the primary goal of this work. Stochastic analysis, Mönch fixed point theorem, fractional calculus, and semigroup theory all support the possibility of a moderate solution for the suggested system. We construct and prove the criteria for the mild solution’s exponential fall. The trajectory (T-) controllability for the fractional Schrödinger equation is established in the latter portion, extending the findings of Durga and Muthukumar (2023); Grecksch and Lisei (2011); Keller (2013); Wang et al. (2012). An example is given to verify the theoretical findings gained. We discuss the use of fractional Schrödinger’s equation in the Oil and Gas Industry.

Tractability of linear ill-posed problems in Hilbert space

We introduce a notion of tractability for ill-posed operator equations in Hilbert space. For such operator equations the asymptotics of the best possible rate of reconstruction in terms of the underlying noise level is known in many cases. However, the relevant question is, which level of discretization, again driven by the noise level, is required in order to achieve this best possible accuracy. The proposed concept adapts the one from Information-based Complexity. Several examples indicate the relevance of this concept in the light of the curse of dimensionality.

Thermodiffusively unstable laminar hydrogen flame in a sufficiently large 3D computational domain – Part II: NOx formation mechanism and flamelet modeling

In this work, the NOx formation mechanism in three-dimensional (3D) thermodiffusively unstable premixed hydrogen flames is investigated through direct numerical simulations (DNS) and a reaction path analysis based on the DNS dataset. In part I of this study (Wen et al., 2024), the characteristic patterns of the instabilities were studied using the same dataset. Here, first, the effects of the computational setup (2D vs. 3D) and curvature on the flux ratio of nitrogen atom are quantified. In addition, the composition space model (CSM) proposed in previous works is extended to account for the NOx chemical reaction mechanism to predict NOx formation in thermodiffusively unstable premixed hydrogen flames. The flamelet solutions are obtained from the premixed flamelet equations in composition space so that the wide range of curvatures associated with the strongly corrugated flame front can be considered. The performance of the CSM in predicting the NOx species and the important radicals involved in the NOx formation pathways is evaluated through an a priori analysis. To investigate the effects of curvature on the NOx formation pathways and the performance of the composition space model, the positively- and negatively-curved regions in the DNS are studied separately. The results show that different from the 2D simulation, the NNH reaction pathway becomes dominant in the 3D simulation due to the increased range of curvature, which promotes the accumulation of the highly diffusive H radical in the positively-curved regions. The NNH reaction pathway is dominant in the positively-curved regions, while the N2O reaction pathway is more important in the negatively-curved regions. The flamelet model based on the composition space solutions that consider the effects of curvature yields accurate predictions for the radicals that are sensitive to the effects of curvature.Novelty and significance statementThe novelty of the present work includes the following aspects,(i) Analysis of NOx formation mechanism based on the large-scale 3D DNS dataset of a laminar fuel-lean premixed hydrogen flame on a sufficiently large domain;(ii) Quantification of the effects of computational setup (2D and 3D) and curvature on the NOx formation pathways;(iii) Extension of the previous composition space model (CSM) by incorporating the NOx chemical reaction mechanism to consider the effects of curvature on the NOx formation.

The classical double copy in curved spacetimes: perturbative Yang-Mills from the bi-adjoint scalar

We formulate a version of the double copy for classical fields in curved spacetimes. We provide a correspondence between perturbative solutions to the bi-adjoint scalar equations and those of the Yang-Mills equations in position space. At the linear level, we show that there exists a map between these solutions for maximally symmetric spacetime backgrounds, that provides every Yang-Mills solution by the action of an appropriate differential operator on a bi-adjoint scalar solution. Given the existence of a linearized map, we show that it is possible to cast the solutions of the Yang-Mills equations at arbitrary perturbation order in terms of the corresponding bi-adjoint scalar solutions. This all-order map is reminiscent of the flat space BCJ double copy, and works for any curved spacetime where the perturbative expansion holds. We show that these results have the right flat space limit, and that the correspondence is agnostic to the choice of gauge.

Convergence problem of the generalized Kadomtsev–Petviashvili II equation in anisotropic Sobolev space

Convergence problem of the generalized Kadomtsev–Petviashvili II equation in anisotropic Sobolev space

Analytical solution of forced vibration of rectangular plates with part through surface crack based on wave propagation method

An analytical wave propagation framework is developed in this study for steady state forced vibration of plates with part through surface cracks for the first time. Rectangular thin plates with partial-width and full-width cracks are considered. The analytical waves are obtained by solving the dual equations of the plate established considering the line–spring model (LSM) of crack. The dual equations formed here are characterized by simultaneously solving kinematic variables and their dual dynamic variables. Both conventional and refined LSMs are considered. Two ways are proposed to generate the system equation in wave space based on the relationships of wave propagation, wave reflection and wave scattering. Compared with the traditional methods, most of them can only give exact analytical solutions under the condition of opposite-side simply supported boundaries, the developed analysis framework can treat arbitrary combinations of simple boundary condition of the cracked plate. Comparisons of results of the proposed method with those from the literature and finite element method (FEM) validate the effectiveness of the proposed method. A series of parametric studies for investigating the effects of length, position and depth of crack on the forced response of cracked plate are also presented. The proposed framework are effective for predicting the forced responses and natural frequencies of thin plates with part-through surface cracks.

Meshless track assimilation (MTA) of 3D PTV data

We propose a data assimilation algorithm to improve the resolution of 3D particle tracking velocimetry (PTV) data and, as a byproduct, it estimates the pressure field. The proposed method, called meshless tracking assimilation (MTA), does not rely on a mesh to compute derivatives. Instead, it exploits the analytical formulation of the obtained field to extract the gradients of the involved quantities in arbitrary points. MTA actively constraints PTV data points to satisfy the Navier–Stokes equations in both time and space at user-defined spatial locations. Three test cases are investigated. The first one is a synthetic dataset reconstructed from a direct numerical simulation and it serves to ensure that the method is properly implemented and working. The second one is the 2020 data assimilation challenge which gives a qualitative comparison with other data assimilation methods. Finally, MTA is used on a Tomographic PTV experiment of a Jet Flow with ReD≃3000 to ensure it can be implemented successfully on experimental data. The first and third test cases show a significant performance boost compared to the classic linear interpolation. This improvement allows for a higher level of structural resolution without sacrificing the temporal and spatial smoothness of the solution. The second test case generally aligns well with other state-of-the-art data assimilation techniques in terms of performance, but there are some noticeable differences.

Homogenized color-gradient lattice Boltzmann model for immiscible two-phase flow in multiscale porous media

In this paper, a homogenized multiphase lattice Boltzmann (LB) model is established for parallelly simulating immiscible two-phase flow in both solid-free regions (pore scale) and porous areas (continuum scale). It combines the color-gradient multiphase model with the Darcy–Brinkman–Stokes method by adding a term that includes surface force and drag force of porous matrix to multiple-relaxation-time LB equation in moment space. Moreover, an improved algorithm is proposed to characterize and implement the apparent wettability in the locally homogenized porosity field. Validations and test cases are given to demonstrate the accuracy and robustness of this new model, as well as its applicability for trans-scale fluid simulation of transport and sorption behavior from porous (Darcy flow) area to free (Stokes flow) area. For practicality, the two-phase seepage flow in a composite rock structure with multiscale pores is simulated by this new model, and the effects of viscosity ratio and wettability on the displacement process are discussed.

Existence and regularity of mild solutions to backward problem for nonlinear fractional super-diffusion equations in Banach spaces

Existence and regularity of mild solutions to backward problem for nonlinear fractional super-diffusion equations in Banach spaces