Equations In Spaces Research Articles

Analysis of the three possible event horizons of anti-de Sitter space–time in the Klein–Gordon equation

In this paper, we calculate exactly the solution of Klein–Gordon equation in anti-de Sitter space–time. Due to spherical symmetry, the angular wave functions are given by spherical harmonics. The radial wave functions are given by local Heun functions. We analyze the behavior of the radial wave function in regions near three possible event horizon positions [Formula: see text] and investigate the decay rate, Hawking radiation and Hawking temperature for these three cases. Comparing the results, we verify that if the decay rate and Hawking radiation are greater, then the closer the event horizon is to the singularity of the black hole. Finally, we analyze the special case of a small black hole limit.

On the Large-x Asymptotic of the Classical Solutions to the Non-Linear Benjamin Equation in Fractional Sobolev Spaces

In this work, we study the large-x asymptotic of classical solutions to the non-linear Benjamin equation modeling propagation of small amplitude internal waves in a two fluid system. In our analysis, we extend known Hs-well-posedness results to the case of the variable-weight Sobolev spaces. The spaces provide a direct control over the asymptotics of classical solutions and their weak derivatives, and permit us to compute the bulk large-x asymptotic of classical solutions explicitly in terms of input data. The asymptotic formula provides a precise description of the qualitative behaviour of classical solutions in weighted spaces and yields a number of weighted persistence and continuation results automatically.

Optimal decay of the Boltzmann equation

The Boltzmann equation is a typical example of partially dissipative equations, where the linearized collision operator is positive definite with respect to the microscopic part and the dissipation of the hydrodynamic part is discovered from the coupling structure between the transport operator and the linearized collision operator. Guo and Wang (Comm. Partial Differential Equations, 37, 2012) developed a general energy method for proving the optimal time decay rates of the solution to such type of equations in the whole space; however, the decay rate of the highest order spatial derivatives of the solution is not optimal. In this paper, by incorporating the high‐low frequency decomposition in the energy estimates, both linearly and nonlinearly, we prove the optimal decay rates of any high order spatial derivatives of the low frequency part of the solution to the Boltzmann equation and the almost exponential decay rate of the high frequency part, which imply in particular the optimal decay rate of the highest order spatial derivatives of the solution. Moreover, the velocity‐weighted assumption of the initial data required in Guo and Wang (Comm. Partial Differential Equations, 37, 2012) is removed by capturing the time‐weighted dissipation estimates via the time‐weighted energy method. The method can be applied to the compressible Navier–Stokes equations and many partially dissipative equations in kinetic theory and fluid dynamics.

A SOH estimation method of lithium-ion batteries based on partial charging data

Accurate and reliable assessment of lithium-ion battery SOH is critical to improving the safety performance of electric vehicles. In the existing data-driven methods, the models are often complex, the calculation is large and the interpretation ability is not strong, so it is difficult to realize the commercial application. In this paper, a novel SOH estimation method is proposed. Firstly, the incremental capacity (IC) curve is used to extract the aging features from part of the charging data, and a linear battery aging state space equation is established based on the features. The aging state of the battery can be directly observed under the condition of small calculation amount. Second, the aging state of the battery can be tracked and corrected by double Kalman filter to improve estimation accuracy. The performance of the proposed method is verified on CACLE dataset, Oxford dataset and DUT dataset respectively. The estimated errors are all <2.5 %, and the mean RMSE is 0.0066. The results show that the method can be effectively applied to different lithium-ion batteries. In order to further demonstrate the advantages of this estimator, we also compare it with literature methods using the same dataset. In the CACLE data set and Oxford data, the average time of this method is 0.848 s and the average RMSE is 0.0050, which is in a dominant position. The proposed method provides a simple and feasible way to estimate the SOH of electric vehicles.

Integrating the full four-loop negative geometries and all-loop ladder-type negative geometries in ABJM theory

The decomposition of the four-point ABJM amplituhedron into negative geometries produces compact integrands of logarithmic of amplitudes such that the infrared divergence only comes from the last loop integration, from which we can compute the cusp anomalous dimension of the ABJM theory. In this note, we integrate L – 1 loop momenta of the L-loop negative geometries for all four-loop negative geometries and a special class of all-loop ladder-type negative geometries by a method based on Mellin transformation, and from these finite quantities we extract the corresponding contribution to the cusp anomalous dimension. We find that the infrared divergence of a box-type negative geometry at L = 4 is weaker than other negative geometries, then only tree-type negative geometries contribute to the cusp anomalous dimension at L = 4. For the all-loop ladder-type negative geometries, we prove and conjecture some recursive structures as integral equations in Mellin space and find that they cannot contribute zeta values like ζ3, ζ5 to the cusp anomalous dimension.

Global solutions for bi-harmonic inhomogeneous non-linear Schrödinger equations in Lebesgue spaces

This paper studies the inhomogeneous bi-harmonic nonlinear Schrödinger equation iu˙+Δ2u±|x|−τ|u|p−1u=0,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mbox{i}\\dot{u} +\\Delta ^{2} u\\pm |x|^{-\ au}|u|^{p-1}u=0, $$\\end{document} where u:=u(t,x):R×RN→C\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$u:=u(t,x):\\mathbb{R}\ imes \\mathbb{R}^{N}\ o \\mathbb{C}$\\end{document}, τ>0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\ au >0$\\end{document} and p>1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$p>1$\\end{document}. One proves the existence of global solutions with datum in Lebesgue space Lϱ(RN)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$L^{\\varrho}(\\mathbb{R}^{N})$\\end{document}, for a certain 1<ϱ<2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$1<\\varrho <2$\\end{document}. This work complements the known global well-posedness result in the mass-sub-critical regime, namely 1<p<1+8−2τN\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$1< p<1+\\frac{8-2\ au}{N}$\\end{document} and with finite mass. This work aims to develop a local theory in non-Hilbert Lebesgue spaces. Moreover, one complements the local solution to a global one under some suitable assumptions on the parameters. To this end, one uses a Strichartz-type estimate based on Lϱ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$L^{\\varrho}$\\end{document}. Moreover, one breaks down the data into two parts: the first is in L2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$L^{2}$\\end{document}, and the second has a small Strichrtz norm under the free bi-harmonic Schrödinger propagator. Then, one uses a standard fix point argument coupled with Strichartz estimates. First, one resolves the above problem in L2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$L^{2}$\\end{document} and then takes the considered equation with a perturbed source term. The local solution is complemented by a global one with induction. Since the free associated kernel ei⋅Δ2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$e^{\\mbox{i}\\cdot \\Delta ^{2}}$\\end{document} is unitary in L2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$L^{2}$\\end{document}, the Schrödinger equations seem more adapted to be mathematically studied in functional spaces based on L2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$L^{2}$\\end{document}, such as the Sobolev spaces Ws,2(RN)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$W^{s,2}(\\mathbb{R}^{N})$\\end{document}. The main novelty here is investigating the above inhomogeneous bi-harmonic nonlinear Schrödinger problem in some Lebesgue spaces different from L2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$L^{2}$\\end{document}. One essential challenge is overcoming the lack of conservation laws, namely the mass and the energy, representing standard tools in the Schrödinger context. The second technical difficulty is the presence of an inhomogeneous source term, namely the singular decaying term |x|−τ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$|x|^{-\ au}$\\end{document}, which causes some serious complications. In order to deal with the inhomogeneous term, one uses Lorentz spaces with the property |x|−τ∈LNτ,∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$|x|^{-\ au}\\in L^{\\frac {N}{\ au},\\infty}$\\end{document}.

Solutions of differential equations in bicomplex space using Sadik transforms

Solutions of differential equations in bicomplex space using Sadik transforms

A new error estimate of a finite difference scheme for a fractional transport-advection equation with zero order term

In this work, we propose a finite difference scheme for Caputo’s fractional derivative transport equation in time and space, with a zero-order term. A new error estimation of the approximate solution has been demonstrated. By introducing an approximation of the Caputo derivative, we proved that the convergence is of order 2-α in time, and 2-β in space, for 0<α;β<1. Results of conditional stability and convergence of the numerical method are discussed. Finally a numerical implementation was presented to show the conformity between the theoretical and numerical approach of the finite difference scheme.

Rotational Taylor dispersion in linear flows

The coupling between advection and diffusion in position space can often lead to enhanced mass transport compared with diffusion without flow. An important framework used to characterize the long-time diffusive transport in position space is the generalized Taylor dispersion theory. In contrast, the dynamics and transport in orientation space remains less developed. In this work we develop a rotational Taylor dispersion theory that characterizes the long-time orientational transport of a spheroidal particle in linear flows that is constrained to rotate in the velocity-gradient plane. Similar to Taylor dispersion in position space, the orientational distribution of axisymmetric particles in linear flows at long times satisfies an effective advection–diffusion equation in orientation space. Using this framework, we then calculate the long-time average angular velocity and dispersion coefficient for both simple shear and extensional flows. Analytic expressions for the transport coefficients are derived in several asymptotic limits including nearly spherical particles, weak flow and strong flow. Our analysis shows that at long times the effective rotational dispersion is enhanced in simple shear and suppressed in extensional flow. The asymptotic solutions agree with full numerical solutions of the derived macrotransport equations and results from Brownian dynamics simulations. Our results show that the interplay between flow-induced rotations and Brownian diffusion can fundamentally change the long-time transport dynamics.

Research on Longitudinal Control of Electric Vehicle Platoons Based on Robust UKF–MPC

In a V2V communication environment, the control of electric vehicle platoons faces issues such as random communication delays, packet loss, and external disturbances, which affect sustainable transportation systems. In order to solve these problems and promote the development of sustainable transportation, a longitudinal control algorithm for the platoon based on robust Unscented Kalman Filter (UKF) and Model Predictive Control (MPC) is designed. First, a longitudinal kinematic model of the vehicle platoon is constructed, and discrete state–space equations are established. The robust UKF algorithm is derived by enhancing the UKF algorithm with Huber-M estimation. This enhanced algorithm is then used to estimate the state information of the leading vehicle. Based on the vehicle state information obtained from the robust UKF estimation, feedback correction and compensation are added to the MPC algorithm to design the robust UKF–MPC longitudinal controller. Finally, the effectiveness of the proposed controller is verified through CarSim/Simulink joint simulation. The simulation results show that in the presence of communication delay and data loss, the robust UKF–MPC controller outperforms the MPC and UKF–MPC controllers in terms of MSE and IAE metrics for vehicle spacing error and acceleration tracking error and exhibits stronger robustness and stability.

Representations of the solutions for volterra integro-differential equations in hilbert spaces

Representations of the solutions for volterra integro-differential equations in hilbert spaces

Distributed estimation of state-of-charge of supercapacitor packs: A consensus-based approach

It is crucial to estimate the state-of-charge (SoC) of a supercapacitor pack as cells are typically connected to form packs in practical applications. However, most of the existing studies adopted a centralized approach, which is to estimate all the single cells, and then calculate the overall SoC. When the total number of the cells comes to a high level, the burden of communication and the price of a single fault rise rapidly. To address this challenge, in this article, we propose a decentralized method for estimation of pack SoC, which is free of the central observer. We use the state space equation to model the supercapacitors in the physical modeling. In the cyber modeling, we construct their communication topology and apply a definition of pack SoC. A two-layer structure is proposed in which we design a Luenberger observer for each cell to estimate SoC of each cell. We propose a consensus-based algorithm to estimate its pack SoC. A laboratory testbed has been built to verify the effectiveness of the proposed method. Experimental results show that the proposed method can effectively implement the estimation and is more robust to communication failure than the centralized method.

Nonlinear twofold saddle point-based mixed finite element methods for a regularized μ(I)-rheology model of granular materials

We propose and analyze new mixed finite element methods for a regularized μ(I)-rheology model of granular flows with an equivalent viscosity depending nonlinearly on the pressure and the Euclidean norm of the symmetric part of the velocity gradient. To this end, and besides the velocity, the pressure and the aforementioned strain rate, we introduce a modified stress tensor that includes the convective term, and the skew-symmetric vorticity, as auxiliary tensor unknowns, thus yielding a mixed variational formulation within a Banach spaces framework. Then, the pressure is obtained through an iterative postprocess suggested by the incompressibility condition of the fluid, which allows us to express this unknown in terms of the aforementioned stress and the velocity. A fixed-point strategy combined with a solvability result for a class of nonlinear twofold saddle point operator equations in Banach spaces, is employed to show, along with the classical Banach fixed-point theorem, the well-posedness of the continuous and discrete formulations. In particular, PEERS and AFW elements of order ℓ⩾0 for the stress, the velocity, and the skew-symmetric vorticity, and piecewise polynomials of degree ⩽ℓ+n (resp. ⩽ℓ+1) for the strain rate with PEERS (resp. with AFW), yield stable Galerkin schemes. Optimal a priori error estimates are derived and associated rates of convergence are established. Finally, numerical results confirming the latter and illustrating the good performance of the methods, are reported.

Existence of time-periodic strong solutions to the Navier-Stokes equation in the whole space

In this paper, the existence of time-periodic strong solutions to the Navier-Stokes equation in Rn is established under a suitable smallness condition on the external force. Our analysis is based on splitting periodic solutions into steady and purely periodic parts. One advantage of this decomposition is the availability of slightly more regularity in time of the purely periodic part. We apply this property to construct time-periodic solutions of the Navier-Stokes equation with information on the classes of their steady and purely periodic parts. It is also shown that the small solution v constructed in our existence theorem is unique within a class of time-periodic, not necessarily small, solutions having the same integrability properties as v.

Global well-posedness for the higher order non-linear Schrödinger equation in modulation spaces

We consider the initial value problem (IVP) associated with a higher order non-linear Schrödinger (h-NLS) equation∂tu+ia∂x2u+b∂x3u=2ia|u|2u+6b|u|2∂xu,x,t∈R, with given data in the modulation space Ms2,p(R). Using ideas from Killip, Visan, Zhang, Oh and Wang, we prove that the IVP associated with the h-NLS equation is globally well-posed in the modulation spaces Ms2,p(R) for s≥14 and p≥2.

Bifurcation and dynamics of periodic solutions of MEMS model with squeeze film damping

In this paper, we study the oscillations of an idealized mass–spring model of micro-electro-mechanical system (MEMS) with squeeze film damping. The model consists of two parallel electrodes separated by a gap d: one of them is fixed, and another one is movable and attached to a linear spring with stiffness coefficient k>0. The oscillation, under the influence of AC–DC voltage V(t)=vdc+vaccos2πTt, is ruled by the following singular differential equation my′′+[A(d−y)3+Ad−y]y′+ky=θ0A2V2(t)(d−y)2.Here, y is the vertical displacement of the moving plate (y is always assumed to be less than d), m>0 is its mass, A>0 is the electrode area, and θ0>0 is the absolute dielectric constant of vacuum. Taking d as the parameter, we show the existence of saddle–node bifurcation of T-periodic solutions to the equation in the parameter space. This answers, from certain point of view, the open problem proposed by Torres in his monograph, see Torres (2015, Open Problem 2.1, p. 18). Further, we prove that the equation has exactly two classes of T-periodic solutions: as d tends to +∞, one of them uniformly tends to +∞ at the rate of d, while the minimum values of the second class tend to, or cross, 0.

Stability Analysis of l-Power Reciprocal Functional Equation in Intuitionistic Fuzzy Banach Spaces

In this study, we explore the generalized Hyers-Ulam-Rassias stability of a l-power reciprocal functional equation in Intuitionistic Fuzzy Banach Spaces.

Backward Uniqueness for 3D Navier–Stokes Equations With Non-Trivial Final Data and Applications

Abstract Presented is a backward uniqueness result of bounded mild solutions of 3D Navier–Stokes Equations in the whole space with non-trivial final data. A direct consequence is that a solution must be axi-symmetric in $[0, T]$ if it is so at time $T$. The proof is based on a new weighted estimate that enables to treat terms involving Calderon–Zygmund operators. The new weighted estimate is expected to have certain applications in control theory when classical Carleman-type inequality is not applicable.

Well-posedness and continuity properties of the Fornberg–Whitham equation in the Besov space $$B^1_{\infty ,1}(\mathbb {R})$$

Well-posedness and continuity properties of the Fornberg–Whitham equation in the Besov space $$B^1_{\infty ,1}(\mathbb {R})$$

On stress ratio equations in three-dimensional stress space for modelling soil behaviour

On stress ratio equations in three-dimensional stress space for modelling soil behaviour