Equations In Spaces Research Articles

A strong maximum principle for quasilinear elliptic differential equations in a variable exponent Sobolev space

A strong maximum principle is an important property for elliptic and parabolic equations. In this paper, we consider a strong maximum principle for a class of quasilinear elliptic equations containing p(⋅)-Laplacian and mean curvature operator.

Exploring New Traveling Wave Solutions for the Spatiotemporal Evolution of a Special Reaction–Diffusion Equation by Extended Riccati Equation Method

In this work, we aim to explore new exact traveling wave solutions for the reaction–diffusion equation, which describes complex nonlinear phenomena such as cell growth and chemical reactions in nature. Obtaining exact solutions to this equation is crucial for understanding aspects such as reaction activity and the diffusion coefficient. We solve the reaction–diffusion equation by using the Riccati equation as an auxiliary equation. By controlling the parameters in the Riccati equation, we obtained a large number of traveling wave solutions, many of which were not formerly recorded in other documents. Numerical simulations demonstrate the evolution of various traveling waves of the reaction–diffusion equation in time and space. These rich exact solutions and wave phenomena help to expand our knowledge of this equation.

Non-uniform dependence on initial data for the Euler equations in Besov spaces

In this paper, we consider the initial value problem to the higher dimensional Euler equations in the whole space. Based on the local well-posedness result and the lifespan, we prove that the data-to-solution map of this problem is not uniformly continuous in nonhomogeneous Besov spaces. Our obtained result improves considerably the previous results given by Himonas-Misiołek (2010) [8] and Pastrana (2021) [21].

$$L^1-$$decay of higher-order norms of solutions to the Navier–Stokes equations in the upper-half space

$$L^1-$$decay of higher-order norms of solutions to the Navier–Stokes equations in the upper-half space

Quantizing Galilean spacetime: a reconstruction of Maxwell’s equations in empty space

As was recently shown, non-relativistic quantum theory can be derived by means of a projection method from a continuum of classical solutions for (massive) particles. In this paper, we show that Maxwell’s equations in empty space can be derived using the same method. In this case, the starting point is a continuum of solutions of equations of motion for massless particles describing the structure of Galilean space-time. As a result of the projection, the space-time structure itself is changed by the appearance of a new fundamental constant c with the dimension of a velocity. This maximum velocity c, derived here for massless particles, is analogous to the accuracy limit ħ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\hbar $$\\end{document} derived earlier for massive particles. The projection method can thus be interpreted as a generalized quantization. We suspect that all fundamental fields can be traced back to continuous sets of particle trajectories, and that in this sense, the particle concept is more fundamental than the field concept.

Optimal robust online tracking control for space manipulator in task space using off-policy reinforcement learning

This study addresses the demands for adaptability, uncertainty management, and high performance in the control of space manipulators, and the inadequacies in achieving optimal control and handling external uncertainty in task space in previous research. Based on off-policy reinforcement learning, a model-free and time-efficient method for online robust tracking control in task space is devised. To address the complexity of dynamic equations in task space, a mixed-variable approach is adopted to transform the multivariable coupled time-varying problem into a single-variable problem. Subsequently, the optimal control policy is derived with the disturbance convergence, stability, and optimality of the control method being demonstrated. This marks the first instance of achieving robust optimal tracking control in task space for space manipulators. The efficacy and superiority of the presented algorithm are validated through simulation.

Latent-SDE: guiding stochastic differential equations in latent space for unpaired image-to-image translation

Score-based diffusion models have shown promising results in unpaired image-to-image translation (I2I). However, the existing methods only perform unpaired I2I in pixel space, which requires high computation costs. To this end, we propose guiding stochastic differential equations in latent space (Latent-SDE) that extracts domain-specific and domain-independent features of the image in the latent space to calculate the loss and guides the inference process of a pretrained SDE in the latent space for unpaired I2I. To refine the image in the latent space, we propose a latent time-travel strategy that increases the sampling timestep. Empirically, we compare Latent-SDE to the baseline of the score-based diffusion model on three widely adopted unpaired I2I tasks under two metrics. Latent-SDE achieves state-of-the-art on Cat →\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\rightarrow $$\\end{document} Dog and is competitive on the other two tasks. Our code will be freely available for public use upon acceptance at https://github.com/zhangXJ147/Latent-SDE.

Uniqueness and non-degeneracy of solutions for nonlinear fractional Schrödinger equation with perturbation

This paper concerned a fractional Schrödinger equation in whole space, for ɛ > 0 is a small parameter, ɛ2s(−Δ)su + V(x)u = |u|p−2u, where 12<s<1, N > 1 and 2<p<2NN−2s. We prove the non-degeneracy and uniqueness of bubble solutions by using local Pohozaev identity and finite dimensional reduction, which are the cornerstones to construct different type solutions.

Convection-Diffusion Equations in Uniformly Local Lebesgue Spaces

Convection-Diffusion Equations in Uniformly Local Lebesgue Spaces

Analytical solution for forced vibration of multilayer structures composed of plates with different geometric dimensions

Multilayer system composed of parallel plate components with different geometric dimensions is frequently used to describe engineering objects, such as electronic assembly. In this work, an analytical method was proposed for forced vibration of this type of multilayer system. The proposed method overcomes the limitation that the traditional wave method is only applicable to all plate components must have the same in-plane dimensions. The proposed analytical method has an efficiency advantages in parameter analysis than element-based methods such as finite element method (FEM). The connection joints between two adjacent plate components, such as ball grid array (BGA) solder interconnect, are represented by elastic springs. The vibration of each component are described in terms of general and physical analytical waves, respectively, and the dynamic coupling between them are established by an equivalent dynamic flexibility matrix. The forced responses of the multilayer system are analytically calculated by solving the system equation in wave space. In the numerical examples, the effectiveness of the proposed method is validated by comparing the present results with the FEM results. The influence of number of the defective solder joints on vibration response is also investigated.

Global regularity for critical SQG in bounded domains

AbstractWe prove the existence and uniqueness of global smooth solutions of the critical dissipative SQG equation in bounded domains in . We introduce a new methodology of transforming the single nonlocal nonlinear evolution equation in a bounded domain into an interacting system of extended nonlocal nonlinear evolution equations in the whole space. The proof then uses the method of the nonlinear maximum principle for nonlocal operators in the extended system.

Noncommutative of Klein–Gordon and Schrödinger equations in the background of the improved Hua plus modified Eckart potential model in 3D-(R/NR)NCQS symmetries: Spectrum and thermodynamic properties

The impact of deformation space on the physical characteristics of diverse physics systems has been thoroughly investigated in research papers. In this work, we study the deformed Klein–Gordon equation (DKGE) in the three-dimensional relativistic non-commutative quantum space (3D-RNCQS) regime by using the improved Hua plus modified Eckart potential (IHPMEP) model. For this consideration, the DKGE in the 3D-RNCQS regime is solved using the standard perturbation theory and the well-known Bopp’s shifts method with the Greene–Aldrich approximation to the centrifugal barrier. The new relativistic energy equation and eigenfunction for the IHPMEP in the presence of deformation space-space for the heterogeneous (CO, HF, and NO) and homogeneous (N2, H2, and Li2) diatomic molecules are obtained to be sensitive to the atomic quantum numbers ([Formula: see text] and [Formula: see text]), the mixed potential depths ([Formula: see text], and [Formula: see text]), the inverse of the screening parameter [Formula: see text], and non-commutativity parameters ([Formula: see text], [Formula: see text], and [Formula: see text]). Analysis is performed on the non-relativistic limit of new energy spectra. By appropriately adjusting the combined potential parameters, we analyze the obtained new bound state eigenvalues of the DKGE and deformed Schrödinger equation with the IHPMEP in 3D-NCQS symmetries and obtain the new modified Eckart potential, the modified Hua potential, the modified Morse potential, and the modified Pöschl–Teller potential. Within the framework of the 3D-NRNCQS regime, the homogeneous and heterogeneous composite systems under IHPMEP models are examined. A thorough investigation is carried out into the impact of space-space deformation on the thermal parameters of the IHPMEP, including the partition function, mean energy, free energy, specific heat, and entropy. This work is of a fundamental absorbability nature and pedagogical interest in atomic and molecular physics.

Distributed finite-time attack detection and estimation of intelligent connected vehicle platoon

ABSTRACT This paper presents a distributed finite-time observer design approach for the problem of intelligent connected vehicles (ICVs) containing both attack signals and unknown disturbances. Firstly, an analysis is conducted on the two types of network attacks that ICVs may encounter, followed by the construction of a dynamic vehicle model and its transformation into a state space equation. Secondly, an intermediate variable is introduced to design a distributed attack estimation observer, which both ensures accurate estimation of the vehicle safety state and attack signal within a limited time and overcomes the constraints of traditional observers. Additionally, sufficient conditions for the finite-time boundedness of the observer error system with H ∞ performance are provided in the form of linear matrix inequalities. Furthermore, an adaptive threshold attack detection method is designed. Finally, the simulation results validate the efficiency of the design methods.

Hyperstability Results for the General Linear Functional Equation in Non-Archimedean 2-Banach Spaces

Hyperstability Results for the General Linear Functional Equation in Non-Archimedean 2-Banach Spaces

Existence of Weak Solution for Black-Scholes Partial Differential Equation and Application of Energy Estimate Theorem in Sobolev Space

Purpose:This paper aims to solve the BS second-order parabolic equation in Sobolev spaces to obtain weak solutions for financial applications, extending previous work in this field.Methodology:This paper constructs a set of functions that transforms the Black-Scholes partial differential equation into weak formulations. This study focuses on the Mathematics of finance, particularly the evolution of option pricing. An option's underlying asset involves agreements to buy or sell at a future strike price. The Black-Scholes (BS) equation, widely used in financial applications, models this.Findings:The analytical solutions, existence, uniqueness and other estimates were also obtained in weak form using boundary conditions to establish the effects of their financial implications in Sobolev spaces.Implication:The problem's regularity conditions were considered, and the coefficients and boundary of the domain are all smooth functions. To this end, this paper illustrates the definitions and assumptions that led to valuable assertions.

Stability and Instability of an Apollonius-Type Functional Equation

For the inner product space, we have Appolonius’ identity. From this identity, Park and Th. M. Rassias induced and investigated the quadratic functional equation of the Apollonius type. And Park and Th. M. Rassias first introduced an Apollonius-type additive functional equation. In this work, we investigate an Apollonius-type additive functional equation in 2-normed spaces. We first investigate the stability of an Apollonius-type additive functional equation in 2-Banach spaces by using Hyers’ direct method. Then, we consider the instability of an Apollonius-type additive functional equation in 2-Banach spaces.

On the Continuity Equation in Space–Time Algebra: Multivector Waves, Energy–Momentum Vectors, Diffusion, and a Derivation of Maxwell Equations

Historically and to date, the continuity equation (C.E.) has served as a consistency criterion for the development of physical theories. In this paper, we study the C.E. employing the mathematical framework of space–time algebra (STA), showing how common equations in mathematical physics can be identified and derived from the C.E.’s structure. We show that, in STA, the nabla equation given by the geometric product between the vector derivative operator and a generalized multivector can be identified as a system of scalar and vectorial C.E.—and, thus, another form of the C.E. itself. Associated with this continuity system, decoupling conditions are determined, and a system of wave equations and the generalized analogous quantities to the energy–momentum vectors and the Lorentz force density (and their corresponding C.E.) are constructed. From the symmetry transformations that make the C.E. system’s structure invariant, a system with the structure of Maxwell’s field equations is derived. This indicates that a Maxwellian system can be derived not only from the nabla equation and the generalized continuity system as special cases, but also from the symmetries of the C.E. structure. Upon reduction to well-known simpler quantities, the results found are consistent with the usual STA treatment of electrodynamics and hydrodynamics. The diffusion equation is explored from the continuity system, where it is found that, for decoupled systems with constant or explicitly dependent diffusion coefficients, the absence of external vector sources implies a loss in the diffusion equation structure, transforming it into Helmholtz-like and wave equations.

Recurrence formula for some higher order evolution equations

Riccati’s differential equation is formulated as abstract equation in finite or infinite dimensional Banach spaces. Since the Riccati’s differential equation with the Cole–Hopf transform shows a relation between the first order evolution equations and the second order evolution equations, its generalization suggests the existence of recurrence formula leading to a sequence of differential equations with different order. In conclusion, by means of the logarithmic representation of operators, a transform between the first order evolution equations and the higher order evolution equation is presented. Several classes of evolution equations with different orders are given, and some of them are shown as examples.

Efficient parallel solver for rarefied gas flow using GSIS

Recently, the general synthetic iterative scheme (GSIS) has been proposed to find the steady-state solution of the Boltzmann equation in the whole range of gas rarefaction, where its fast-converging and asymptotic-preserving properties lead to the significant reduction of iteration numbers and spatial cells in the near-continuum flow regime. However, the efficiency and accuracy of GSIS have only been demonstrated in two-dimensional problems with small numbers of spatial cells and discrete velocities. Here, a large-scale parallel computing strategy is designed to extend the GSIS to three-dimensional flow problems, including the supersonic flows which are usually difficult to solve by the discrete velocity method. Since the GSIS involves the calculation of the mesoscopic kinetic equation which is defined in six-dimensional phase-space, and the macroscopic high-temperature Navier–Stokes–Fourier equations in three-dimensional physical space, the proper partition of the spatial and velocity spaces, and the allocation of CPU cores to the mesoscopic and macroscopic solvers, are the keys to improving the overall computational efficiency. These factors are systematically tested to achieve optimal performance, up to 100 billion spatial and velocity grids. For hypersonic flows around the Apollo reentry capsule, the X38-like vehicle, and the space station, our parallel solver can obtain the converged solution within one hour.

Complex potential energy surfaces with projected CAP technique: Vibrational excitation of N2.

The projected complex absorbing potential (CAP) technique is one of the methods that allow one to extend the bound state methods for computing resonances' energies and widths. Here, we explore the accuracy of the potential energy curves generated with different electronic structure theory methods in combination with the projected CAP technique by considering resonant vibrational excitation (RVE) of N2 by electron impact as a model process. We report RVE cross sections computed using the boomerang model with potential energy curves obtained with CAP-based extended multistate complete active space perturbation theory (XMS-CASPT2) and equation of motion coupled-cluster method for electron attachment with single and double substitution (EOM-EA-CCSD) methods. We also compare potential energy curves computed with several electronic structure methods, including XMS-CASPT2, EOM-EA-CCSD, multireference configuration interaction with singles (MR-CIS) and singles and doubles (MR-CISD). A good agreement is observed between the experiment and simulated RVE cross sections obtained with the potential energy curves generated with XMS-CASPT2 and EOM-EA-CCSD methods, thus highlighting the potential of the projected CAP technique combined with accurate electronic structure methods for dynamical simulations of the processes that proceed through metastable electronic states.