Potential Equation Research Articles

A Priori Estimates for Solutions to Landau Equation Under Prodi–Serrin Like Criteria

In this paper, we introduce Prodi–Serrin like criteria which enable us to provide a priori estimates for the solutions to the spatially homogeneous Landau equation for all classical soft potentials and dimensions d≧3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d \\geqq 3$$\\end{document}. The physical case of Coulomb interaction in dimension d=3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d=3$$\\end{document} is included in our analysis; this generalizes the work of Silvestre (J Differ Equ 262:3034–3055, 2017). Our approach is quantitative and does not require a preliminary knowledge of elaborate tools for nonlinear parabolic equations.

Inverse problem in energy-dependent potentials using semiclassical methods

Wave equations with energy-dependent potentials appear in many areas of physics, ranging from nuclear physics to black hole perturbation theory. In this work, we use the semiclassical Wentzel-Kramers-Brillouin (WKB) method to first revisit the computation of bound states of potential wells and reflection/transmission coefficients in terms of the Bohr-Sommerfeld rule and the Gamow formula. We then discuss the inverse problem, in which the latter observables are used as a starting point to reconstruct the properties of the potentials. By extending known inversion techniques to energy-dependent potentials, we demonstrate that so-called width-equivalent or WKB-equivalent potentials are not isospectral anymore. Instead, we explicitly demonstrate that constructing quasi-isospectral potentials with the inverse techniques is still possible. Those reconstructed, energy-independent potentials share key properties with the width-equivalent potentials. We report that including energy-dependent terms allows for a rich phenomenology, particularly for the energy-independent equivalent potentials. Published by the American Physical Society 2024

Determination of the Energy Eigenvalues of the Varshni-Hellmann Potential

In this paper, we solve the bound state problem for the Varshni-Hellmann potential via a useful technique. In our technique, we obtain the bound state solution of the Schrödinger equation for the Varshni-Hellmann potential via ansatz method. We obtain the energy eigenvalues and the corresponding eigenfunctions. Also, the behavior of the energy spectra for both the ground and the excited state of the two body systems is illustrated graphically. The similarity of our results to the accurate numerical values is indicative of the efficiency of our technique.

Towards the Analytical Generalization of the Transcendental Energy Equation, Group Velocity, and Effective Mass in One-Dimensional Periodic Potential Wells with a Computational Application to Common Coupled Potentials

The analytical generalization for N periodic potential wells coupled to a probe rectangular-like potential and a zero potential is extremely important in the study of one-dimensional periodic potentials in solid state physics, e.g., in the calculation of transport, optical, and magnetic properties. These findings raise the possibility of calculating equations for the generalization of N arbitrary potentials related to any potential V(x) using special functions as a solution. In this work, a novel analytical generalization of the transcendental energy equation, group velocity, and effective mass for N-coupled potentials to a probe one-dimensional potential V=V(x) was proposed. Initially, two well-known linear periodic potentials V=V(x) were employed to obtain analytical solutions for rectangular-like and Dirac-delta potentials. Python libraries were used to easily represent the equations for one or two rectangular-like potentials coupled with an arbitrary potential, highlighting the transcendental energy, group velocity, and effective mass. The results showed that the group velocity behavior changed its orientation due to the sign of the potential, whereas the width of the potential V(x) strongly influenced the group velocity behavior. The effective mass was also modified by the potential shapes, and their combinations, both effective mass and group velocity, exhibited similar physical behaviors to those found in ordinary rectangular-like potentials.

Nonlocal modelling of piezoelectric solid with cracks based on peridynamic differential operator

The paper establishes a nonlocal model for piezoelectric solids using the Peridynamic differential operator. The method transforms the displacement-electric potential equation, constitutive equation, and boundary conditions of piezoelectric solids from a local differential form to a nonlocal integral form. The solution for the displacement and electric potential distributions in piezoelectric solids is obtained using variational principles and the Lagrange multiplier method. The correctness and effectiveness of the model are validated through a comparison of the deformation analysis of piezoelectric solid square plates with analytical solutions. Analysis of piezoelectric solid square plates with edge cracks proves the applicability of this model for solving discontinuous problems. Furthermore, the paper explores the suitability of the model in analysing problems involving multiple cracks in piezoelectric solid plates, highlighting its capability to address defective piezoelectric solid problems. This model provides a new perspective for solving problems involving defects under electro-mechanical coupling conditions.

Validation of the added mass effect in the narrow gaps of nuclear reactor pressure vessels

Nuclear reactor pressure vessels (RPVs) must be robust to external environmental excitations such as seismic loads. Precise numerical models that predict the dynamic responses to external loads are crucial for evaluating the safety of RPVs. Notably, in RPVs submerged in reactor coolant, it is necessary to examine the added mass effect of the fluid via fluid-structure interaction (FSI) analysis, employing either an analytical approach or a numerical analysis. The present study focuses on modeling and analyzing the added mass effect in the narrow gaps associated with the dense structural layout in the reactor. In this context, this paper validates analytical and numerical models that accurately describe the added mass effect for two cases: (i) simple concentric cylinders with a narrow gap filled with fluid and (ii) a complex small-scale RPV with narrow gaps also filled with fluid. Initially, the analytical model, employing Lagrange's equation of motion and velocity potential, and the numerical model, utilizing FE modeling in acoustic FSI through ANSYS, were validated through experimental modal testing for the added mass effect in the narrow gap between simple concentric cylinders. The analytical model for the concentric cylinders indicates that the added mass diverges to infinity as the gap approaches zero, which should be examined carefully. Through experiments for the concentric cylinders, the analytical and FEM-based added mass effects were validated with a natural frequency error of 3 %. Subsequently, based on the results for the concentric cylinders, a complex numerical model of the small-scale reactor with complex boundary conditions was developed and cross-validated through corresponding experimental modal testing of the small-scale RPV. The added mass model for the narrow gaps in the RPV was validated by substantial agreement of eight representative modes, as demonstrated by a 5 % natural frequency error.

Optimal [formula omitted] error estimates of stabilizer-free weak Galerkin finite element method for the drift-diffusion problem

We continue our effort in Li et al. (2024) to develop a stabilizer-free weak Galerkin finite element method (SFWG-FEM) for the two-dimensional unsteady-state drift-diffusion (DD) model. The transient state equation includes both the first derivative convection and the second derivative diffusion terms, and it couples with a Poisson potential equation. The piecewise Pk(k≥1) polynomials elements in the interior and on the boundaries are utilized for both electron concentration and electric potential functions. The vector function of [Pj(K)]2(j≥j0) for the discrete weak gradient space is used on each element K. The main difficulty is the treatment of the non-linearity and coupling of the system. Based on an introduced Ritz projection and two L2 projection operators, optimal error estimates of O(Δt+hk+1) in the L2 norm and O(Δt+hk) in the energy-like norm are derived. Numerical experiments are presented to illustrate the theoretical analysis.

MODELLING OF MASS TRANSFER IN WASTEWATER FACILITIES

Problem statement. The design of wastewater treatment systems is a complex process and requires the use of special mathematical models. As a rule, empirical models are used at the stage of designing structures of water drainage systems, which allow obtaining only an “integral” characteristic of the efficiency of wastewater treatment. But in a number of cases, it is important to have information about the spatial distribution of the impurity concentration in the structure. To solve this problem, you need to have three-dimensional mathematical models. In the future, there is a shortage of such models, so the creation of three-dimensional multifactorial models for the analysis of the efficiency of drainage system structures is an urgent task. The purpose of the article. Development of a three-dimensional numerical model for the analysis of the mass transfer process to determine the impurity concentration in the clarifier. Methodology. The analysis of impurity concentration fields in the clarifier is carried out by numerical integration of the three-dimensional equation for the velocity potential and the three-dimensional equation of the convective-diffusion transport of the impurity. For the numerical integration of the Laplace equation for the velocity potential, the variable-triangular method and the Liebmann method are used. Finite-difference splitting schemes are used for numerical integration of the three-dimensional equation of convective-diffusion transport of impurities. Scientific novelty. A dynamic multifactorial numerical model was created for the analysis of the process of mass transfer of impurities in a settling tank by conducting a computational experiment. Practical value. The built multifactorial numerical model makes it possible to analyze the efficiency of wastewater treatment in clarifiers that have a complex geometric shape and cannot be calculated on the basis of existing engineering methods. Conclusions. On the basis of the developed three-dimensional numerical model, a computer code was created, which allows you to quickly obtain information about the distribution of the impurity concentration in the settling tank.

Numerical Solutions of Potential Flow Equations using Finite Differences

This article delves into the numerical solutions of potential flow equations using finite differences, aiming to enhance our understanding of fluid dynamics. The general objective is to obtain numerical solutions to potential flow equations using finite differences, with specific objectives including the investigation of potential flow equations, the solutions of associated PDE and the analysis of the stability of employed numerical schemes. The study employs a combination of numerical methods to achieve its objectives; MATLAB is utilized as a computational tool, while the Gauss-Seidel and Jacobi’s iterative methods are implemented for solving PDEs. Central differences are employed for discretization. The study yields valuable insights into the behaviour of potential flow systems. The significance of this research lies in its contribution to advancing our comprehension of fluid dynamics with potential applications. Generally, this work provides a foundation for further exploration and application of numerical methods in the study of potential flow.

Phase-field modeling of fracture for ferromagnetic materials through Maxwell’s equation

Electro-active materials are classified as electrostrictive and piezoelectric materials. They deform under the action of an external electric field. Piezoelectric material, as a special class of active materials, can produce an internal electric field when subjected to mechanical stress or strain. In return, there is the converse piezoelectric response, which expresses the induction of the mechanical deformation in the material when it is subjected to the application of the electric field. This work presents a variational-based computational modeling approach for failure prediction of ferromagnetic materials. In order to solve this problem, a coupling between magnetostriction and mechanics is modeled, then the fracture mechanism in ferromagnetic materials is investigated. Furthermore, the failure mechanics of ferromagnetic materials under the magnetostrictive effects is studied based on a variational phase-field model of fracture. Phase-field fracture is numerically challenging since the energy functional may admit several local minima, imposing the global irreversibility of the fracture field and dependency of regularization parameters related discretization size. Here, the failure behavior of a magnetoelastic solid body is formulated based on the Helmholtz free energy function, in which the strain tensor, the magnetic induction vector, and the crack phase-field are introduced as state variables. This coupled formulation leads to a continuity equation for the magnetic vector potential through well-known Maxwell’s equations. Hence, the energetic crack driving force is governed by the coupled magneto-mechanical effects under the magneto-static state. Several numerical results substantiate our developments.

Involvement of ANO1 currents in pacemaking of PDGFRα-positive specialised smooth muscle cells in rat caudal epididymis.

The epididymal duct exhibits spontaneous phasic contractions (SPCs) to store and transport sperm. Here, we explored molecular identification of pacemaker cells driving SPCs in the caudal epididymal duct and also investigated properties of pacemaker currents underlying SPCs focusing on ANO1 Ca2+-activated Cl- channels (CaCCs). Immunohistochemistry was performed to visualise the distribution of platelet-derived growth factor receptor α (PDGFRα)- or ANO1-positive cells in the rat caudal epididymal duct. Perforated whole-cell patch clamp technique was applied to enzymatically isolated epididymal cells, while SPCs were recorded with video edge-tracking technique. Immunohistochemistry revealed the distribution of α-smooth muscle actin (α-SMA)-positive cells co-expressing both PDGFRα and ANO1 in the innermost smooth muscle layer. Approximately one-third of isolated epididymis cells exhibited spontaneous transient inward currents (STICs) at the holding potential -60 mV. The reversal potential for STICs was close to the calculated chloride equivalent potential depending on intracellular Cl- concentrations. Ani9 (3 µM), the ANO1 specific inhibitor, decreased both amplitude and frequency of STICs, while cyclopiazonic acid (CPA, 30 µM), a sarco-/endoplasmic reticulum Ca2+-ATPase (SERCA) inhibitor, abolished STICs. Ani9 (3 or 10 µM) reduced the frequency of SPCs without changing their amplitude. Thus, PDGFRα+, ANO1+ specialised smooth muscle cells (SMCs) appear to function as pacemaker cells to electrically drive epididymal SPCs by generating ANO1-dependnet STICs. STICs arising from spontaneous Ca2+ release from intracellular Ca2+ store and subsequent opening of ANO1 result in depolarisations that spread into adjacent SMCs where L-type voltage-dependent Ca2+ channels are activated to develop SPCs.

ANALYSIS OF THE EFFICIENCY OF THE USE OF PROTECTIVE SCREENS TO REDUCE THE LEVEL OF AIR POLLUTION

The task of assessing the areas of chemical pollution near the highway, where protective screens of different geometric shapes are located, is considered. The purpose of the work is to develop numerical models for calculating pollution zones formed near protective screens, as well as conducting a laboratory experiment to analyze the patterns of formation of pollution zones near screens of a complex geometric shape. For mathematical modeling of the process of formation of pollution zones near the protective screen, the equation of convective-diffusion transfer of impurities is used. This equation takes into account atmospheric diffusion, wind speed, emission intensity of a chemically hazardous substance, the location of the emission source, and the shape of the protective screen. The Navier-Stokes equation and the Laplace equation for the velocity potential are used to solve the problem of aerodynamics. Finite-difference methods are used for numerical integration of modeling equations. A package of application programs was created on the basis of the developed numerical models. Numerical models and a package of programs have been built, allowing to study the process of the formation of areas of pollution near the highway in almost real time. The results of the computational experiment are presented.

Approaching certain central-field Schrödinger problems through the harmonic oscillator green's function

We propose a method to analytically solve the Schrödinger equation with central potentials. By means of Sturm–Liouville expansion, the Green's function (GF) of the Schrödinger differential operator can be represented in the GF of the isotropic harmonic oscillator. Applying this straightforward approach, we solve the Schrödinger problem for the Coulomb potential, for the Morse potential, and for the respective inverted potentials. Using the obtained results, we formulate a method to analytically solve Schrodinger equations for multi-well potentials, demonstrating that classically forbidden regions can be treated as inverted harmonic oscillators, whereas harmonic oscillators or Morse potentials can approximate the wells.

Eigenmodes and eigenfunctions of high-power electromagnetic waves in inhomogeneous magnetized plasmas and their stability to stimulated Raman backscattering

Eigenmodes and eigenfunctions of high-power electromagnetic waves in inhomogeneous magnetized plasmas are investigated. Considering the dielectric permittivity in the presence of an external magnetic field and relativistic nonlinearity and considering the inhomogeneity of the electron density in the linear and rippled structures, two nonlinear wave equations are obtained. In the linear regime, eigenmodes and eigenfunctions of the electric field for two-electron densities are obtained. In the nonlinear regime, results indicate that by increasing the external magnetic field, the amplitude of the electric field increases, which affects the dielectric permittivity. The conditions of the stimulated Raman backscattering and the equation of the space-charge potential are obtained. Results also indicate that in the presence of the external magnetic field, plasma inhomogeneity, and relativistic nonlinearity, the stimulated Raman backscattering becomes unstable.

An interface formulation for the Poisson equation in the presence of a semiconducting single-layer material

In this paper, we consider a semiconducting device with an active zone made of a single-layer material. The associated Poisson equation for the electrostatic potential (to be solved in order to perform self-consistent computations) is characterized by a surface particle density and an out-of-plane dielectric permittivity in the region surrounding the single-layer. To avoid mesh refinements in such a region, we propose an interface problem based on the natural domain decomposition suggested by the physical device. Two different interface continuity conditions are discussed. Then, we write the corresponding variational formulations adapting the so called three-fields formulation for domain decomposition and we approximate them using a proper finite element method. Finally, numerical experiments are performed to illustrate some specific features of this interface approach.

Lump–kink and hybrid solutions of the extended (3+1)-dimensional potential KP equation in fluid mechanics

In this paper, the extended (3+1)-dimensional potential KP equation in fluid mechanics is studied through Hirota bilinear method. Many types of hybrid solutions, such as the lump–kink solution, lump-two kink solution and periodic lump solution are obtained by assuming different functions in the bilinear equation. The interaction solution between lump and triangular periodic wave is also derived by combining sine and cosine functions with quadratic functions. Dynamical structures of these exact solutions are depicted by presenting the corresponding three-dimensional, two-dimensional structures and density graphs. These diverse interaction solutions could be helpful for understanding physical phenomena in fluid mechanics.

A (2+1) -dimensional evolution model of Rossby waves and its resonance Y-type soliton and interaction solutions

In this paper, we derive a Kadomtsev-Petviashvili equation by using the multi-scale expansion and perturbation method, which is a model from the potential vorticity equation in the traditional approximation and describes the Rossby wave propagation properties. The N-soliton solutions, resonance Y-type soliton solutions, resonance X-type soliton solutions and interaction solutions of the equation are obtained with the help of dependent-variable transformation. In addition, the composite graphs are given to view the resonance phenomenon of Rossby waves. The results better enrich the research of Rossby waves in ocean dynamics and atmospheric dynamics.

Numerical treatment of entropy generation and Bejan number into an electroosmotically-driven flow of Sutterby nanofluid in an asymmetric microchannel

This article aims to investigate the electrokinetic peristaltic flow of the Sutterby nanofluid through an asymmetric microchannel with a porous medium and the Joule heating parameter. The magnetic field is applied in the transverse direction and the electric field on the flow direction. The flow model consists of the continuity, momentum, heat, and electric potential equations, with appropriate boundary conditions. The Debye-Hückel linearization approximation is considered. Under the long wavelength and small Reynolds number approximation, the lengthy equations were reduced. The resultant coupled equations are numerically solved by the renowned NDSolve coding using Mathematica software. Entropy and Bejan number are also incorporated in this study. The velocity, temperature, and the phenomenon of entrapment are analyzed in depth with the aid of graphical representations. Streamlines of various waveforms are also discussed. The analysis discovered that the velocity of the fluid enhanced for porosity parameter and reverse effects is observed on the quantity of the magnetic parameter.

The role of above-code labeling programs in reducing CO2e emissions in residential buildings

Residential buildings account for 15 % of total U.S. carbon dioxide (CO2) emissions. Voluntary, above-code labeling programs, such as the ENERGY STAR® and Zero Energy Ready Home™ programs, can play a crucial role in reducing the carbon footprint of the new construction residential building sector by encouraging adoption of more energy-efficient and lower emissions building practices and technologies. It’s important for the industry to understand the carbon dioxide equivalent (CO2e) and energy cost savings potential of these programs. To estimate the potential CO2e emissions savings for homes built to above-code labeling program levels, it is imperative to have reliable and accurate tools for estimating emissions from buildings. The open-source OpenStudio-ERI workflow employs detailed physics-based building energy modeling (BEM) to estimate energy usage and operational CO2e emissions. In this study we use the OpenStudio-ERI workflow and regional, time-varying emissions factors from the Cambium database to compare the CO2e emissions and energy costs of homes built to performance levels required by these above-code labeling programs relative to homes built to model energy codes. We present the results of our analysis and provide a range of emissions and annual energy cost savings for the above-code labeling programs.

Analysis of Self-Gravitating Fluid Instabilities from the Post-Newtonian Boltzmann Equation.

Self-gravitating fluid instabilities are analysed within the framework of a post-Newtonian Boltzmann equation coupled with the Poisson equations for the gravitational potentials of the post-Newtonian theory. The Poisson equations are determined from the knowledge of the energy-momentum tensor calculated from a post-Newtonian Maxwell-Jüttner distribution function. The one-particle distribution function and the gravitational potentials are perturbed from their background states, and the perturbations are represented by plane waves characterised by a wave number vector and time-dependent small amplitudes. The time-dependent amplitude of the one-particle distribution function is supposed to be a linear combination of the summational invariants of the post-Newtonian kinetic theory. From the coupled system of differential equations for the time-dependent amplitudes of the one-particle distribution function and gravitational potentials, an evolution equation for the mass density contrast is obtained. It is shown that for perturbation wavelengths smaller than the Jeans wavelength, the mass density contrast propagates as harmonic waves in time. For perturbation wavelengths greater than the Jeans wavelength, the mass density contrast grows in time, and the instability growth in the post-Newtonian theory is more accentuated than the one of the Newtonian theory.