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Wavelets and multiwavelets have lately been adopted in quantum chemistry to overcome challenges presented by the two main families of basis sets: Gaussian atomic orbitals and plane waves. In addition to their numerical advantages (high precision, locality, fast algorithms for operator application, linear scaling with respect to system size, to mention a few), they provide a framework that narrows the gap between the theoretical formalism of the fundamental equations and the practical implementation in a working code. This realization led us to the development of the Python library called VAMPyR (Very Accurate Multiresolution Python Routines). VAMPyR encodes the binding to a C++ library for multiwavelet calculations (algebra and integral and differential operator application) and exposes the required functionality to write a simple Python code to solve, among others, the Hartree–Fock equations, the generalized Poisson equation, the Dirac equation, and the time-dependent Schrödinger equation up to any predefined precision. In this study, we will outline the main features of multiresolution analysis using multiwavelets and we will describe the design of the code. A few illustrative examples will show the code capabilities and its interoperability with other software platforms. [ABSTRACT FROM AUTHOR]

Keywords: Rational normal forms; Birkhoff normal forms; Nekhoroshev theorem; Gevrey norm; Hamiltonian PDEs.GraphBy Joackim Bernier; Nicolas Camps; Benoît Grébert and Zhiqiang WangReported by Author; Author; Author; Author [Extracted from the article]

Non-axisymmetric frictionless JKR-type adhesive contact between a rigid body and a thin incompressible elastic layer bonded to a rigid base is considered in the framework of the leading-order asymptotic model, which has the form of an overdetermined boundary value problem. Based on the first-order perturbation of the Neumann operator in the Dirichlet problem for Poisson's equation, the decohesion initiation problem is formulated in the form of a variational inequality. The asymptotic model assumes that the contact zone and its boundary contour during the detachment process are unknown. The absence of the solvability theorem is illustrated by an example of the instability of an axisymmetric flat circular contact. [ABSTRACT FROM AUTHOR]

Nonlinear equations called the Stenflo equations are usually used for the analytical description of the propagation of internal gravity waves in the Earth's upper atmosphere. Solutions in the form of dipole vortices, tripole vortices, and vortex chains are previously obtained by these equations. The Stenflo equations also describe rogue waves, breathers, and dark solitons. If disturbances cease to be small, then their profiles are usually deformed, and, presumably, they cannot be considered plane waves. This study shows that this is not always the case for internal gravity waves and that these waves can propagate as plane waves even with large amplitudes. An exact solution of the system of nonlinear Stenflo equations for internal gravity waves that contain nonlinear terms in the form of Poisson brackets is given. The solution is obtained in the form of plane waves with arbitrary amplitude. To find a solution, the original system of equations is transformed. It is split into equations for the stream and vorticity functions as well as equations for the perturbed density. To solve the obtained equations, the procedure of the successive zeroing of Poisson brackets is applied. As a result, linear equations that allow one to find the accurate analytical solutions for internal gravity waves in the form of plane waves with arbitrary amplitude are obtained. By solving these linear equations in two different ways, we have analytically found expressions for the perturbed quantities and the dispersion equation. The nonlinear equations obtained for the current, vorticity, and perturbed density functions can be used to find other nonlinear solutions. The given solutions in the form of plane waves with arbitrary amplitude may be of interest for the analysis of the propagation of internal gravity waves in the Earth's atmosphere and the interpretation of experimental data. [ABSTRACT FROM AUTHOR]

The functional equation given by y = (I - T) x is called Poisson’s equation. For a (weakly) mean ergodic operator T, this equation can be solved for a given y if and only if x n : = 1 n ∑ k = 1 n ∑ j = 0 k - 1 T j y (weakly) converges. In this paper, we use Abel summability of (T n) in order to solve the equation y = (I - T) x , where T is a bounded linear operator on a Banach space X. [ABSTRACT FROM AUTHOR]

Since 1966, China has been using apparent resistivity observation to forecast strong aftershocks of the Xingtai earthquake. Retrospective studies of subsequent strong earthquakes have shown that anomalies in apparent resistivity observation before earthquakes usually exhibit anisotropic characteristics. In addition to the anisotropic changes in apparent resistivity before earthquakes, factors such as subway operation near the observation area, metal pipeline networks, and changes in water levels have also been found to cause anisotropic changes. These factors are called environmental interference factors. Therefore, distinguishing between anisotropic changes before earthquakes and anisotropic changes caused by interference and eliminating the effects of interference is crucial for using apparent resistivity observations for forecasting. Taking the observation of Hefei seismic station in Anhui Province as an example, a model is constructed using the finite element method to try to establish a method for analyzing anisotropy in apparent resistivity before earthquakes, and the data from other provincial stations are used for verification. In the modeling process, the influence coefficient is a measure of the relationship between the variation in apparent resistivity and the changes in the medium of the measurement area. The following results are obtained by calculating the influence coefficient using the finite element method: the influence coefficient between the power supply electrode and the measuring electrode of the apparent resistivity observation is negative, and the rest are positive, and the distribution of the influence coefficient shows obvious symmetry, with the axis of symmetry being the line connecting the electrodes and its midline, and the absolute value of the influence coefficient is inversely proportional to the distance from the electrodes. In addition, according to the constructed finite element model, the amplitude of anisotropic changes caused by interference can be quantitatively calculated. Given that interference is ubiquitous in various regions of the world, this study can provide a reference for international earthquake forecasters to quantitatively remove environmental interference in anisotropy. Moreover, when building apparent resistivity stations in seismic areas for earthquake prediction, it is best to avoid areas with larger local influence coefficients to ensure that the anomalous data before the earthquake is true and reliable. [ABSTRACT FROM AUTHOR]

This paper proposed analytical modeling of a Schottky tunnel field‐effect transistor (STFET)—based biosensor with adjusted gate oxide. This model is developed by resolving the Poisson's equation and calculating the parabolic potential lateral to the channel depth. The special property of the bio‐transistor, which serves as a biosensor, is then included in the analytical modeling of drain current. After the biomolecule interacts with the bio‐transistor, a change in the drain current was employed as a metric to determine the sensitivity. The advanced analytical modeling explored several device restrictions. A device simulation is used to maintain and validate the established and planned characteristic trend. Consequently, the suggested model can be the right solution for the best design and fabrication of a biosensor. [ABSTRACT FROM AUTHOR]

We prove optimal Lipschitz regularity of solutions to Poisson's equation with measure data supported on a C 1 , Dini interface and with C 0 , Dini density. We achieve this by deriving pointwise gradient estimates on the interface, further showing the piecewise differentiability of solutions up to this surface. Our approach relies on perturbation arguments and estimates for the Green's function of the Laplacian. Additionally, we provide sharp counterexamples highlighting the minimality of our assumptions. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with global existence of large solutions for a dissipative model arising from electro-hydrodynamics, which is the nonlinear nonlocal system coupled by the Poisson–Nernst–Planck equations and the incompressible Navier–Stokes equations through charge transport and external forcing terms. By introducing some proper weighted functions and fully using the algebraic structure of the system, we prove that, under some conditions imposed on the indices p, p1, q, r, α, there exist two positive constants c0, C0 such that if the initial data u 0 = ( u 0 h , u 0 3 ) and (v0, w0) satisfy ‖ u 0 h ‖ B ̇ p 1 , ∞ − 1 + 3 p 1 + ‖ u 0 h ‖ B ̇ p 1 , ∞ − 1 + 3 p 1 α ‖ u 0 3 ‖ B ̇ p 1 , ∞ − 1 + 3 p 1 1 − α + K 0 ≤ c 0 with K 0 ≔ ‖ v 0 ‖ B ̇ q , 1 − 2 + 3 q ⁡ exp C 0 ‖ u 0 ‖ B ̇ p , 1 − 1 + 3 p + C 0 ‖ w 0 ‖ B ̇ r , 1 − 2 + 3 r + 1 exp C 0 ‖ u 0 ‖ B ̇ p , 1 − 1 + 3 p , then the system admits a unique global solution. Moreover, the global existence of large solution was also established in two dimensional case. [ABSTRACT FROM AUTHOR]

A way to improve the accuracy of the spectral properties in density functional theory (DFT) is to impose constraints on the effective, Kohn–Sham (KS), local potential [J. Chem. Phys. 136, 224109 (2012)]. As illustrated, a convenient variational quantity in that approach is the "screening" or "electron repulsion" density, ρrep, corresponding to the local, KS Hartree, exchange and correlation potential through Poisson's equation. Two constraints, applied to this minimization, largely remove self-interaction errors from the effective potential: (i) ρrep integrates to N − 1, where N is the number of electrons, and (ii) ρrep ≥ 0 everywhere. In this work, we introduce an effective "screening" amplitude, f, as the variational quantity, with the screening density being ρrep = f2. In this way, the positivity condition for ρrep is automatically satisfied, and the minimization problem becomes more efficient and robust. We apply this technique to molecular calculations, employing several approximations in DFT and in reduced density matrix functional theory. We find that the proposed development is an accurate, yet robust, variant of the constrained effective potential method. [ABSTRACT FROM AUTHOR]

Besides being impacted by quantum confinement effects, the channel electrostatics of ultra-thin-body silicon-on-insulator (SOI) MOS devices, with channel thicknesses less than 10 nm, are also likely to be impacted by interface trap states. In this work, we comprehensively investigated the effect of band edge energy (surface passivation energy) on the band structure of the silicon channel. We propose to utilize this band edge energy (Δ E e d g e ) to study the effect of interface traps on device electrostatics, which is generally used to passivate the channel/oxide interface. First, by using s p 3 d 5 s ∗ semi-empirical tight-binding methodology with a fully passivated interface (Δ E e d g e > --> 5 eV) and by including suitable bandgap correction for different device temperatures, the band structure is obtained, which is solved self-consistently with Poisson's equation to accurately determine the channel electrostatics, without the effect of trap states. Interface trap states are now seen in the band structure through suitably varying the edge energy (− 5 eV < Δ E e d g e < 5 eV) based on which the interface trap density (D i t ) and the interface trap charge density (Q i t ) are determined. Through incorporating Q i t in the boundary condition for solving Poisson's equation self-consistently with the band structure, channel electrostatics is recomputed to analyze the effect of traps for a wide range of device conditions. Finally, the degradation in the integrated charge density due to interface traps is accurately modeled for different SOI channel thickness and device temperatures. [ABSTRACT FROM AUTHOR]

In this study, we characterize $ LP $-Kenmotsu manifolds admitting $ * $-Ricci–Yamabe solitons ($ * $-RYSs) and gradient $ * $-Ricci–Yamabe solitons (gradient $ * $-RYSs). It is shown that an $ LP $-Kenmotsu manifold of dimension $ n $ admitting a $ * $-Ricci–Yamabe soliton obeys Poisson's equation. We also determine the necessary and sufficient conditions under which the Laplace equation is satisfied by $ LP $-Kenmotsu manifolds. Finally, by using a non-trivial example of an $ LP $-Kenmotsu manifold, we verify some results of our paper.

We propose a novel and simple spectral method based on the semi-discrete Fourier transforms to discretize the fractional Laplacian (− Δ) α 2 . Numerical analysis and experiments are provided to study its performance. Our method has the same symbol | ξ | α as the fractional Laplacian (− Δ) α 2 at the discrete level, and thus it can be viewed as the exact discrete analogue of the fractional Laplacian. This unique feature distinguishes our method from other existing methods for the fractional Laplacian. Note that our method is different from the Fourier pseudospectral methods in the literature which are usually limited to periodic boundary conditions (see Remark 1.1). Numerical analysis shows that our method can achieve a spectral accuracy. The stability and convergence of our method in solving the fractional Poisson equations were analyzed. Our scheme yields a multilevel Toeplitz stiffness matrix, and thus fast algorithms can be developed for efficient matrix-vector multiplications. The computational complexity is O (2 N log ⁡ (2 N)) , and the memory storage is O (N) with N the total number of points. Extensive numerical experiments verify our analytical results and demonstrate the effectiveness of our method in solving various problems. [ABSTRACT FROM AUTHOR]

In this paper, we consider the generalized sine-Gordon equation ψ t x = (1 + a ∂ x 2) sin ψ and the sinh-Poisson equation u x x + u y y + σ sinh u = 0 , where a is a real parameter, and σ is a positive parameter. Under different conditions, e.g., a = 0 , a ≠ 0 , and σ > 0 , the periods of the periodic wave solutions for the above two equations are discussed. By the transformation of variables, the generalized sine-Gordon equation and sinh-Poisson equations are reduced to planar dynamical systems whose first integral includes trigonometric terms and exponential terms, respectively. We successfully handle the trigonometric terms and exponential terms in the study of the monotonicity of the period function of periodic solutions. [ABSTRACT FROM AUTHOR]

We present the construction of the ground state of the Gross–Pitaevskii–Poisson equations using genetic algorithms. By employing numerical solutions, we develop an empirical formula for the density that works within the considered parameter space. Through the analysis of both numerical and empirical solutions, we investigate the stability of these ground-state solutions. Our findings reveal that while the numerical solution outperforms the empirical formula, both solutions lead to similar oscillation modes. We observe that the stability of the solutions depends on specific values of the central density and the nonlinear self-interaction term and establish an empirical criterion delineating the conditions under which the solutions exhibit stability or instability. [ABSTRACT FROM AUTHOR]

In this study, we characterize LP-Kenmotsu manifolds admitting *-Ricci-Yamabe solitons (*-RYSs) and gradient *-Ricci-Yamabe solitons (gradient *-RYSs). It is shown that an LP-Kenmotsu manifold of dimension n admitting a *-Ricci-Yamabe soliton obeys Poisson's equation. We also determine the necessary and sufficient conditions under which the Laplace equation is satisfied by LP-Kenmotsu manifolds. Finally, by using a non-trivial example of an LP-Kenmotsu manifold, we verify some results of our paper. [ABSTRACT FROM AUTHOR]

This paper is devoted to finding a necessary and sufficient condition for the solvability of the overdetermined problem for Poisson's equation with both the Dirichlet and Neumann conditions on the entire boundary. The proof is based on the boundary condition formula for the Newton potential. The obtained results are also extended to general second-order linear elliptic equations. As a byproduct, we present a characterization of the Schiffer property. It gives a definitive answer to the Schiffer problem. [ABSTRACT FROM AUTHOR]

This paper considers the following fractional Kirchhoff–Schrödinger–Poisson equation: a+b∫ℝ3(−Δ)α2u2dx(−Δ)αu+ϕu=h(u)+λuinℝ3,(−Δ)βϕ=u2inℝ3,$$ \left\{\begin{array}{lll}& \left(a&#x0002B;b{\int}_{{\mathrm{\mathbb{R}}}&#x0005E;3}{\left&#x0007C;{\left(-\Delta \right)}&#x0005E;{\frac{\alpha }{2}}u\right&#x0007C;}&#x0005E;2\mathrm{d}x\right){\left(-\Delta \right)}&#x0005E;{\alpha }u&#x0002B;\phi u&#x0003D;h(u)&#x0002B;\lambda u& \mathrm{in}\kern0.4em {\mathrm{\mathbb{R}}}&#x0005E;3,\\ {}& {\left(-\Delta \right)}&#x0005E;{\beta}\phi &#x0003D;{u}&#x0005E;2& \mathrm{in}\kern0.4em {\mathrm{\mathbb{R}}}&#x0005E;3,\end{array}\right. $$where a,b>0,0<α,β<1,2α+2β>3,λ∈ℝ,h∈C(ℝ,ℝ)$$ a,b>0,0<\alpha, \beta <1,2\alpha &#x0002B;2\beta >3,\lambda \in \mathrm{\mathbb{R}},h\in C\left(\mathrm{\mathbb{R}},\mathrm{\mathbb{R}}\right) $$. For the case of h(u)∼|u|p−2u$$ h(u)\sim {\left&#x0007C;u\right&#x0007C;}&#x0005E;{p-2}u $$ with p∈(8α+4β−3α+β,8α3+2)∪(8α3+2,2α∗),2α∗=63−2α$$ p\in \left(\frac{8\alpha &#x0002B;4\beta -3}{\alpha &#x0002B;\beta },\frac{8\alpha }{3}&#x0002B;2\right)\cup \left(\frac{8\alpha }{3}&#x0002B;2,{2}_{\alpha}&#x0005E;{\ast}\right),{2}_{\alpha}&#x0005E;{\ast }&#x0003D;\frac{6}{3-2\alpha } $$, the nonlinear term h$$ h $$ satisfies more general conditions, we obtain the existence of normalized solutions for the above system by using Pohozaev manifold and variational method. [ABSTRACT FROM AUTHOR]

The primary objective of this study is to explore the spatial rotary movements of a symmetrically charged rigid body (RB) that is rotating around a fixed point, akin to Lagrange's scenario as a novel scenario where its center of mass experiences a slight displacement from the symmetry dynamic axis. The body's movement is presumed to be affected by a gyrostatic moment and a force from an electromagnetic field, attributed to the presence of a located point charge on this axis. The regulating equations of motion that are pertaining to the equations Euler–Poisson are solved through the utilization of Poincaré's small parameter method along with its adaptations when the scenario of irrational frequencies is considered. The three angles of Euler are derived and graphed to ascertain the body's position at any point throughout the motion. The temporal evolutions of the achieved outcomes are drawn to showcase the significant impact of the selected parameters on the motion. The phase plane diagrams have been generated to illustrate the stability of the body during the motion. The novelty of studying the rotatory motion of a charged RB under these specific conditions lies in the intricate interplay of gyrostatic effects, magnetic interactions, and nonlinear dynamics. This research can push the boundaries of theoretical mechanics and provide valuable insights and tools for both theoretical advancements and practical applications. Moreover, the achieved results from this analysis can be utilized to improve the dynamic performance of diverse engineering applications, particularly those dependent on gyroscopic theory. This includes enhancing the functionality of satellites, compasses, submarines, and automatic pilots used in aircraft. Essentially, the findings have practical implications for optimizing the performance and stability of these systems. [ABSTRACT FROM AUTHOR]

This study presents a semi‐implicit MPM to adequately characterize the mechanical behavior of unsaturated soil based on Biot's mixture theory. To represent the dependency of the degree of saturation on the suction, we employ the VG model along with a soil‐water characteristic curve, which determines a functional form of permeability called the Mualem model. Hencky's hyperelastic model and the Drucker‐Prager model assuming nonassociativity are adopted for elastic and plastic deformations, respectively. The novelty of this study is the incorporation of the fractional‐step method into the MPM framework so that the pore water pressure is obtained by implicitly solving the pressure Poisson's equation, which reduces numerical instability and improves computational efficiency. Also, because the drag force between solid and liquid phases is evaluated using the intermediate velocity of pore water relative to the intermediate velocity of solid skeleton, the time increment can be chosen without considering the magnitude of water permeability. In addition, to suppress "odd‐even" oscillation, we employ a sub‐grid method in which two grids with different spatial resolutions are used for the velocities and pore water pressure. Furthermore, considering the advantages and disadvantages of two different interpolation schemes for pore water pressure, we suggest switching the schemes depending on the model conditions. Several numerical examples are presented to demonstrate the performance of the proposed method. Specifically, unidirectional consolidation and leak flow analyses are performed for verification purposes, followed by validation analysis of a model experiment of infiltration‐induced landslides. [ABSTRACT FROM AUTHOR]

In this paper, the solvability of a new class of nonlocal boundary value problems for the Poisson equation is studied. Nonlocal conditions are specified in the form of a connection between the values of the unknown function at different points of the boundary. In this case, the boundary operator is determined using matrices of involution-type mappings. Theorems on the existence and uniqueness of solutions to the studied problems are proved. Using Green's functions of the classical Dirichlet and Neumann boundary value problems, Green's functions of the studied problems are constructed and integral representations of solutions to these problems are obtained. [ABSTRACT FROM AUTHOR]

Research on two-dimensional material-based phototransistors has recently become a topic of great interest. However, the high number of design features, which impact the performance of these devices, and the multi-physical nature of the device operation make the accurate analysis of these devices a challenge. Here, we present a simple yet effective numerical framework to overcome this challenge. The one-dimensional framework is constructed on the drift-diffusion equations, Poisson's equation, and wave propagation in multi-layered medium formalism. We apply this framework to study phototransistors made from monolayer molybdenum disulfide ( MoS 2 ) placed on top of a back-gated silicon-oxide-coated silicon substrate. Numerical results, which show good agreement with the experimental results found in the literature, emphasize the necessity of including the inhomogeneous background for accurately calculating device metrics such as quantum efficiency and bandwidth. For the first time in literature, we calculate the phase noise of these phototransistors, which is a crucial performance metric for many applications where precise timing and synchronization are critical. We determine that applying a low drain-to-source voltage is the key requirement for low phase noise. [ABSTRACT FROM AUTHOR]

We study the existence of positive solutions for a class of systems which strongly couple a quasilinear Schrödinger equation driven by a weighted N$N$‐Laplace operator and without the mass term, and a higher‐order fractional Poisson equation. Since the system is set in RN$\mathbb {R}^N$, the limiting case for the Sobolev embedding, we consider nonlinearities with exponential growth. Existence is proved relying on the study of a corresponding Choquard equation in which the Riesz kernel is a logarithm, hence sign‐changing and unbounded from above and below. This is in turn solved by means of a variational approximating procedure for an auxiliary Choquard equation, where the logarithm is uniformly approximated by polynomial kernels. Our results are new even in the planar case N=2$N=2$. [ABSTRACT FROM AUTHOR]

Physics-informed neural networks (PINNs) are a powerful class of numerical solvers for partial differential equations, employing deep neural networks with successful applications across a diverse set of problems. However, their effectiveness is somewhat diminished when addressing issues involving singularities, such as point sources or geometric irregularities, where the approximations they provide often suffer from reduced accuracy due to the limited regularity of the exact solution. In this work, we investigate PINNs for solving Poisson equations in polygonal domains with geometric singularities and mixed boundary conditions. We propose a novel singularity enriched PINN, by explicitly incorporating the singularity behavior of the analytic solution, e.g., corner singularity, mixed boundary condition, and edge singularities, into the ansatz space, and present a convergence analysis of the scheme. We present extensive numerical simulations in two and three dimensions to illustrate the efficiency of the method, and also a comparative study with several existing neural network based approaches. [ABSTRACT FROM AUTHOR]

We propose and analyze a novel approach to construct structure preserving approximations for the Poisson-Nernst-Planck equations, focusing on the positivity preserving and mass conservation properties. The strategy consists of a standard time marching step with a projection (or correction) step to satisfy the desired physical constraints (positivity and mass conservation). Based on the L² projection, we construct a second order Crank-Nicolson type finite difference scheme, which is linear (exclude the very efficient L² projection part), positivity preserving, and mass conserving. Rigorous error estimates in the L² norm are established, which are both second order accurate in space and time. The other choice of projection, e.g., H¹ projection, is discussed. Numerical examples are presented to verify the theoretical results and demonstrate the efficiency of the proposed method. [ABSTRACT FROM AUTHOR]

A bosonic Laplacian is a conformally invariant second‐order differential operator acting on smooth functions defined on domains in Euclidean space and taking values in higher‐order irreducible representations of the special orthogonal group. In this paper, we firstly introduce the motivation for study of the generalized Maxwell operators and bosonic Laplacians (also known as the higher spin Laplace operators). Then, with the help of connections between Rarita–Schwinger type operators and bosonic Laplacians, we solve Poisson's equation for bosonic Laplacians. A representation formula for bounded solutions to Poisson's equation in Euclidean space is also provided. In the end, we provide Green's formulas for bosonic Laplacians in scalar‐valued and Clifford‐valued cases, respectively. These formulas reveal that bosonic Laplacians are self‐adjoint with respect to a given L2 inner product on certain compact supported function spaces. [ABSTRACT FROM AUTHOR]

Physics‐informed neural networks (PINNs) provide a new class of mesh‐free methods for solving differential equations. However, due to their long training times, PINNs are currently not as competitive as established numerical methods. A promising approach to bridge this gap is transfer learning (TL), that is, reusing the weights and biases of readily trained neural network models to accelerate model training for new learning tasks. This work applies TL to improve the performance of PINNs in the context of magnetostatic field simulation, in particular to resolve boundary value problems with geometrical variations of the computational domain. The suggested TL workflow consists of three steps. (a) A numerical solution based on the finite element method (FEM). (b) A neural network that approximates the FEM solution using standard supervised learning. (c) A PINN initialized with the weights and biases of the pre‐trained neural network and further trained using the deep Ritz method. The FEM solution and its neural network‐based approximation refer to an computational domain of fixed geometry, while the PINN is trained for a geometrical variation of the domain. The TL workflow is first applied to Poisson's equation on different 2D domains and then to a 2D quadrupole magnet model. Comparisons against randomly initialized PINNs reveal that the performance of TL is ultimately dependent on the type of geometry variation considered, leading to significantly improved convergence rates and training times for some variations, but also to no improvement or even to performance deterioration in other cases. [ABSTRACT FROM AUTHOR]

Poisson's equation frequently emerges in many fields, yet its exact solution is rarely feasible, making the numerical approach notably valuable. This study aims to provide a tutorial-level guide to numerically solving Poisson's equation, focusing on estimating the electrostatic field and potential resulting from an axially symmetric Gaussian charge distribution. The Finite Difference Method is utilized to discretize the desired spatial domain into a grid of points and approximate the derivatives using finite difference approximations. The resulting system of linear equations is then tackled using the Successive Over-Relaxation technique. Our results suggest that the potential obtained from the direct integration of the distance-weighted charge density is a reasonable choice for Dirichlet boundary conditions. We examine a scenario involving a charge in free space; the numerical electrostatic potential is estimated to be within a tolerable error range compared to the exact solution. [ABSTRACT FROM AUTHOR]

Recently, the physics-informed neural network (PINN) has attracted much attention in solving partial differential equations (PDEs). The success is due to the strong generalization ability of the neural network (NN), which is supported by the universal approximation theorem, and its mesh-free implementation. In this paper, we propose a multi-head NN enriched PINN (MultiPINN) for solving differential equations. The trial function is built based on the radial basis function (RBF)-interpolation, which makes NN training parameters partially interpre. The loss function is constructed by embedding the physics information of differential equations and boundary conditions. Then the parameters in MultiPINN are trained using the ADAM optimizer. A significant feature of MultiPINN is that it combines the traditional RBF interpolation method with machine learning (ML) techniques. The ML technique is employed to learn the basis feature enrichment that provides global information. The multi-head mechanism is used so that each node has multiple bases, which can improve the accuracy of the MultiPINN solution. Two ordinary differential equations and three partial differential equations, i.e. the convection equation, the Burgers equation, and the Poisson equation, are used in the numerical experiments. The experimental outcomes demonstrate that MultiPINN produces solutions consistent with both analytical solutions and solutions obtained through traditional numerical methods. Additionally, MultiPINN shows robustness and adaptability over the other NN-based methods in the implementations. [ABSTRACT FROM AUTHOR]

This work aims to provide a unique analytical model designed for calculating the magnetic field of a surface-mounted permanent magnet Vernier motor (SPMVM). The model is created utilizing the separation variable method to solve Laplace or Poisson equations of the vector magnetic potential in a quasi-Cartesian coordinate system subdomain model. The theoretical justifications for the final analytical equations of electromagnetic torque, no-load back-EMF, stator inductance, and magnetic flux density distribution are given in detail. Then the solution of the equation is expressed in the form of a hyperbolic function. Furthermore, the improved analytical method in the study is generic and can be applied to estimating the parameters of machines with different design concepts. Finally, FEM is employed as the reference for comparison between other models. Compared to the original subdomain model, the calculation results of the improved analytical model are more time-efficient and closer to those of FEM. In addition, the improved analytical method can reduce the dimension of the coefficient matrix and improve the calculation speed. The SPMVM prototype is then built to demonstrate the validity and feasibility of the improved subdomain method and serves as a guide for future SPMVM development. [ABSTRACT FROM AUTHOR]

The nonlinear Schrödinger–Poisson equation with gravitational potential field is examined, along with its many soliton interactions and travelling wave solutions. By applying Hirota Bilinear scheme, we will give a brief study on transformation of solitons, also by Sine–Gordon expansion scheme solitary travelling wave solutions will be obtained in terms of trigonometric and hyperbolic functions. The innovative Hirota bilinear method is a powerful and standard technique, play a crucial role in producing soliton solutions as well as lump solutions. With the assistance of this method one can transformed integrable equation into Hirota bilinear form under dependent variable transformation. Soliton solutions obtained by Hirota bilinear method are significant in mathematical physics, may be superimposed in fibers. These solitons are applicable in optical communications, which enable to produce faster, richer, more secure and more flexible communication systems. Also we will investigate the gravitational potential's behaviour on soliton solutions, such as parabolic, bright, dark, anti-dark, combination and S-shaped solitons. Contour plots and three-dimensional soliton solutions are provided. These solitons are important in nonlinear processes, magnetic devices, and lasers beams because of variations in gravitational field. [ABSTRACT FROM AUTHOR]

The rational development of organic electronic devices can greatly benefit from a molecular‐level description, where an accurate description of the Coulombic interactions among charge carriers holds paramount importance. However, the molecular‐level organic field‐effect transistor (OFET) simulations do not fully include the short‐range charge‐carrier Coulombic interactions, thus limiting their accuracy. Here, an efficient solution is demonstrated to the 3D Poisson's equation with mixed boundary conditions, optimized for integration into the simulations of organic electronic devices. It leverages the finite difference method for discretization, the Krylov subspace iterative methods for solving the linear system, the algebraic multigrid algorithm for preconditioning, and OpenMP and CUDA parallelization for acceleration. For the organic field‐effect transistors of micrometer sizes or smaller dimensions, obtaining the electric potential within the device takes less than 0.1 s, which is sufficiently efficient for molecular‐level modeling. As a result, both short‐ and long‐range electrostatic interactions are successfully incorporated in the molecular‐level modeling of OFET devices. This integration significantly enhances the robustness of OFET modeling for future applications. [ABSTRACT FROM AUTHOR]

The paper is devoted to the study of the solvability of some nonlocal boundary value problems for the Poisson equation in the unit ball. Nonlocal conditions are specified in the form of a connection between the values of the unknown function at different points of the boundary. In this case, the boundary operator is determined using matrices of involution-type mappings. Theorems on the existence and uniqueness of solutions to the studied problems are proved. An explicit form of the Green's function and an integral representation of solutions to the studied problems are constructed. [ABSTRACT FROM AUTHOR]

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This paper concerns the structural stability of subsonic steady-states to the bipolar Euler-Poisson equations under small perturbation of doping profiles. Here, the electron density is imposed with degenerate sonic boundary and considered in interiorly subsonic case, while the hole density is considered in fully subsonic case. Unlike the unipolar model, we show that the structural stability in bipolar model holds regardless of the type of doping profile and propose a new version of comparison principle, which captures the intrinsic negative correlation between electrons and holes. To overcome the difficulty caused by the degenerate effect at the sonic boundary, we introduce two different weight functions to handle the singularity near both sides of endpoints separately. The approaches adopted to prove the structural stability include the local singularity analysis, the monotonicity method, the continuation argument and the squeezing skill. Several numerical simulations are performed which support our theoretical results. [ABSTRACT FROM AUTHOR]

This paper investigates the electromagnetic fields being scattered by a metal spherical object in a vacuum environment, providing a numerical implementation of the obtained analytical results. A time-harmonic magnetic dipole source, far enough, emits the incident field at low frequencies, oriented arbitrarily in the three-dimensional space. The aim is to find a detailed solution to the scattering problem at spherical coordinates, which is useful for data inversion. Based on the theory of low frequencies, the Maxwell-type problem is transformed into Laplace's or Poisson's interconnected equations, accompanied by the proper boundary conditions on the perfectly conducting sphere and the radiation conditions at infinity, which are solved gradually. Broadly, the static and the first three dynamic terms are sufficient, while the terms of a higher order are negligible, which is confirmed by the field graphical representation. [ABSTRACT FROM AUTHOR]

Assuming spherical symmetry and weak field, it is shown that if one solves the Poisson equation or the Einstein field equations sourced by a topological defect, i.e. a singularity of a very specific form, the result is a localized gravitational field capable of driving flat rotation (i.e. Keplerian circular orbits at a constant speed for all radii) of test masses on a thin spherical shell without any underlying mass. Moreover, a large-scale structure which exploits this solution by assembling concentrically a number of such topological defects can establish a flat stellar or galactic rotation curve, and can also deflect light in the same manner as an equipotential (isothermal) sphere. Thus, the need for dark matter or modified gravity theory is mitigated, at least in part. [ABSTRACT FROM AUTHOR]

This paper explores the utilization of randomized SVD (rSVD) in the context of kernel matrices arising from radial basis functions (RBFs) for the purpose of solving interpolation and Poisson problems. We propose a truncated version of rSVD, called trSVD, which yields a stable solution with a reduced condition number in comparison to the non-truncated variant, particularly when manipulating the scale or shape parameter of RBFs. Notably, trSVD exhibits exceptional proficiency in capturing the most significant singular values, enabling the extraction of critical information from the data. When compared to the conventional truncated SVD (tSVD), trSVD achieves comparable accuracy while demonstrating improved efficiency. Furthermore, we explore the potential of trSVD by employing scale parameter strategies, such as leave-one-out cross-validation and effective condition number. Then, we apply trSVD to solve a 2D Poisson equation, thereby showcasing its efficacy in handling partial differential equations. In summary, this study offers an efficient and accurate solver for RBF problems, demonstrating its practical applicability. The code implementation is provided to the scientific community for their access and reference. [ABSTRACT FROM AUTHOR]

For the spatial discretization of elliptic and parabolic partial differential equations (PDEs), we provide a Matrix-Oriented formulation of the classical Finite Element Method, called MO-FEM, of arbitrary order k ∈ N. On a quite general class of 2D domains, namely separable domains , and even on special surfaces, the discrete problem is then reformulated as a multiterm Sylvester matrix equation where the additional terms account for the geometric contribution of the domain shape. By considering the IMEX Euler method for the PDE time discretization, we obtain a sequence of these equations. To solve each multiterm Sylvester equation, we apply the matrix-oriented form of the Preconditioned Conjugate Gradient (MO-PCG) method with a matrix-oriented preconditioner that captures the spectral properties of the Sylvester operator. Solving the Poisson problem and the heat equation on some separable domains by MO-FEM-PCG, we show a gain in computational time and memory occupation wrt the classical vector PCG with same preconditioning or wrt a LU based direct method. As an application, we show the advantages of the MO-FEM-PCG to approximate Turing patterns on some separable domains and cylindrical surfaces for a morphochemical reaction-diffusion PDE system for battery modelling. [ABSTRACT FROM AUTHOR]

We study the Schrödinger–Poisson–Slater equation where p ∈ (2 , 3) , λ > 0 , V (x) ∈ C (R 3 , R +) and I 2 (x) : = (4 π | x |) - 1 is the Newtonian potential. We prove the nonexistence of nontrivial solutions, existence of positive nonradial (and radial) ground-state solutions, and mountain-pass-type solutions to (SPS), depending on the values of parameters p and λ . To our knowledge, this is the first study of the existence of ground-state solutions at positive energy levels when p ∈ (2 , 3) . Furthermore, we show that a symmetry breaking occurs for the ground-state solutions, which is a purely nonlocal phenomenon that cannot be observed in the local prototype case. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence of normalized solutions for the nonautonomous Schrödinger–Poisson equations - Δ u + λ u + | x | - 1 ∗ | u | 2 u = A (x) | u | p - 2 u , in R 3 , where λ ∈ R , A ∈ L ∞ (R 3) satisfies some mild conditions. Due to the nonconstant potential A, we use Pohozaev manifold to recover the compactness for a minimizing sequence. For p ∈ (2 , 3) , p ∈ (3 , 10 3) and p ∈ (10 3 , 6) , we adopt different analytical techniques to overcome the difficulties due to the presence of three terms in the corresponding energy functional which scale differently. [ABSTRACT FROM AUTHOR]

P- and S-wave separation plays an important role in elastic reverse-time migration. It can reduce the artifacts caused by crosstalk between different modes and improve image quality. In addition, P- and Swave separation can also be used to better understand and distinguish wave types in complex media. At present, the methods for separating wave modes in anisotropic media mainly include spatial nonstationary filtering, low-rank approximation, and vector Poisson equation. Most of these methods require multiple Fourier transforms or the calculation of large matrices, which require high computational costs for problems with large scale. In this paper, an efficient method is proposed to separate the wave mode for anisotropic media by using a scalar anisotropic Poisson operator in the spatial domain. For 2D problems, the computational complexity required by this method is 1/2 of the methods based on solving a vector Poisson equation. Therefore, compared with existing methods based on pseudo-Helmholtz decomposition operators, this method can significantly reduce the computational cost. Numerical examples also show that the P and S waves decomposed by this method not only have the correct amplitude and phase relative to the input wavefield but also can reduce the computational complexity significantly. [ABSTRACT FROM AUTHOR]

We describe a pressure projection scheme for the simulation of incompressible flow in cubic domains with open boundaries based on fast Fourier transforms. The scheme is implemented in flow_solve, a numerical code designed for process studies of rotating, density‐stratified flow. The main algorithmic features of the open‐boundary code are the near‐spectral accuracy of the discrete differentiation and a dynamic two‐dimensional domain decomposition that scales efficiently to large numbers of processors. The simulated flows are not required to be periodic or to satisfy symmetry conditions at the open boundaries owing to the use of mixed series expansions combining cosine and singular Bernoulli polynomial basis functions. These expansions facilitate the imposition of inhomogeneous boundary conditions and allow the code to be used for offline, one‐way nesting within an arbitrarily embedded subdomain of a larger scale simulation. The projection scheme is designed to exploit a simple and powerful numerical engine: inversion of Poisson's equation with homogeneous Neumann boundary conditions using fast cosine transforms. Here, we describe the mathematical transformations used to accommodate the imposition of space‐ and time‐varying boundary conditions. The utility of the approach for process studies and for nesting within submesoscale‐resolving ocean models is demonstrated with simulations of wind‐driven near‐inertial waves in the upper ocean. Plain Language Summary: Internal gravity waves in the upper ocean are affected by the presence of features such as fronts, eddies, and geographically constrained winds and currents. Ocean models that are able to accurately capture geographic realism often lack the capacity to resolve the intricacies of the internal waves that may be generated. This paper presents a mathematical framework that combines the realism of ocean models with high‐fidelity simulation of small‐scale dynamics. The framework is demonstrated through a series of examples of increasing complexity, including wave trapping in a wind‐perturbed anticyclonic vortex and simulation of near‐inertial waves near a density front in the Gulf of Lion. The proposed methodology has the potential to take advantage of the increasing horizontal resolution and fine‐scale realism of ocean models, while also relaxing the approximations that limit their ability to resolve the detailed dynamics of the internal wave field. Key Points: A computational framework based on fast spectral transforms is developed for computing nonhydrostatic flow in domains with open boundariesMixed basis function expansions are formulated to differentiate discrete data lacking symmetry or periodicity with near‐spectral accuracyNested simulations of wave trapping in a vortex and near‐inertial waves near a density front in the Gulf of Lion demonstrate the approach [ABSTRACT FROM AUTHOR]

The issues of modeling the electrical characteristics of a silicon conical field-effect gate-all-around (GAA) nanotransistor are discussed. An analytical model of the drain current of a transistor with a fully encompassing conical gate with a stacked subgate SiO2/HfO2 oxide, taking into account the influence of the charge of the interfacial trap at the Si/SiO2 interface, is developed. To model the potential distribution in a conical working area under the condition of a constant trap density, an analytical solution of the Poisson equation is obtained using the parabolic approximation method in the cylindrical coordinate system with the corresponding boundary conditions. The potential model is used to develop an expression for the drain current of a GAA nanotransistor with a stacked subgate oxide. The key electrical and physical characteristics are numerically studied depending on the density of the traps and the thickness of the SiO2 and HfO2 layers. [ABSTRACT FROM AUTHOR]

Induced polarization (IP) tomography measurements, as a near-surface geophysical method, can provide information about the degree of chargeability of subsurface materials and are commonly used in mineral exploration, engineering studies (e.g., sediment/bedrock interface identification, crushed zones and faults detection, and landslide and soil properties imaging), as well as in environmental investigations (contaminant plums identification and landfill characterization). The purpose of these measurements is to obtain the distribution of polarizability characteristics inside an object, generally below the surface, at the boundary of the object, or outside the area in question. The results of such measurements can be mathematically modeled for the specific polarizability properties by solving the Poisson’s equation restricted by appropriate boundary conditions. In this paper, we focused on the importance of simulating induced-polarization responses and retrieving chargeability distributions in geo-materials to enhance the characterization of subsurface structures. We presented the methods for forward modelling and non-linear inversion of IP measurements. To this end, in the first step, the Poisson’s equation for a two-dimensional ground with an arbitrary distribution of conductivity is solved using the finite difference numerical method. In the next step, based on the existing relations between conductivity and chargeability (the Siegel’s formulation), the apparent IP response is calculated. Finally, we solved the non-linear chargeability inversion problem following a non-linear apparent resistivity inversion. This is achieved by imposing physical constraints to prevent the estimation of unrealistic model parameters, using a Newton-based optimization method. To evaluate the efficiency of the proposed methodology, we utilized the proposed algorithm on two simulated examples and a real data set. Our numerical results show that the algorithm reliably represents the main features and structure of the Earth’s subsurface in terms of resistivity and chargeability models. All the algorithms presented in this paper have written in the MATLAB programming language. [ABSTRACT FROM AUTHOR]

Trap density (Nt), density of carriers (no) and mobility are extracted in Al/beetroot/Cu Schottky diode using the Poisson's equation. The trap density and carrier concentration were found to be 1.25 × 109 and 1.8 × 109 cm–3, respectively, with the mobility 124.54 cm2 v−1 s−1 for Al/beetroot/Cu. The device shows highly asymmetric current–voltage characteristics with a good degree of nonlinearity. The diode shows an acceptable low-voltage large signal and small signal nonlinearities with asymmetry of 17.6 at 0.85 V, maximum rate of change of nonlinearity of 6 and sensitivity of 25.6 A W–1 at 0.7 V. For Al/Beet/Cu the calculated zero bias resistance is 23.3 kΩ, which is much smaller than previously reported MIM diodes. For this device fNL > 3 and fSENS > 7 A W–1, so this present device has a great potential as a rectifier. We also proposed a one-dimensional Al/organic semiconductor/Cu diode with low capacitance. The cut-off frequency of the one-dimensional Al/beetroot/Cu device is estimated as 20.8 × 1012 Hz and if the beet layer is replaced by Cur-M then the estimated frequency becomes 29 × 1012 Hz. This article also reports the comparative study between a series of natural dyes-based diodes and finally the comparison with our fabricated inorganic Al/TiO2/Cu diode. The sensitivity of Al/indigo/Cu, Al/turmeric/Cu and Al/beetroot/Cu are 0.4, 4.3 and 25.7 A W–1 with their zero-bias resistances 2.57 MΩ, 12.8 kΩ and 23.26 kΩ. Again, it was proven by the parabolic behaviour of the dI/dV vs. V plot that all-natural dye-based Schottky diodes may be affected in some way by the quantum tunnelling phenomena. Finally, the simulation was utilized to develop predicted behaviour and a better understanding of the physical mechanisms determining the effectiveness of the device under research. [ABSTRACT FROM AUTHOR]

To solve the Laplacian problems, we adopt a meshless method with the multiquadric radial basis function (MQRBF) as a basis whose center is distributed inside a circle with a fictitious radius. A maximal projection technique is developed to identify the optimal shape factor and fictitious radius by minimizing a merit function. A sample function is interpolated by theMQ-RBF to provide a trial coefficient vector to compute the merit function. We can quickly determine the optimal values of the parameters within a preferred rage using the golden section search algorithm. The novel method provides the optimal values of parameters and, hence, an optimal MQ-RBF; the performance of the method is validated in numerical examples. Moreover, nonharmonic problems are transformed to the Poisson equation endowed with a homogeneous boundary condition; this can overcome the problem of these problems being ill-posed. The optimal MQ-RBF is extremely accurate. We further propose a novel optimal polynomial method to solve the nonharmonic problems, which achieves high precision up to an order of 10-11. [ABSTRACT FROM AUTHOR]

We are concerned with a global existence theory for finite‐energy solutions of the multidimensional Euler‐Poisson equations for both compressible gaseous stars and plasmas with large initial data of spherical symmetry. One of the main challenges is the strengthening of waves as they move radially inward towards the origin, especially under the self‐consistent gravitational field for gaseous stars. A fundamental unsolved problem is whether the density of the global solution forms a delta measure (i.e., concentration) at the origin. To solve this problem, we develop a new approach for the construction of approximate solutions as the solutions of an appropriately formulated free boundary problem for the compressible Navier‐Stokes‐Poisson equations with a carefully adapted class of degenerate density‐dependent viscosity terms, so that a rigorous convergence proof of the approximate solutions to the corresponding global solution of the compressible Euler‐Poisson equations with large initial data of spherical symmetry can be obtained. Even though the density may blow up near the origin at a certain time, it is proved that no delta measure (i.e., concentration) in space‐time is formed in the vanishing viscosity limit for the finite‐energy solutions of the compressible Euler‐Poisson equations for both gaseous stars and plasmas in the physical regimes under consideration. [ABSTRACT FROM AUTHOR]

The purpose of this study is to present a subgeometric convergence formula for the stationary distribution of the finite-level M/G/1-type Markov chain when taking its infinite-level limit, where the upper boundary level goes to infinity. This study is carried out using the fundamental deviation matrix , which is a block-decomposition-friendly solution to the Poisson equation of the deviation matrix. The fundamental deviation matrix provides a difference formula for the respective stationary distributions of the finite-level chain and the corresponding infinite-level chain. The difference formula plays a crucial role in the derivation of the main result of this paper, and the main result is used, for example, to derive an asymptotic formula for the loss probability in the MAP/GI/1/ N queue. [ABSTRACT FROM AUTHOR]