Singular Source Research Articles

Limiting profile of the blow-up solutions for the fourth-order nonlinear Emden-Fowler equation with a singular source

We consider the anisotropic Emden-Fowler equation with exponential nonlinearity in fourth-dimensional with a singular source, given by \begin{document}$ \begin{equation*} \left\{\begin{array}{rclll}\Delta(a(x)\Delta u)-V(x)\,div(a(x)\nabla\,u) & = & \rho^{4} a(x)\,e^{u} -32 \pi^2\;\alpha\,a(x)\,\delta_{p}&\mbox{in} \; \; \Omega\subset \mathbb R^{4}\\\\u = \Delta u & = &0 &\mbox{on} \; \; \partial\Omega. \end{array} \right. \end{equation*} $\end{document} The leading part $ \Delta $ is, usually, called Laplacian operator. $ V = V(|x|) $ is a smooth bounded radial potential and $ a = a(|x|) $ is a given smooth radial function over $ \bar{\Omega} $, called the Schrödinger wave function. Namely, we extend the result of [8] by allowing the presence in the equation a singular source given by Dirac measure with pole at point $ p $ and a generalization in higher dimensional, of the interest problem treated in [11]. We construct a family of solutions $ u_{\rho} $ that concentrate at $ p $ as $ \rho $ tends to $ 0 $.

Sixth-Order Compact Finite Difference Method for 2D Helmholtz Equations with Singular Sources and Reduced Pollution Effect

Sixth-Order Compact Finite Difference Method for 2D Helmholtz Equations with Singular Sources and Reduced Pollution Effect

Modeling Supercontinuum Lasers to Simulate Damage Thresholds on the Retina

Laser damage thresholds depend on the amount of energy absorbed, and in this specific case, energy absorbed by the retina. The goal of our research is to simulate a supercontinuum laser source and compare the resulting damage to that of the current ANSI approach. To calculate retinal damage we utilize the Arrhenius integral, which returns numerical values for laser-induced damage with values ranging from zero to one depending on the severity. Our simulations use numerical analysis to iterate through different coordinates in space and time to return data representative of the physical situation after we expose the eye to a gaussian laser beam. We have performed simulations ranging from a singular laser source up to a three-laser source and are currently working towards an automated method of generating these complex configurations in the hopes of more efficiently replicating the damage properties of a specific laser product and comparing it to the existing ANSI approach.

Continuous Monitoring of Hyperthermia Treatment of Breast Tumors With Singular Sources Method

Qualitative microwave imaging methods provide a fast and accurate reconstruction of the shapes of the targets from scattered electric field measurements. In this paper, a qualitative imaging method, the singular sources method (SSM), is analyzed for two-dimensional transverse magnetic electromagnetic (2D-TM EM) inverse scattering scenarios. The contribution of the paper can be briefly summarized as: (i) The SSM was previously proposed for the far-field scenario, here we extend the SSM in the near field - inhomogeneous background setup. While making the extension each step (which includes an integral equation) is explained by exploiting the linearity and reciprocity principles to give physical insights. (ii) The relation between the electrical parameters of the scatterers and the indicator of SSM is derived. (iii) To analyze the performance of SSM in real-world problems, the proposed method is tested for monitoring hyperthermia treatment problems with a realistic breast model. Obtained results show that the SSM can handle realistic breast phantoms for monitoring hyperthermia problems.

An ACA-BM-SBM for 2D acoustic sensitivity analysis

<abstract> <p>In this paper, we present a novel computational approach (named ACA-BM-SBM) for the calculation of 2D acoustic sensitivity by combining the Burton-Miller-type singular boundary method (BM-SBM) with the adaptive cross-approximation (ACA) algorithm. The BM-SBM circumvents the source singularities of the fundamental solutions by introducing the origin intensity factors, and it eliminates the fictitious frequency problem in external acoustic fields by introducing the Burton-Miller formula. As a semi-analysis meshless method, the BM-SBM can accurately solve the external acoustic problem governed by the Helmholtz equation. Nevertheless, the computational inefficiency introduced by the dense coefficient matrix renders this method suboptimal, particularly for large-scale simulations. As the number of nodes increases, the computation time and store memory increase dramatically. ACA is a purely algebraic method based on hierarchical matrices which can be used to partition the coefficient matrix step by step. By employing ACA, the BM-SBM can be effectively accelerated, and this results in less computation time, as well as fewer memory requirements. Numerical experiments, including Dirichlet and Neumann boundary conditions, illustrate that the proposed approach is an accurate, efficient and fast numerical method for acoustic sensitivity analysis.</p> </abstract>

Ship Air Pollution Estimation by AIS Data: Case Port of Klaipeda

Ships operating on fossil fuel release pollutant emissions into the atmosphere. Released pollutants have a negative effect on the environment and human health, especially in port cities. For this reason, it is very important to properly evaluate these emissions so they can be managed. The current and most common methodologies for shipping pollution evaluation are used for whole port areas or larger terminals over a long period of time and are not analyzed in terms of detailed activity, which may lead to underestimations in certain areas. This study aims to evaluate emissions from ships in port by combining ships’ technical, AIS and EMEP data that allow us to evaluate emissions in port, not as a singular area source but enables individual ship emissions evaluation at any given point in time. To achieve this emission calculation, an algorithm was compiled by using EMEP/EEA Tier 3 methodology. The developed method presents a way to evaluate emissions in a detailed manner not only for groups of ships but also for individual ships if that is required. This method also lets us analyze shipping emissions’ intensity throughout all port territory and identify the most excessive pollution sources. However, the method adds additional work for researchers because of the huge data arrays required for complex calculations.

Generalized Multiscale Finite Element Method for Highly Heterogeneous Compressible Flow

In this paper, we study the generalized multiscale finite element method (GMsFEM) for single phase compressible flow in highly heterogeneous porous media. We follow the major steps of the GMsFEM to construct permeability dependent offline basis for fast coarse-grid simulation. The offline coarse space is efficiently constructed only once based on the initial permeability field with parallel computing. A rigorous convergence analysis is performed for two types of snapshot spaces. The analysis indicates that the convergence rates of the proposed multiscale method depend on the coarse meshsize and the eigenvalue decay of the local spectral problem. To further increase the accuracy of multiscale method, residual driven online multiscale basis is added to the offline space. The construction of online multiscale basis is based on a carefully design error indicator motivated by the analysis. We find that online basis is particularly important for the singular source. Rich numerical tests on typical 3D highly heterogeneous medias are presented to demonstrate the impressive computational advantages of the proposed multiscale method.

Transient Dynamic Response Analysis of Two-Dimensional Saturated Soil with Singular Boundary Method

In this paper, the singular boundary method (SBM) in conjunction with the exponential window method (EWM) is firstly extended to simulate the transient dynamic response of two-dimensional saturated soil. The frequency-domain (Fourier space) governing equations of Biot theory is solved by the SBM with a linear combination of the fundamental solutions. In order to avoid the perplexing fictitious boundary in the method of fundamental solution (MFS), the SBM places the source point on the physical boundary and eliminates the source singularity of the fundamental solution via the origin intensity factors (OIFs). The EWM is carried out for the inverse Fourier transform, which transforms the frequency-domain solutions into the time-domain solutions. The accuracy and feasibility of the SBM-EWM are verified by three numerical examples. The numerical comparison between the MFS and SBM indicates that the SBM takes a quarter of the time taken by the MFS.

Twisted Light in a Single‐Crystal Fiber: Toward Undistorted Femtosecond Vortex Amplification

AbstractIntense femtosecond optical vortices with spatially structured amplitude and spiral phase front give rise to novel phenomena in light–matter interactions and strong‐field physics. However, current femtosecond vortex sources exhibit a poor power handling capability and amplification remains an open challenge due to a number of inherent technical difficulties. Here, it is demonstrated that a single‐crystal fiber laser amplifier is particularly well‐suited to directly amplify a femtosecond optical vortex without pulse stretching and compression in the time domain, while still maintaining the spatial properties associated with a clear central singularity and a spiral phase front, i.e., a well‐defined amount of orbital angular momentum (OAM).The optical nonlinearity experienced by such twisted light is verified to be substantially weaker compared to a fundamental mode beam where supercontinuum generation and spatial distortion are observed. The simple design and straightforward power scaling capability pave the way toward ultrahigh‐intensity femtosecond singular laser sources with an arbitrary topological charge. Such ultrafast OAM light sources are expected to help reveal complex physical phenomena in light–matter interactions and expand the applications to attoscience with X‐ray vortices, laser plasma acceleration, and micromachining.

Numerical simulation of anti-plane wave propagation in heterogeneous media

This paper formulates a semi-analytical boundary meshless method, the singular boundary method (SBM), to simulate the anti-plane wave motion in the heterogeneous media. Two different continuous inhomogeneity variations are studied, and the corresponding fundamental solutions are derived by means of a simple variable transformation. The derivation reveals that the fundamental solutions are determined by the wavenumbers of the resulted governing equations after the variable transformation. The source singularities of these fundamental solutions are overcome by the novel origin intensity factors (OIFs). The fundamental solutions and OIFs make the SBM applicable for the simulation of the anti-plane wave scattering and diffraction in nonhomogeneous media. Numerical experiments, including the finite and semi-infinite cases with different variations and boundary conditions, are presented to illustrate the convergence, accuracy and effectiveness of the proposed SBM.

Convergence of a diffuse interface Poisson-Boltzmann (PB) model to the sharp interface PB model: A unified regularization formulation

Both the sharp interface and diffuse interface Poisson-Boltzmann (PB) models have been developed in the literature for studying electrostatic interaction between a solute molecule and its surrounding solvent environment. In the mathematical analysis and numerical computation for these PB models, a significant challenge is due to singular charge sources in terms of Dirac delta distributions. Recently, based on various regularization schemes for the sharp interface PB equation, the first regularization method for the diffuse interface PB model has been developed in [S. Wang, E. Alexov, and S. Zhao, Mathematical Biosciences and Engineering, 18, 1370–1405, (2021)] for analytically treating the singular charges. This work concerns with the convergence of a diffuse interface PB model to the sharp interface PB model, as the diffused Gaussian-convolution surface (GCS) approaches to the sharp solvent accessible surface (SAS). Due to the limitation in numerical algorithm and mesh resolution, such a convergence is impossible to be verified numerically. Through analyzing the weak solution for the regularized PB equations, the convergences for both the reaction-field potential and electrostatic free energy are rigorously proved in this work. Moreover, this study provides a unified regularization for both sharp interface and diffuse interface PB models, and clarifies the connection between this unified formulation and the existing regularizations. This lays a theoretical foundation to develop regularization for more complicated PB models.

A shallow Ritz method for elliptic problems with singular sources

In this paper, a shallow Ritz-type neural network for solving elliptic equations with delta function singular sources on an interface is developed. There are three novel features in the present work; namely, (i) the delta function singularity is naturally removed, (ii) level set function is introduced as a feature input, (iii) it is completely shallow, comprising only one hidden layer. We first introduce the energy functional of the problem and then transform the contribution of singular sources to a regular surface integral along the interface. In such a way, the delta function singularity can be naturally removed without introducing a discrete one that is commonly used in traditional regularization methods, such as the well-known immersed boundary method. The original problem is then reformulated as a minimization problem. We propose a shallow Ritz-type neural network with one hidden layer to approximate the global minimizer of the energy functional. As a result, the network is trained by minimizing the loss function that is a discrete version of the energy. In addition, we include the level set function of the interface as a feature input of the network and find that it significantly improves the training efficiency and accuracy. We perform a series of numerical tests to show the accuracy of the present method and its capability for problems in irregular domains and higher dimensions.

Classification and a priori estimates for the singular prescribing Q-curvature equation on 4-manifold

On (M,g) a compact riemannian 4-manifold we consider the prescribed Q-curvature equation defined on M with finite singular sources. We first prove a classification theorem for singular Liouville equations defined on R4 and perform a concentration compactness analysis. Then we derive a quantization result for bubbling solutions and establish a priori estimate under the assumption that certain conformal invariant does not take some quantized values. Furthermore we prove a spherical Harnack inequality around singular sources provided their strength is not an integer. Such an inequality implies that in this case singular sources are isolated simple blow up points.

Erbium-doped ZBLAN fiber laser pumped at 1.7 µm for emission at 2.8 µm.

In this paper, a novel, to the best of our knowledge, efficient pump scheme for an erbium-doped fluoride fiber laser with emission at 2.8 µm in the mid-infrared region is proposed and demonstrated. A singular pump source at 1.7 µm is used to excite Er3+ ions from ground state 4I15/2 to lower laser level 4I13/2, and then further boost the ions to 4I9/2, where a non-radiation transition occurs for the Er3+ ions to reach upper laser level 4I11/2. This scheme can efficiently recycle ions on the lower laser level 4I13/2 by excited-state absorption, therefore realizing population inversion and enhancing laser efficiency. In our demonstration, a 660-mW laser output at 2.8 µm is achieved from a 1.7-µm core-pumped erbium-doped fluoride fiber laser, where the slope efficiency versus launched pump power is 23.7%. The proposed innovative pump scheme shows great potential to realize high-power, high-efficiency erbium-doped fiber lasers at 2.8 µm.

A half-space singular boundary method for predicting ground-borne vibrations

In this study, the singular boundary method (SBM) is extended to simulate the dynamic response of the half-space saturated soils in the frequency domain. The SBM employs the fundamental solutions of the governing equations as basis functions. Thus, the domain of interest can be discretized into several source points merely on the boundary, which makes the SBM available to describe the wave propagation in the half-space soil medium. The source singularity caused by the coincidence of collocation points and source points is overcome by the origin intensity factors (OIFs), which is given for the unclosed boundary. Two numerical examples exhibit the feasibility and advantages of the proposed SBM.

Two Time-Stepping Schemes for Sub-Diffusion Equations with Singular Source Terms

Singular source terms in sub-diffusion equations may lead to the unboundedness of solutions, which will bring a severe reduction of convergence order of existing time-stepping schemes. In this work, we propose two efficient time-stepping schemes for solving sub-diffusion equations with a class of source terms mildly singular in time. One discretization is based on the Gr{\"u}nwald-Letnikov and backward Euler methods. First-order error estimate with respect to time is rigorously established for singular source terms and nonsmooth initial data. The other scheme derived from the second-order backward differentiation formula (BDF) is proved to possess second-order accuracy in time. Further, piecewise linear finite element and lumped mass finite element discretizations in space are applied and analyzed rigorously. Numerical investigations confirm our theoretical results.

Fourier–Chebyshev approximation of Kochin functions in the Fourier–Kochin approach to panel methods

The dominant computational task involved in the methods, widely called panel methods, currently used to compute 3D potential flows around large floating bodies such as ships and offshore structures (and other classes of dispersive waves such as flexural-gravity waves in very large floating structures or ice sheets) consists in evaluating N2 coefficients, called influence coefficients. These coefficients are associated with the flows created by distributions of singularities (sources, dipoles) over the N panels that approximate the surface of the body (ship, offshore structure) at every panel of the body surface. In contrast to the O(N2) computations required in existing panel methods, the dominant part of the computations of influence coefficients are independent of N in the Fourier–Kochin (FK) method if the Kochin functions in the FK flow representation are approximated by means of Fourier series and Chebyshev polynomials. The numerical analysis reported in this study shows that Fourier–Chebyshev approximations to the Kochin functions are feasible and indeed practical. In particular, the analysis shows that the number NF of basic Fourier integrals (the major and most difficult computational task) that must be evaluated within the approach to the numerical implementation of the FK method considered in the study is given by NF≈(600)2, i.e. is smaller than N2 if 600<N. Thus, one has NF≪N2 for typical panel numbers N=O(104), and the FK approach opens the way for a class of panel methods in which the dominant computations are independent of the number N of panels instead of O(N2) in existing panel methods. Another major advantageous feature of the FK method is that this approach circumvents the notorious difficulties involved in the numerical evaluation and the panel-surface-integration of the complicated Green functions associated with ship and offshore hydrodynamics.

Advanced 1200V SiC MOSFET Concept Based on Singular Point Source MOS (S-MOS) Technology

This paper presents a Singular Point Source MOS (S-MOS) cell concept suitable for SiC MOSFETs targeting low conduction losses, low switching losses and high robustness. The S-MOS concept differs from standard Planar or Trench MOS cells in the manner by which the total channel width per device area is determined. For the proof of concept and device electrical performance evaluation, the paper will provide 2D and 3D TCAD simulations results for 1200V SiC MOSFETs including the S-MOS and reference planar and trench structures.

A regularization approach for solving the super-Gaussian Poisson-Boltzmann model with heterogeneous dielectric functions

A regularization method is introduced for the first time in the literature for treating charge singularities in the heterogeneous dielectric Poisson-Boltzmann (PB) model. In the existing Gaussian and super-Gaussian electrostatic analysis, the singular sources in terms of Dirac delta functions are approximated by trilinear interpolation, which introduces a large error in potential solutions, and has to rely on an error cancellation for delivering acceptable estimates of the electrostatic free energy. To overcome this difficulty, a dual decomposition of potential and dielectric function is carried out in the proposed regularization, so that the charge singularities can be analytically captured by the Coulomb potential, while the reaction field potential satisfies a regularized PB equation with a new source term. A rigorous analysis has been conducted to show that a super-Gaussian density, instead of a Gaussian one, is required to guarantee a small neighborhood around each atom center with a nearly homogeneous dielectric medium, so that the source term can be well defined in the sense of distribution. Moreover, the well-posedness of the regularized formulation has been proved, and the regularity of the weak solution has been clarified. In the numerical implementation, special considerations are given in calculating the source term, by using analytical differentiations, instead of numerical approximations, whenever possible. The accuracy, convergence, efficiency, and robustness of the proposed regularization are numerically verified via benchmark studies. It is found that the regularization is more accurate than the trilinear method, as well as produces a faster convergence in energy estimation. Moreover, the grid artifact or artificial grid energy is completely eliminated in the present finite difference PB model.

Sharp upper bound of the number of solutions for the SU(N + 1) Toda system on torus with non-critical parameters

We study the following SU(N+1) Toda systemΔui+∑j=1Naijeuj=4π∑k=0mni,kδpk onEτ,i=1,⋯,N≥2, where Eτ:=C/(Z+Zτ) with Imτ>0 is a flat torus, (aij) is the standard Cartan matrix of SU(N+1), δpk is the Dirac measure at pk, and ni,k∈N satisfy natural non-critical conditions. It is known from [3,25] that solutions exist. In this paper we show that this system has at most∏k=0m∏j=1N∏i=1j∑l=ij(nl,k+1)(N+1)(∏j=2N(j!))m+1∈N solutions. We have several examples to show that this upper bound should be sharp. As a corollary we obtain the uniqueness of the solution for the case m=0 and (n1,0,⋯,nN−1,0,nN,0)=(0,⋯,0,1). Some results about the symmetry of solutions are also established. As applications, we obtain sharp non-existence and existence results for the SU(N+1) Toda system in the plane with four singular sources.