Exact Integral Equation Research Articles

Boundary integrated neural networks for 2D elastostatic and piezoelectric problems

In this paper, we make the first attempt to adopt the boundary integrated neural networks (BINNs) for the numerical solution of two-dimensional (2D) elastostatic and piezoelectric problems. The proposed BINNs combine the artificial neural networks with the exact boundary integral equations (BIEs) to effectively solve the boundary value problems based on the corresponding partial differential equations (PDEs). The BIEs are utilized to localize all the unknown physical quantities on the boundary, which are approximated by using artificial neural networks and resolved via a training process. In contrast to many traditional neural network methods based on a domain discretization, the present BINNs offer several distinct advantages. Firstly, by embedding the analytical BIEs into the learning procedure, the present BINNs only need to discretize the boundary of the problem domain, which reduces the number of the unknowns and can lead to a faster and more stable learning process. Secondly, the differential operators in the original PDEs are substituted by integral operators, which can effectively eliminate the need for additional differentiations of the neural networks (high-order derivatives of the neural networks may lead to instabilities in the learning process). Thirdly, the loss function of the present BINNs only contains the residuals of the BIEs, as all the boundary conditions have been inherently incorporated within the formulation. Therefore, there is no necessity for employing any weighting functions, which are commonly used in most traditional methods to balance the gradients among the different objective functions. Extensive numerical experiments show that the present BINNs are much easier to train and can usually provide more accurate solutions as compared to many traditional neural network methods.

Dimers, trimers, tetramers, and other multimers in a multiband Bose-Hubbard model

We study the bound states of $N$ identical bosons that are described by a multiband Bose-Hubbard model with generic hoppings and an attractive onsite interaction. Using a variational approach, we first derive exact integral equations for the dimers, trimers, tetramers, and other multimers, and then apply them to a one-dimensional sawtooth model that features two bands. In particular we reveal the presence of not only the offsite dimer states which consist of two monomers on different sites even in the strong-coupling limit but also the offsite trimer states which consist of either a dimer on one site and a monomer on another site or three monomers on three different sites. Our variational calculations for the ground states of onsite dimers, onsite trimers and offsite trimers benchmark perfectly well with the DMRG simulations. We also present DMRG results for the ground states of onsite tetramers, offsite tetramers, onsite pentamers, offsite pentamers, and for those of other multimers.

Instanton effects vs resurgence in the O(3) sigma model

We investigate the ground-state energy of the integrable two dimensional O(3) sigma model in a magnetic field. By determining a large number of perturbative coefficients we explore the closest singularities of the corresponding Borel function. We then confront its median resummation to the high precision numerical solution of the exact integral equation and observe that the leading exponentially suppressed contribution is not related to the asymptotics of the perturbative coefficients. By analytically expanding the integral equation we calculate the leading non-perturbative contributions up to fourth order and find complete agreement. These anomalous terms could be attributed to instantons, while the asymptotics of the perturbative coefficients seems to be related to renormalons.

Nonperturbative Dyson-Schwinger equation approach to strongly interacting Dirac fermion systems

Studying the strong correlation effects in interacting Dirac fermion systems is one of the most challenging problems in modern condensed matter physics. The long-range Coulomb interaction and the fermion-phonon interaction can lead to a variety of intriguing properties. In the strong-coupling regime, weak-coupling perturbation theory breaks down. The validity of $1/N$ expansion with $N$ being the fermion flavor is also in doubt since $N$ equals to 2 or 4 in realistic systems. Here, we investigate the interaction between $(1+2)$- and $(1+3)$-dimensional massless Dirac fermions and a generic scalar boson, and develop an efficient nonperturbative approach to access the strong-coupling regime. We first derive a number of self-consistently coupled Ward-Takahashi identities based on a careful symmetry analysis and then use these identities to show that the full fermion-boson vertex function is solely determined by the full fermion propagator. Making use of this result, we rigorously prove that the full fermion propagator satisfies an exact and self-closed Dyson-Schwinger integral equation, which can be solved by employing numerical methods. A major advantage of our nonperturbative approach is that there is no need to employ any small expansion parameter. Our approach provides a unified theoretical framework for studying strong Coulomb and fermion-phonon interactions. It may also be used to approximately handle the Yukawa coupling between fermions and order-parameter fluctuations around continuous quantum critical points. Our approach is applied to treat the Coulomb interaction in undoped graphene. We find that the renormalized fermion velocity exhibits a logarithmic momentum dependence but is nearly energy independent, and that no excitonic gap is generated by the Coulomb interaction. These theoretical results are consistent with experiments in graphene.

From perturbative to non-perturbative in the O(4) sigma model

We study the resurgent trans-series for the free energy of the two-dimensional O(4) sigma model in a magnetic field. Exploiting integrability, we obtain very high-order perturbative data, from which we can explore non-perturbative sectors. We are able to determine exactly the leading real-valued exponentially small terms, which we check against the direct numerical solution of the exact integral equation, and find complete agreement.

Resurgence in the O(4) sigma model

We analyze the free energy of the integrable two dimensional O(4) sigma model in a magnetic field. We use Volin’s method to extract high number (2000) of perturbative coefficients with very high precision. The factorial growth of these coefficients are regulated by switching to the Borel transform, where we perform several asymptotic analysis. High precision data allowed to identify Stokes constants and alien derivatives with exact expressions. These reveal a nice resurgence structure which enables to formulate the first few terms of the ambiguity free trans-series. We check these results against the direct numerical solution of the exact integral equation and find complete agreement.

A parallel‐plate waveguide antenna radiating through a perfectly conducting wedge

Parallel-plate waveguide antennas hosted in an electrically perfectly conducting wedge are studied. The analysis relies on an exact singular integral equation formulation. Asymptotic extraction techniques are employed to recast the kernel functions as the sum of two terms: one singular in closed form and another having the form of either a rapidly converging eigenfunctions expansion or a computationally efficient diffraction integral. The obtained integral equation is discretised in the framework of the Nystrom method using rapidly converging quadratures that take into account both the singularities of the kernels and the geometrical singularities at the edges. Numerical examples and case studies demonstrate the high accuracy and efficiency of the proposed algorithms. The analysis covers the canonical problem of a parallel-plate waveguide antenna backed on a perfectly conducting wedge as a special case.

Exact Results for the Boundary Energy of One-Dimensional Bosons.

We study bosons in a one-dimensional hard-wall box potential. In the case of contact interaction, the system is exactly solvable by the Bethe ansatz, as first shown by Gaudin in 1971. Although contained in the exact solution, the boundary energy in the thermodynamic limit for this problem is only approximately calculated by Gaudin, who found the leading order result at weak repulsion. Here we derive an exact integral equation that enables one to calculate the boundary energy in the thermodynamic limit at an arbitrary interaction. We then solve such an equation and find the asymptotic results for the boundary energy at weak and strong interactions. The analytical results obtained from the Bethe ansatz are in agreement with the ones found by other complementary methods, including quantum MonteCarlo simulations. We study the universality of the boundary energy in the regime of a small gas parameter by making a comparison with the exact solution for the hard rod gas.

Fredholm theory for the mean first-passage time of integrate-and-fire oscillators with colored noise input.

We develop a method to investigate the effect of noise timescales on the first-passage time of nonlinear oscillators. Using Fredholm theory, we derive an exact integral equation for the mean event rate of a leaky integrate-and-fire oscillator that receives constant input and temporally correlated noise. Furthermore, we show that Fredholm theory provides a unified framework to determine the system scaling behavior for small and large noise timescales. In this framework, the leading-order and higher-order asymptotic corrections for slow and fast noise are naturally emerging. We show the scaling behavior in the both limits is not reciprocal. We discuss further how this approach can be extended to study the first-passage time in a general class of nonlinear oscillators driven by colored noise at arbitrary timescales.

Effective Resistance of Two-Dimensional Truncated Infinite Mesh Structures

Determining the effective resistance of a mesh is a powerful tool for simplifying the analysis of complex electrical structures, ranging from transistors to power delivery systems. A common structure in science and engineering is a two-dimensional resistive grid. Applications of this structure include IR drop analysis and decoupling capacitor allocation in on-chip power and ground networks in VLSI systems, and the analysis of electrical and thermal conductive media, such as a semiconductor substrate. In the case where the effective resistance is evaluated far from the grid edges, an infinite resistive lattice can be used to simplify the analysis of a grid. If however the target nodes are located close to the grid edges, the assumption of infinite dimensions becomes invalid, producing inaccurate results. To bridge this gap, a resistive mesh truncated along a single or two dimensions is discussed here. Integral expressions for the effective resistance within a truncated infinite mesh structure are provided, as well as a closed-form approximation, which exhibits good agreement with the exact integral equation, exhibiting an average error of 0.27% and a maximum error of 4.77%. These expressions significantly improve the accuracy of the effective resistance estimation near the edges and corner of a resistive mesh, providing a tenfold reduction in error. In case studies, the expressions provide four to six orders of magnitude speedup in IR analysis of a $10^{4} \times 10^{4}$ grid, while providing accuracy comparable to nodal analysis.

Fractional Impulsive Differential Equations: Exact Solutions, Integral Equations and Short Memory Case

Abstract Fractional impulsive differential equations are revisited first. Some fundamental solutions of linear cases are given in this study. One straightforward technique without using integral equation is adopted to obtain exact solutions which are given by use of piecewise functions. Furthermore, a class of short memory fractional differential equations is proposed and the variable case is discussed. Mittag–Leffler solutions with impulses are derived which both satisfy the equations and impulsive conditions, respectively.

Exact equations for averaged electromagnetic field and its fluctuations at wave multiple scattering by plane periodic array of magnetic microelements

Electromagnetic wave (EM) multiple scattering by a plane periodic array of magnetic microelements in free space is considered analytically by natural subdividing of the EM wave into the averaged and fluctuation components. Each magnetic element is characterized by magnetic susceptibility tensor and shape. An exact Dyson integral equation is derived for the magnetic field Floquet–Bloch amplitude in-plane averaged over an array unit cell. The mass operator of the Dyson equation is expressed via the T-scattering operator of the array unit cell that satisfies a type of the Lippmann–Schwinger equation. We showed that magnetic field fluctuations are generated by the Bragg–Laue diffraction of an averaged magnetic field on the periodic array and are described inside the array as waves propagating with the Laue wave vectors equal to the difference between the in-plane wave vector of the incident magnetic field and the reciprocal lattice wave vector. We derived, for the first time, an exact quadrature to calculate magnetic field fluctuations from their averaged value. These general results are illustrated by a simple Born approximation. In particular, we revealed a mechanism of discrete waveguide excitation by an incident plane EM wave via the averaged EM wave Brag–Laue diffraction on the magnetic microelement array in the quasi-static approach when the wavelength of incident EM is much larger than the sizes of magnetic elements and periods of the array. The mode energy excitation coefficient at normal incidence of the plane EM wave on the array is evaluated.

Oblique scattering from radially inhomogeneous dielectric cylinders: An exact Volterra integral equation formulation

We study plane-wave electromagnetic scattering by radially and strongly inhomogeneous dielectric cylinders at oblique incidence. The method of analysis relies on an exact reformulation of the underlying field equations as a first-order 4 × 4 system of differential equations and on the ability to restate the associated initial-value problem in the form of a system of coupled linear Volterra integral equations of the second kind. The integral equations so derived are discretized via a sophisticated variant of the Nyström method. The proposed method yields results accurate up to machine precision without relying on approximations. Numerical results and case studies ably demonstrate the efficiency and high accuracy of the algorithms.

Heat flux of driven granular mixtures at low density: Stability analysis of the homogeneous steady state.

The Navier-Stokes order hydrodynamic equations for a low-density driven granular mixture obtained previously [Khalil and Garzó, Phys. Rev. E 88, 052201 (2013)PLEEE81539-375510.1103/PhysRevE.88.052201] from the Chapman-Enskog solution to the Boltzmann equation are considered further. The four transport coefficients associated with the heat flux are obtained in terms of the mass ratio, the size ratio, composition, coefficients of restitution, and the driven parameters of the model. Their quantitative variation on the control parameters of the system is demonstrated by considering the leading terms in a Sonine polynomial expansion to solve the exact integral equations. As an application of these results, the stability of the homogeneous steady state is studied. In contrast to the results obtained in undriven granular mixtures, the stability analysis of the linearized Navier-Stokes hydrodynamic equations shows that the transversal and longitudinal modes are (linearly) stable with respect to long enough wavelength excitations. This conclusion agrees with a previous analysis made for single granular gases.

An exact sum-rule for the Hubbard model: an historical/pedagogical approach

The aim of the present article is to derive an exact integral equation for the Green function of the Hubbard model through an equation-of-motion procedure, like in the original Hubbard papers. Though our exact integral equation does not allow to solve the Hubbard model, it represents a strong constraint on its approximate solutions. An analogous sum rule has been already obtained in the literature, through the use of a spectral moment technique. We think however that our equation-of-motion procedure can be more easily related to the historical procedure of the original Hubbard papers. We also discuss examples of possible applications of the sum rule and propose and analyse a solution, fulfilling it, that can be used for a pedagogical introduction to the Mott–Hubbard metal-insulator transition.

Particle in a box with a time-dependentδ-functionpotential

In quantum information processing, one often considers inserting a barrier into a box containing a particle to generate one bit of Shannon entropy. We formulate this problem as a one-dimensional Schr\"{o}dinger equation with a time-dependent $\delta$-function potential. It is a natural generalization of the particle in a box, a canonical example of quantum mechanics, and we present analytic and numerical investigations on this problem. After deriving an exact Volterra-type integral equation, composed of an infinite sum of modes, we show that approximate formulas with the lowest-frequency modes correctly capture the qualitative behavior of the wave function. If we take into account hundreds of modes, our numerical calculation shows that the quantum adiabatic theorem actually gives a very good approximation even if the barrier height diverges within finite time, as long as it is sufficiently longer than the characteristic time scale of the particle. In particular, if the barrier is slowly inserted at an asymmetric position, the particle is localized by the insertion itself, in accordance with a prediction of the adiabatic theorem. On the other hand, when the barrier is inserted quickly, the wave function becomes rugged after the insertion because of the energy transfer to the particle. Regardless of the position of the barrier, the fast insertion leaves the particle unlocalized so that we can obtain meaningful information by a which-side measurement. Our numerical procedure provides a precise way to calculate the wave function throughout the process, from which one can estimate the amount of this information for an arbitrary insertion protocol.

Three-dimensional BEM for transient dynamic analysis of piezoelectric and anisotropic elastic solids

A boundary element approach based on the Green's functions in integral representation and the convolution quadrature method is presented. Proposed approach is designed for analyzing 3D initial boundary-value problems of the dynamics of general anisotropic elastic and piezoelectric linear homogeneous solids with mixed boundary conditions. Numerical modelling of transient dynamics of elastic anisotropic and piezoelectric three-dimensional solids is carried out to demonstrate the potential of the developed boundary element software. Obtained solutions are compared with the corresponding FEM results and results of the dynamic experiment. A numerical technique based on the exact Laplace-domain boundary integral equations for the direct approach of 3D linear theories of anisotropic elasticity and piezoelectricity is employed. The BEM scheme is constructed using the collocation method and the convolution quadrature method in the form of a stepping method for numerical inversion of integral Laplace transform. Results of the stepped BE-modelling of the problems when a transient force is acting on 3D piezoelectric and anisotropic elastic homogeneous solids are presented.

Exact solution of the ( 0+1 )-dimensional Boltzmann equation for a massive gas

We solve the one-dimensional boost-invariant kinetic equation for a relativistic massive system with the collision term treated in the relaxation time approximation. The result is an exact integral equation which can be solved numerically by the method of iteration to arbitrary precision. We compare predictions for the shear and bulk viscosities of a massive system with those obtained from the exact solution. Finally, we compare the time evolution of the bulk pressure obtained from our exact solution with results obtained from the dynamical equations of second-order viscous hydrodynamics.

The contact problem when there are friction forces in the contact area for a three-component cylindrical base

The plane contact problem of elasticity theory on the interaction when there are friction forces in the contact area of an absolutely rigid cylinder (punch) with an internal surface of a cylindrical base, consisting of two circular cylindrical layers rigidly connected to one another and with an elastic space, is considered. The layers and space have different elastic constants. A vertical force and a counterclockwise torque, act on the punch, and the punch – base system is in a state of limiting equilibrium,. An exact integral equation of the first kind with a kernel represented in an explicit analytical form, is obtained for the first time for this problem using analytical calculation programs. The main properties of the kernel of the integral equation are investigated, and it is shown that the numerator and denominator of the kernel symbols can be represented in the form of polynomials in products of the powers of the moduli of the displacement of the layers and the half-space. A solution of the integral equation is constructed by the direct collocation method, which enables the solution of the problem to be obtained for practically any values of the initial parameters. The contact stress distributions, the dimensions of the contact area, the interconnection between the punch displacement and the forces and torques acting on it are calculated as a function of the geometrical and mechanical parameters of the layers and the space. The results of the calculations in special cases are compared with previously known results.

Asymptotic distribution of the size of a d-dimensional model of an open string

In this paper, we propose a statistical description of the configuration of a d-dimensional model of an open string that avoids self-intersection. For the distribution of the distance between the ends of the string we have obtained an exact integral equation similar to the well-known Dyson equation in quantum field theory. The resulting equation is invariant under continuous transformations of the renormalization group, which allows one to use the renormalization group method to establish the desired asymptotic distribution.