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Thermodynamic potentials play a substantial role in numerous scientific disciplines and serve as basic constructs for describing the behavior of matter. Despite their significance, comprehensive investigations of their topological characteristics and their connections to molecular interactions have eluded exploration due to experimental inaccessibility issues. This study addresses this gap by analyzing the topology of the Helmholtz energy, Gibbs energy, Grand potential, and Null potential that are associated with different isothermal boundary conditions. By employing Monte Carlo simulations in the NVT, NpT, and μVT ensembles and a molecular-based equation of state, methane, ethane, nitrogen, and methanol are investigated over a broad range of thermodynamic conditions. The predictions from the two independent methods are overall in very good agreement. Although distinct quantitative differences among the fluids are observed, the overall topology of the individual thermodynamic potentials remains unaffected by the molecular architecture, which is in line with the corresponding states principle—as expected. Furthermore, a comparative analysis reveals significant differences between the total potentials and their residual contributions. [ABSTRACT FROM AUTHOR]

Molecular-based equations of state for describing the thermodynamics of chain molecules are often based on mean-field like arguments that reduce the problem of describing the interactions between chains to a simpler one involving only nonbonded monomers. While for dense liquids such arguments are known to work well, at low density they are typically less appropriate due to an incomplete description of the effect of chain connectivity on the local environment of the chains' monomer segments. To address this issue, we develop three semi-empirical approaches that significantly improve the thermodynamic description of chain molecules at low density. The approaches are developed for chain molecules with repulsive intermolecular forces; therefore, they could be used as reference models for developing equations of the state of real fluids based on perturbation theory. All three approaches are extensions of Wertheim's first-order thermodynamic perturbation theory (TPT1) for polymerization. The first model, referred to as TPT1-v, incorporates a second-virial correction that is scaled to zero at liquid-like densities. The second model, referred to as TPT1-y, introduces a Helmholtz-energy contribution to account for correlations between next-nearest-neighbor segments within chain molecules. The third approach, called TPT-E, directly modifies TPT1 without utilizing an additional Helmholtz energy contribution. By employing TPT1 at the core of these approaches, we ensure an accurate description of mixtures and enable a seamless extension from chains of tangentially bonded hard-sphere segments of equal size to hetero-segmented chains, fused chains, and chains of soft repulsive segments (which are influenced by temperature). The low-density corrections implemented in TPT1 are designed to preserve these good characteristics, as confirmed through comparisons with novel molecular simulation results for the pressure of various chain fluids. TPT1-v exhibits excellent transferability across different chain types, but it relies on knowing the second virial coefficient of the chain molecules, which is non-trivial to obtain and determined here using Monte Carlo simulation. The TPT1-y model, on the other hand, achieves comparable accuracy to TPT1-v while being fully predictive, requiring no input besides the geometry of the chain molecules. [ABSTRACT FROM AUTHOR]

In this work, we demonstrate how the ion association constant can be attributed to the difference between the full Poisson–Boltzmann equation and its linearized version in very dilute solutions. We follow a pragmatic approach first by deriving an analytical approximated solution to the Poisson–Boltzmann equation, then calculating its respective Helmholtz free energy and activity coefficient, and then finally comparing it to the contribution from the mass action law principle. The final result is the Ebeling association constant. We conclude that electrostatic ion–ion interaction models miss the ion association contribution naturally introduced in higher-order electrostatic theories. We also demonstrate how the negative deviations from the Debye–Hückel limiting law can be physically attributed to the ion association phenomenon. [ABSTRACT FROM AUTHOR]

We numerically investigate the generalized Steklov problem for the modified Helmholtz equation and focus on the relation between its spectrum and the geometric structure of the domain. We address three distinct aspects: (i) the asymptotic behavior of eigenvalues for polygonal domains; (ii) the dependence of the integrals of eigenfunctions on domain symmetries; and (iii) the localization and exponential decay of Steklov eigenfunctions away from the boundary for smooth shapes and in the presence of corners. For this purpose, we implemented two complementary numerical methods to compute the eigenvalues and eigenfunctions of the associated Dirichlet-to-Neumann operator for planar bounded domains. We also discuss applications of the obtained results in the theory of diffusion-controlled reactions and formulate conjectures with relevance in spectral geometry. [ABSTRACT FROM AUTHOR]

The Weyl expansion technique, also known as the angular spectrum expansion, expresses an outgoing spherical wave as a linear combination of plane waves. The scalar spherical waves are the solutions of the homogeneous Helmholtz equation and therefore have direct relation to the scalar multipole fields. This paper gives the Weyl expansion of multipole fields, scalar and vector, of any degree and order for spherical wave functions. The expressions are given in closed form for the scalar, ψℓm(τ)$$ {\psi}_{\mathit{\ell m}}^{\left(\tau \right)} $$, and vector, Mℓm(τ),Nℓm(τ)$$ {\mathbf{M}}_{\mathit{\ell m}}^{\left(\tau \right)},{\mathbf{N}}_{\mathit{\ell m}}^{\left(\tau \right)} $$, Lℓm(τ)$$ {\mathbf{L}}_{\mathit{\ell m}}^{\left(\tau \right)} $$, multipole fields, evaluated across a plane orthogonal to any given direction. In the case of scalar spherical multipoles, the spherical gradient operator has been used, while for the vector spherical multipoles, the vector spherical wave operator has been constructed. [ABSTRACT FROM AUTHOR]

We study a reduced order model (ROM) based waveform inversion method applied to a Helmholtz problem with impedance boundary conditions and variable refractive index. The first goal of this paper is to obtain relations that allow the reconstruction of the Galerkin projection of the continuous problem onto the space spanned by solutions of the Helmholtz equation. The second goal is to study the introduced nonlinear optimization method based on the ROM aimed to estimate the refractive index from reflection and transmission data. Finally we compare numerically our method to the conventional least squares inversion based on minimizing the distance between modelled to measured data. [ABSTRACT FROM AUTHOR]

We consider the inverse interior problem of recovering the surface impedances of the cavity from sources and measurements placed on a curve inside of it. The uniqueness issue is investigated, and a hybrid method is proposed for the numerical solution. The approach takes advantages of both direct and iterative schemes, such as it does not require an initial guess and has an accuracy of a Newton‐type method. Presented numerical experiments demonstrate the feasibility and effectiveness of the approach. [ABSTRACT FROM AUTHOR]

This work is devoted to the numerical solution of the Helmholtz equation with variable wave number and including a point source in appropriately truncated infinite domains. Motivated by a two‐dimensional model, we formulate a simplified one‐dimensional model. We study its well posedness via wave number explicit stability estimates and prove convergence of the finite element approximations. As a proof of concept, we present the outcome of some numerical experiments for various wave number configurations. Our experiments indicate that the introduction of the artificial boundary near the source and the associated boundary condition lead to an efficient model that accurately captures the wave propagation features. [ABSTRACT FROM AUTHOR]

This paper addresses the electromagnetic inverse scattering problem of determining the location and shape of anisotropic objects from near-field Cauchy data. We investigate both cases involving the Helmholtz equation and Maxwell's equations for this inverse problem. Our study focuses on developing efficient imaging functionals that enable a fast and stable recovery of the anisotropic object. The implementation of the imaging functionals is simple and avoids the need to solve an ill-posed problem. The resolution analysis of the imaging functionals is conducted using the Green representation formula. Furthermore, we establish stability estimates for these imaging functionals when noise is present in the data. To illustrate the effectiveness of the methods, we present numerical examples showcasing their performance. [ABSTRACT FROM AUTHOR]

We consider a plane wave, a radiation solution, and the sum of these solutions (total solution) for the Helmholtz equation in an exterior region in . For a ray in this region whose direction is different from the propagation direction of the plane wave, we show that the restriction of the radiation solution to this ray is uniquely determined by the intensity of the total solution on an interval of this ray. As a corollary, we also prove that the restriction of the radiation solution to any plane in the exterior region is uniquely determined by the intensity of the total solution on an open domain in this plane. In particular, these results solve one of the old mathematical questions in holography. [ABSTRACT FROM AUTHOR]

In this paper, we show for the first time the increasing stability of the inverse source problem for the n‐dimensional Helmholtz equation at multiple wave numbers, which is different from the two‐ or three‐dimensional Helmholtz equation. In addition, we develop a new, unified approach to study increasing stability in any dimension. The method is based on the Fourier transform and explicit bounds for analytic continuation. [ABSTRACT FROM AUTHOR]

In the present research work, we construct and examine the self-similarity of optical solitons by employing the Riccati Modified Extended Simple Equation Method (RMESEM) within the framework of non-integrable Coupled Nonlinear Helmholtz Equations (CNHEs). This system models the transmission of optical solitons and coupled wave packets in nonlinear optical fibers and describes transverse effects in nonlinear fiber optics. Initially, a complex transformation is used to convert the model into a single Nonlinear Ordinary Differential Equation (NODE), from which hyperbolic, exponential, rational, trigonometric, and rational hyperbolic solutions are produced. In order to better understand the physical dynamics, we offer several 3D, contour, and 2D illustrations for the independent selections of physical parameter values. These illustrations highlight the graphic behaviour of some optical solitons and demonstrate that, under certain constraint conditions, acquired optical solitons lose their stability when they approach an axis and display periodic-axial perturbations, which lead to the generation of optical fractals. As a framework, the generated optical solitons have several useful applications in the field of telecommunications. Furthermore, our suggested RMESEM demonstrates its use by broadening the spectrum of optical soliton solutions, offering important insights into the dynamics of the CNHEs, and suggesting possible applications in the management of nonlinear models. [ABSTRACT FROM AUTHOR]

This paper investigates inverse source problems for the Helmholtz and Navier equations in multi-layered media, considering both two and three-dimensional cases respectively. The study reveals a consistent increase in stability for each scenario, characterized by two main terms: a Hölder-type term associated with data discrepancy, and a logarithmic-type term that diminishes as more frequencies are considered. In the two-dimensional case, measurements on interfaces and far-field data are essential. By employing the fundamental solution in free-space as the test function and utilizing the asymptotic behavior of the solution and continuation principle, stability results are obtained. In the three-dimensional case, measurements on interfaces and artificial boundaries are taken, and the stability result can be derived by applying the arguments for inverse source problems in homogeneous media. [ABSTRACT FROM AUTHOR]

We consider an inverse problem for the Helmholtz equation of reconstructing a solution from measurements taken on a segment inside a semi-infinite strip. Homogeneous Neumann conditions are prescribed on both side boundaries of the strip and an unknown Dirichlet condition on the remaining part of the boundary. Additional complexity is that the radiation condition at infinity is unknown. Our aim is to find the unknown function in the Dirichlet boundary condition and the radiation condition. Such problems appear in acoustics to determine acoustical sources and surface vibrations from acoustic field measurements. The problem is split into two sub-problems, a well-posed and an ill-posed problem. We analyse the theoretical properties of both problems; in particular, we show that the radiation condition is described by a stable non-linear problem. The second problem is ill-posed, and we use the Landweber iteration method together with the discrepancy principle to regularize it. Numerical tests show that the approach works well. [ABSTRACT FROM AUTHOR]

Open stubs in a strip (microstrip) transmission line are one of the most common elements of planar circuits used in numerous devices in the various types of wireless systems. Therefore, the urgent problem is to develop an analyzing method for discontinuities in the form of the open stub in a microstrip transmission line at frequencies at which the high-frequency effects must be considered. In the paper, a technique of scattering characteristics calculating on a symmetrical microstrip open stub by transverse resonance method is presented. Boundary value problems for a rectangular volume resonator based on a microstrip transmission line with a symmetric open stub are solved for the three options boundary conditions in the symmetry plane and on the longitudinal boundaries. The intersection of the spectral curves obtained by the numerical solution of the "electric" and "magnetic" boundary value problems determines the minima of a reflection or transmission coefficients of fundamental wave on discontinuities. To algebraize the boundary value problems for the eigen frequencies of volume resonator with discontinuity, the corresponding two-dimensional functions of the magnetic potential are constructed, through which the components of the current density on the strip are determined. The functions of magnetic potential were defined by decomposing them into expansion by Fourier series, which ensures stable convergence of the series and numerical calculation algorithm. The developed technique has been tested by calculating the eigenfrequency spectra of an open microstrip stub using the transverse resonance method on the example of an open stub in a microstrip transmission line with a resonant frequency of about 3.0 GHz. Also, a technique for numerical solutions of "electric" and "magnetic" boundary-value problems for resonators with two electrodynamically coupled symmetric open stubs in a microstrip transmission line is developed. [ABSTRACT FROM AUTHOR]

In this paper, we develop a hypercomplex operator calculus to treat fully analytically boundary value problems for the homogeneous and inhomogeneous fractional Helmholtz equation where fractional derivatives in the sense of Caputo and Riemann–Liouville are applied. Our method extends the recently proposed fractional reduced differential transform method (FRDTM) by using fractional derivatives in all directions. For the special separable case in three dimensions, we obtain completely explicit representations for the fundamental solution. This allows us to interpret and to understand the appearance of spatial steady‐state solutions or spatial blow‐ups of the fractional Helmholtz equation in a better way. More precisely, we were able to present explicit conditions for the parameters in the representation formulas of the fundamental solutions under which we obtain bounded or spatial decreasing steady‐solutions and when spatial blow‐ups occur. We also illustrate this with some representative numerical examples. Furthermore, we show that it is possible to recover the recently studied cases as well as the classical cases as particular limit cases within our more general setting. Using the hypercomplex operator approach also allows us to factorize the fractional Helmholtz operator and obtain some interesting duality relations between left and right derivatives, Caputo and Riemann–Liouville derivatives, and eigensolutions of antipodal eigenvalues in terms of a generalized Borel–Pompeiu formula. This factorization, in turn, allows us to tackle inhomogeneous fractional Helmholtz problems. [ABSTRACT FROM AUTHOR]

This study discusses infiltration in six irrigation channel types with root water uptake in four types of roots. The mathematical model for the infiltration problem is the Richards equation. This equation is then transformed into a modified Helmholtz equation using the Kirchhoff transformation, dimensionless variables. Subsequently, a numerical solution of the modified Helmholtz equation is obtained using the Dual Reciprocity Boundary Element Method (DRBEM) with a predictor–corrector scheme to result in the numerical values of suction potential, water content, and root water uptake function. In addition, the amount of water absorbed by each root and the water distribution pattern in the channel can be obtained and compared. The results indicate that the minimum water content values occur in both impermeable rectangular and trapezoidal channels, and the highest water uptake values are also observed in the impermeable channels. This is consistent with the physical conditions; as in impermeable channels, water loss downward is limited, and water tends to flow toward the plants. [ABSTRACT FROM AUTHOR]

We present a deep learning--based iterative approach to solve the discrete heterogeneous Helmholtz equation for high wavenumbers. Combining classical iterative multigrid solvers and convolutional neural networks (CNNs) via preconditioning, we obtain a faster, learned neural solver that scales better than a standard multigrid solver. Our approach offers three main contributions over previous neural methods of this kind. First, we construct a multilevel U-Net-like encoder-solver CNN with an implicit layer on the coarsest grid of the U-Net, where convolution kernels are inverted. This alleviates the field of view problem in CNNs and allows better scalability. Second, we improve upon the previous CNN preconditioner in terms of the number of parameters, computation time, and convergence rates. Third, we propose a multiscale training approach that enables the network to scale to problems of previously unseen dimensions while still maintaining a reasonable training procedure. Our encoder-solver architecture can be used to generalize over different slowness models of various difficulties and is efficient at solving for many right-hand sides per slowness model. We demonstrate the benefits of our novel architecture with numerical experiments on various heterogeneous two-dimensional problems at high wavenumbers. [ABSTRACT FROM AUTHOR]

Accurate calculations of ocean acoustic propagation are of significant importance in various aspects such as marine environmental monitoring, oceanic communication, and navigation. Various mature simplified models have been developed, such as the normal mode model, parabolic approximation model, and wavenumber integration model. The above-mentioned models and algorithms all have limitations, namely, "applicable domain," which restrict their range of application. Instead of simplifying model, the algorithm described in this article aims at solving the Helmholtz equation, which governs ocean acoustic propagation. It solves the ocean acoustic propagation without being constrained by model application conditions and has a wide range of applications, which is a versatile method for solving acoustic propagation. The algorithm utilizes Chebyshev tau matrix method to directly solve the two-dimensional acoustic Helmholtz equation, which could solve the situation of uneven media within irregular domain, addressing the issue of traditional spectral method being unsuitable for non-rectangular domain. Compared with the mature programs, the new spectral method offers higher computational accuracy and is capable of calculating acoustic propagation in more complex ocean environments. [ABSTRACT FROM AUTHOR]

Autoproducts are quadratic or higher products of frequency-domain acoustic fields that can mimic genuine fields at frequencies within or outside the original field's bandwidth. Past studies have found a variety of interesting autoproduct properties but have been limited to quadratic autoproducts. This paper presents cubic autoproduct theory and documents how noise suppression may be attained with the cubic frequency-difference autoproduct, a product of three frequency-domain acoustic fields. The cubic autoproduct's field equations, developed from the inhomogeneous Helmholtz equation, and analytical results in single- and two-path environments justify interpretating the cubic autoproduct as a pseudofield and underscore its similarities to the quadratic autoproducts. For nonzero field bandwidth, many frequency triplets satisfy the relationship for a single cubic autoproduct frequency. Thus, bandwidth averaging can lead to serendipitous noise suppression and is shown herein to facilitate field-phase-structure recovery from ideal free space fields corrupted by Gaussian noise. Cubic-autoproduct-based direction of arrival (DOA) estimation using signal and noise recordings collected in the ocean are found to be more accurate than conventional DOA estimates from the same data. In particular, cubic autoproduct results showed a 3–5 dB noise suppression advantage for 4- and 6-kHz direct- and reflected-path sounds broadcast 200 m to a four-element receiving array. [ABSTRACT FROM AUTHOR]

An analysis is presented of reflection from a marine sediment consisting of a homogeneous mud layer overlying a sand-mud basement, the latter with an upward-refracting, inverse-square sound speed profile. Such layering is representative of the sediment at the New England Mud Patch (NEMP). By applying appropriate integral transforms and their inverses to the Helmholtz equations for the ocean and the two sediment layers, along with the boundary conditions, a Sommerfeld–Weyl type of wavenumber integral is obtained for the cylindrical-wave reflection coefficient of the sediment, R. A stationary phase evaluation of this integral yields a closed-form expression for the plane-wave reflection coefficient, R0. In the absence of attenuation, the plane-wave solution exhibits total reflection up to a critical grazing angle, ac, but when attenuation in the sediment is introduced, the region of total reflection in |R0| is replaced by a sequence of contiguous peaks. With realistic levels of sediment attenuation, the cylindrical-wave solution, |R|, exhibits a quasi-critical grazing angle, less than ac, which is strongly dependent on the source-plus-receiver height above the seabed, which is mildly dependent on the depth of the mud layer but is essentially independent of frequency. Such behavior is consistent with independent experimental observations at the NEMP. [ABSTRACT FROM AUTHOR]

For effectively solving the nonsymmetric and indefinite linear system originating from the discretized Helmholtz equation with complex wavenumber, we propose a two-parameter modified matrix splitting iteration method. We establish the asymptotic convergence theory for this method and demonstrate its convergence under certain conditions. The proposed iteration method leads to a new preconditioner that can be efficiently inverted by using the discrete sine transform. Numerical experiments are carried out to show that the new preconditioner significantly improves the convergence rate of the GMRES method, which results in effective preconditioned GMRES method for solving the Helmholtz equation of high wavenumber. [ABSTRACT FROM AUTHOR]

In this paper, the Cauchy problem for the Helmholtz equation, also known as the continuation problem, is considered. The continuation problem is reduced to a boundary inverse problem for a well-posed direct problem. A generalized solution to the direct problem is obtained and an estimate of its stability is given. The inverse problem is reduced to an optimization problem solved using the gradient method. The convergence of the Landweber method with respect to the functionals is compared with the convergence of the Nesterov method. The calculation of the gradient in discrete form, which is often used in the numerical solutions of the inverse problem, is described. The formulation of the conjugate problem in discrete form is presented. After calculating the gradient, an algorithm for solving the inverse problem using the Nesterov method is constructed. A computational experiment for the boundary inverse problem is carried out, and the results of the comparative analysis of the Landweber and Nesterov methods in a graphical form are presented. [ABSTRACT FROM AUTHOR]

This study delves into the investigation of exact analytical solutions for the plane stress and displacement fields within linear homogeneous anisotropic nano-beam models of gradient elasticity. It focuses on solving the Helmholtz equation, which encompasses a second-order non-homogeneous linear partial differential equation in plane gradient elasticity theory, utilizing polynomial series-type solutions. The analysis centers on the utilization of gradient Airy stress functions to derive stress fields in the gradient theory. The research yields closed-form analytical solutions for Airy stress functions, stress, and strain fields in both classical and gradient theories. The study considers five distinct types of two-dimensional functionally graded cantilever beams with various boundary conditions: a cantilever anisotropic nano-beam subjected to a concentrated force at the free end, a cantilever anisotropic nano-beam under a uniform load, a simple anisotropic nano-beam under a uniform load, a propped cantilever anisotropic nano-beam under a uniform load, and a fixed end cantilever anisotropic nano-beam under a uniform load. General analytical solutions for the gradient stress and displacement fields of two-dimensional and one-dimensional anisotropic nano-beams under different boundary conditions are provided. The study showcases significant strain gradient size effects at the nano-scale through the derived analytical solutions for anisotropic beams. Additionally, it demonstrates that the strain gradient theory results for the limit case of the gradient coefficient c precisely align with results for isotropic and anisotropic materials in elasticity theory and classical theory. Furthermore, real-world applications are discussed, considering stress and displacement fields in real anisotropic materials such as TiSi 2 single crystals and orthotropic materials, which are special cases of anisotropic materials like wood and epoxy as documented in the literature. • Exact solutions for stress/displacement in gradient elasticity models for anisotropic functionally graded nano-beams. • Analysis focuses on constructing stress fields using "gradient Airy stress function" notation. • Solutions reveal significant strain gradient effects in anisotropic nano-beams. • Literature solutions for CT beams match findings by taking gradient coefficient c to zero. • Gradient stress fields vanish except in y-direction as c approaches infinity. [ABSTRACT FROM AUTHOR]

Starting from Kirchhoff-Huygens representation and Duhamel’s principle of time-domain wave equations, we propose novel butterfly-compressed Hadamard integrators for self-adjoint wave equations in both time and frequency domain in an inhomogeneous medium. First, we incorporate the leading term of Hadamard’s ansatz into the Kirchhoff-Huygens representation to develop a short-time valid propagator. Second, using Fourier transform in time, we derive the corresponding Eulerian short-time propagator in the frequency domain; on top of this propagator, we further develop a time-frequency-time (TFT) method for the Cauchy problem of time-domain wave equations. Third, we further propose a time-frequency-time-frequency (TFTF) method for the corresponding point-source Helmholtz equation, which provides Green’s functions of the Helmholtz equation for all angular frequencies within a given frequency band. Fourth, to implement the TFT and TFTF methods efficiently, we introduce butterfly algorithms to compress oscillatory integral kernels at different frequencies. As a result, the proposed methods can construct wave field beyond caustics implicitly and advance spatially overturning waves in time naturally with quasi-optimal computational complexity and memory usage. Furthermore, once constructed the Hadamard integrators can be employed to solve both time-domain wave equations with various initial conditions and frequency-domain wave equations with different point sources. Numerical examples for two-dimensional wave equations illustrate the accuracy and efficiency of the proposed methods. [ABSTRACT FROM AUTHOR]

Waves are mainly generated by wind via the transfer of wind energy to the water through friction. When the wind subsides, the waves transition into swells and eventually dissipate. Friction plays a crucial role in the generation and dissipation of waves. Numerous wave theories have been developed based on the assumption of inviscid flow, but these theories are inadequate in explaining the transformation of waves into swells. This study addressed these limitations by analytically deriving general solutions to the Navier–Stokes equations. By expressing the velocity field as the product of a solution to the Helmholtz equation and a time-dependent univariate function, the Navier–Stokes equations are decomposed into an ordinary differential equation and the Euler equations, which are solved using tensor calculus. This paper provides solutions for viscous flow with shear currents when applied to the water wave problem. These solutions were validated through their application to the vorticity equation. The decay modulus of water waves was compared with experimental data, showing a significant degree of concordance. In contrast to other wave theories, this study clarified the process through which waves evolve into swells.

Two new assertions characterizing analytically disks in the Euclidean plane $\mathbb{R}^2$ are proved. Weighted mean value property of positive solutions to the Helmholtz and modified Helmholtz equations are used for this purpose; the weight has a logarithmic singularity. The obtained results are compared with those without weight that were found earlier.

Introduction. Optical fibers are widely used for high-bandwidth transmitting communication signals over long distances. The key feature enabling this performance is signal low attenuation, that is signals experience minimal power loss propagating along the optical fiber. One of the factors influencing power loss during information transmission is the fiber bending. Bending can increase the signal transmission power loss of an optical fiber because of both macrobending and microbending. Studying the dependence of signal power losses when bending on waveguide parameters makes it possible to control the signal power losses of an optical fiber during information transmission. Aim of the Study. The study ia aimed at evaluating the effect of microbending on the field and energy of a weakly guiding optical fiber with a gradient refractive index profile in a single-mode regime. Materials and Methods. From the equations for the fields of straight and curved sections of weakly guiding fiber for an arbitrary gradient profile of the refractive index with the help of the subsequent solution of inhomogeneous Helmholtz equations by the Green’s function method, there were obtained expressions for relative energy: the ratio of the field energy of the fiber curved section to the field energy of the fiber straight section (in the first approximation for a single-mode regime). Results. The obtained expression for the relative energy depends on two parameters: the waveguide parameter and the ratio of the optical fiber radius to the radius of curvature. For the quadratic case of a power-law profile, as the closest to the actually used one, numerical calculations were used to construct the dependences of the relative energy on the parameter characterizing the bending for different values of the waveguide parameter. Discussion and Conclusion. It has been shown that in the case of microbending, the longer the wavelength or the smaller the fiber radius, the lower the losses. The results obtained can be used in calculating optical fiber profiles designed to operate in a bent state and eliminate expensive experimental modeling of light guides and in designing waveguides to solve specific applications, in particular, to increase energy efficiency, reliability and performance of the measuring instruments.

We present a method to compute the photoionization spectra of atoms and molecules in linear-response, time-dependent density functional theory. The electronic orbital variations corresponding to ionized electrons are expanded on a basis set of delocalized functions, obtained as the solution of the inhomogeneous Helmholtz equation, with gaussian basis set functions as the right-hand side. The resulting scheme is able to reproduce the photoionization spectra without any need for artificial regularization or localization. We demonstrate that this Green's function-based approach is able to produce accurate spectra for semilocal exchange-correlation functionals, even using relatively small standard gaussian basis sets. [ABSTRACT FROM AUTHOR]

We present a wavenumber-explicit convergence analysis of the hp Finite Element Method applied to a class of heterogeneous Helmholtz problems with piecewise analytic coefficients at large wavenumber k. Our analysis covers the heterogeneous Helmholtz equation with Robin, exact Dirichlet-to-Neumann, and second order absorbing boundary conditions, as well as perfectly matched layers. [ABSTRACT FROM AUTHOR]

Multiple scattering methods are widely used to reduce the computational complexity of acoustic or electromagnetic scattering problems when waves propagate through media containing many identical inclusions. Historically, this numerical technique has been limited to situations in which the inclusions (particles) can be covered by nonoverlapping disks in two dimensions or spheres in three dimensions. This allows for the use of separation of variables in cylindrical or spherical coordinates to represent the solution to the governing partial differential equation. Here, we provide a more flexible approach, applicable to a much larger class of geometries. We use a Green's representation formula and the associated layer potentials to construct incoming and outgoing solutions on rectangular enclosures. The performance and flexibility of the resulting scattering operator formulation in two-dimensions is demonstrated via several numerical examples for multi-particle scattering in free space as well as in layered media. The mathematical formalism extends directly to the three dimensional case as well, and can easily be coupled with several commercial numerical PDE software packages. [ABSTRACT FROM AUTHOR]

For h-FEM discretisations of the Helmholtz equation with wavenumber k, we obtain k-explicit analogues of the classic local FEM error bounds of Nitsche and Schatz (Math. Comput. 28(128), 937–958 1974), Wahlbin (1991, §9), Demlow et al.(Math. Comput. 80(273), 1–9 2011), showing that these bounds hold with constants independent of k, provided one works in Sobolev norms weighted with k in the natural way. We prove two main results: (i) a bound on the local H 1 error by the best approximation error plus the L 2 error, both on a slightly larger set, and (ii) the bound in (i) but now with the L 2 error replaced by the error in a negative Sobolev norm. The result (i) is valid for shape-regular triangulations, and is the k-explicit analogue of the main result of Demlow et al. (Math. Comput. 80(273), 1–9 2011). The result (ii) is valid when the mesh is locally quasi-uniform on the scale of the wavelength (i.e., on the scale of k - 1 ) and is the k-explicit analogue of the results of Nitsche and Schatz (Math. Comput. 28(128), 937–958 1974), Wahlbin (1991, §9). Since our Sobolev spaces are weighted with k in the natural way, the result (ii) indicates that the Helmholtz FEM solution is locally quasi-optimal modulo low frequencies (i.e., frequencies ≲ k ). Numerical experiments confirm this property, and also highlight interesting propagation phenomena in the Helmholtz FEM error. [ABSTRACT FROM AUTHOR]

Numerical simulations of seismic wave propagation in heterogeneous 3-D media are central to investigating subsurface structures and understanding earthquake processes, yet are computationally expensive for large problems. This is particularly problematic for full-waveform inversion (FWI), which typically involves numerous runs of the forward process. In machine learning there has been considerable recent work in the area of operator learning, with a new class of models called neural operators allowing for data-driven solutions to partial differential equations. Recent work in seismology has shown that when neural operators are adequately trained, they can significantly shorten the compute time for wave propagation. However, the memory required for the 3-D time domain equations may be prohibitive. In this study, we show that these limitations can be overcome by solving the wave equations in the frequency domain, also known as the Helmholtz equations, since the solutions for a set of frequencies can be determined in parallel. The 3-D Helmholtz neural operator is 40 times more memory-efficient than an equivalent time-domain version. We use a Helmholtz neural operator for 2-D and 3-D elastic wave modelling, achieving two orders of magnitude acceleration compared to a baseline spectral element method. The neural operator accurately generalizes to variable velocity structures and can be evaluated on denser input meshes than used in the training simulations. We also show that when solving for wavefields strictly at the free surface, the accuracy can be significantly improved via a graph neural operator layer. In leveraging automatic differentiation, the proposed method can serve as an alternative to the adjoint-state approach for 3-D FWI, reducing the computation time by a factor of 350. [ABSTRACT FROM AUTHOR]

Directional interpolation is a fast and efficient compression technique for high‐frequency Helmholtz boundary integral equations, but requires a very large amount of storage in its original form. Algebraic recompression can significantly reduce the storage requirements and speed up the solution process accordingly. During the recompression process, weight matrices are required to correctly measure the influence of different basis vectors on the final result, and for highly accurate approximations, these weight matrices require more storage than the final compressed matrix. We present a compression method for the weight matrices and demonstrate that it introduces only a controllable error to the overall approximation. Numerical experiments show that the new method leads to a significant reduction in storage requirements. [ABSTRACT FROM AUTHOR]

In this study, we explore the effectiveness of elliptic partial differential equations (PDEs) in two and three dimensional space based on Helmholtz equation for simulations of acoustic sound in a very complex environment of propagation. For this purpose, we use an advance and robust numerical technique by utilizing the properties of shifted Chebyshev spectral collocation method. This technique is an extension of the traditional Chebyshev polynomials, incorporating a shift in their argument to enhance flexibility across a wider domain, while retaining an extraordinary numerical characteristic such as orthogonality and spectral convergence making them exceptionally effective in finding the approximate solutions. The exponential order of convergence of the proposed approach is shown both through theoretical and numerical approaches. We provide a number of numerical experiments to verify the theoretical results. The spectral convergence has been substantially enhanced by these numerical examples. The exponential order is further validated by numerical error behaviour in both L2 and L∞ norms. [ABSTRACT FROM AUTHOR]

The main focus of this research is to address the Cauchy problem of the multi-dimensional Helmholtz equation with mixed boundary conditions. This problem is known to be ill-posed according to Hadamard’s definition. To tackle this issue, we propose the mollification regularization method based on exponential decay. Using prior rule and posterior rule to create a regular approximation solution and convergence of the solution is also provided. Furthermore, the proposed method is shown to be robust against data disruption noise, making it a reliable approach for solving the Cauchy problem of the multi-dimensional Helmholtz equation with mixed boundary conditions. [ABSTRACT FROM AUTHOR]

The Helmholtz equation is an important differential equation. It has a wide range of uses in physics, including acoustics, electro-statics, optics, and quantum mechanics. In this article, a hybrid approach called the Shehu transform decomposition method (STDM) is implemented to solve space-fractional-order Helmholtz equations with initial boundary conditions. The fractional-order derivative is regarded in the Caputo sense. The solutions are provided as series, and then we use the Mittag-Leffler function to identify the exact solutions to the Helmholtz equations. The accuracy of the considered problem is examined graphically and numerically by the absolute, relative, and recurrence errors of the three problems. For different values of fractional-order derivatives, graphs are also developed. The results show that our approach can be a suitable alternative to the approximate methods that exist in the literature to solve fractional differential equations. [ABSTRACT FROM AUTHOR]

In this work, we define the notions of Wronskian and simplified Wronskian for Stieltjes derivatives and study some of their properties in a similar manner to the context of time scales or the usual derivative. Later, we use these tools to investigate second-order linear differential equations with Stieltjes derivatives to find linearly independent solutions, as well as to derive the variation of parameters method for problems with gg-continuous coefficients. This theory is later illustrated with some examples such as the study of the one-dimensional linear Helmholtz equation with piecewise-constant coefficients.

We study the Helmholtz equation with periodic coefficients in a closed waveguide. A functional analytic approach is used to formulate and solve the radiation problem in a self‐contained exposition. In this context, we simplify the non‐degeneracy assumption on the frequency. Limiting absorption principles (LAPs) are studied, and the radiation condition corresponding to the chosen LAP is derived; we include an example to show different LAPs lead, in general, to different solutions of the radiation problem. Finally, we characterize the set of all bounded solutions to the homogeneous problem. [ABSTRACT FROM AUTHOR]

The interior inverse elastic scattering problem is considered in this paper. The scattering in the cavity is illuminated by an time-harmonic elastic point source at fixed frequency. The boundary of the cavity is assumed to be a small and smooth perturbation of some circle. In this paper, we are interested in the reconstruction of the boundary according to the measured data on a closed curve inside the cavity. In implementation, we split the displacement field into compressional and shear waves by the Helmholtz decomposition. Then we use coordinate transformation, power series expansion based on perturbed parameters and Fourier expansion of corresponding power series coefficients to simplify the original boundary value problem to a successive sequence of one-dimensional boundary value problem. We linearize the inverse problem by eliminating the higher-order terms in the power series expansion; and design a reconstruction algorithm by the explicit relationship between the Fourier coefficients of the boundary function and the total field. In order to obtain better resolution, we also use a nonlinear correction method in the numerical experiments. The numerical results have shown the effectiveness and the stableness of the proposed method. [ABSTRACT FROM AUTHOR]

A new concept of surface waves of interference nature is described in detail for the case of creeping waves propagating along a smooth strictly concave surface embedded in 3D Euclidean space. In numerous papers devoted to whispering gallery and to creeping waves, it was assumed that they propagate along boundaries formed by smooth plane curves. However, the process of surface wave propagation along smooth surfaces is much more complicated than along plane curves. Indeed, the surface waves slide along geodesic lines on the surface where they typically form numerous caustics and that, in turn, generates singularities in the wave field asymptotics. In addition, the geodesic lines themselves are not plane curves in 3D and therefore their torsion has to be taken into account. Our approach allows us to resolve both of these specific problems of wave propagation along smooth surfaces embedded in ℝ3. It is based on the consideration of the geodesic flow on the surface, which is associated with the surface wave generated by a source. For each geodesic line, we construct an asymptotic solution of the Helmholtz equation localized in a tube vicinity of the geodesic line and having no singularities on caustics. The surface wave under consideration is then presented as a superposition (integral) of the localized solutions. [ABSTRACT FROM AUTHOR]

The dispersion of elastic shear waves in multilayered bodies is a topic of extensive research due to its significance in contemporary science and engineering. Anti-plane shear motion, a two-dimensional mathematical model in solid mechanics, effectively captures shear wave propagation in elastic bodies with relative mathematical simplicity. This study models the vibration of elastic waves in a multilayered inhomogeneous circular membrane using the Helmholtz equation with fractional-order infusion, effectively leveraging the anti-plane shear motion equation to avoid the computational complexity of universal plane motion equations. The method of the separation of variables and the conformable Bessel equation are utilized for the analytical examination of the model's resulting vibrational displacements, as well as the dispersion relation. Additionally, the influence of various wave phenomena, including the dependencies of the wavenumber on the frequency and the phase speed on the wavenumber, respectively, with the variational effect of the fractional order on wave dispersion is considered. Numerical simulations of prototypical cases validate the formulated model, illustrating its applicability and effectiveness. The study reveals that fractional-order infusion significantly impacts the dispersion of elastic waves in both single- and multilayer membranes. The effects vary depending on the membrane's structure and the wave propagation regime (long-wave vs. short-wave). These findings underscore the potential of fractional-order parameters in tailoring wave behavior for diverse scientific and engineering applications. [ABSTRACT FROM AUTHOR]

The paper describes a sparse direct solver for the linear systems that arise from the discretization of an elliptic PDE on a two-dimensional domain. The scheme decomposes the domain into thin subdomains, or “slabs” and uses a two-level approach that is designed with parallelization in mind. The scheme takes advantage of H 2 -matrix structure emerging during factorization and utilizes randomized algorithms to efficiently recover this structure. As opposed to multi-level nested dissection schemes that incorporate the use of H or H 2 matrices for a hierarchy of front sizes, SlabLU is a two-level scheme which only uses H 2 -matrix algebra for fronts of roughly the same size. The simplicity allows the scheme to be easily tuned for performance on modern architectures and GPUs. The solver described is compatible with a range of different local discretizations, and numerical experiments demonstrate its performance for regular discretizations of rectangular and curved geometries. The technique becomes particularly efficient when combined with very high-order accurate multidomain spectral collocation schemes. With this discretization, a Helmholtz problem on a domain of size 1000 λ × 1000 λ (for which N = 100 M ) is solved in 15 min to 6 correct digits on a high-powered desktop with GPU acceleration. [ABSTRACT FROM AUTHOR]

Magnetic resonance electrical propert tomography promises to retrieve electrical properties (EPs) quantitatively and non‐invasively in vivo, providing valuable information for tissue characterization and pathology diagnosis. However, its clinical implementation has been hindered by, for example, B1 measurement accuracy, reconstruction artifacts resulting from inaccuracies in underlying models, and stringent hardware/software requirements. To address these challenges, we present a novel approach aimed at accurate and high‐resolution EPs reconstruction based on water content maps by using a physics‐informed network (PIN‐wEPT). The proposed method utilizes standard clinical protocols and conventional multi‐channel receive arrays that have been routinely equipped in clinical settings, thus eliminating the need for specialized RF sequence/coil configurations. Compared with the original wEPT method, the network generates accurate water content maps that effectively eliminate the influence of B→1+ and B→1− by incorporating data mismatch with electrodynamic constraints derived from the Helmholtz equation. Subsequent regression analysis develops a broad relationship between water content and EPs across various types of brain tissue. A series of numerical simulations was conducted at 7 T to assess the feasibility and performance of the method, which encompassed four normal head models and models with tumorous tissues incorporated, and the results showed normalized mean square error below 1.0% in water content, below 11.7% in conductivity, and below 1.1% in permittivity reconstructions for normal brain tissues. Moreover, in vivo validations conducted over five healthy subjects at both 3 and 7 T showed reasonably good consistency with empirical EPs values across the white matter, gray matter, and cerebrospinal fluid. The PIN‐wEPT method, with its demonstrated efficacy, flexibility, and compatibility with current MRI scanners, holds promising potential for future clinical application. [ABSTRACT FROM AUTHOR]

We introduce new finite-dimensional spaces specifically designed to approximate the solutions to high-frequency Helmholtz problems with smooth variable coefficients in dimension d. These discretization spaces are spanned by Gaussian coherent states, that have the key property to be localised in phase space. We carefully select the Gaussian coherent states spanning the approximation space by exploiting the (known) micro-localisation properties of the solution. For a large class of source terms (including plane-wave scattering problems), this choice leads to discrete spaces that provide a uniform approximation error for all wavenumber k with a number of degrees of freedom scaling as k d - 1 / 2 , which we rigorously establish. In comparison, for discretization spaces based on (piecewise) polynomials, the number of degrees of freedom has to scale at least as k d to achieve the same property. These theoretical results are illustrated by one-dimensional numerical examples, where the proposed discretization spaces are coupled with a least-squares variational formulation. [ABSTRACT FROM AUTHOR]

In this paper, we consider isotropic Mindlin–Toupin strain gradient elasticity theory, in which the equilibrium equations contain two additional length-scale parameters and have the fourth order. For this theory, we developed an extended form of Boussinesq–Galerkin (BG) and Papkovich–Neuber (PN) general solutions. The obtained form of BG solution allows to define the displacement field through a single vector function that obeys the eight-order bi-harmonic/bi-Helmholtz equation. The developed PN form of the solution provides an additive decomposition of the displacement field into the classical and gradient parts that are defined through the standard Papkovich stress functions and modified Helmholtz decomposition, respectively. Relations between different stress functions and the completeness theorem for the derived general solutions are established. As an example, it is shown that a previously known fundamental solution within the strain gradient elasticity can be derived by using the developed PN general solution. [ABSTRACT FROM AUTHOR]

In the present work, we give an explicit solution of the Dirichlet boundary problem for the Helmholtz equation in a nonconvex angle with periodic boundary data. We present uniqueness and existence theorems in an appropriate functional class and we give an explicit formula for the solution in the form of the Sommerfeld integral. The method of complex characteristics [12] is used. [ABSTRACT FROM AUTHOR]

The use of boundary elements in two-dimensional acoustic analysis is presented in this study, along with a detailed explanation of how to derive the final discrete equations from the fundamental fluctuation equations. In order to overcome the fictitious eigenfrequency problem that might arise during the examination of the external sound field, this work employs the Burton-Miller approach. Additionally, this work uses the Taylor expansion to extract the frequency-dependent component from the BEM function, which speeds up the computation and removes the frequency dependency of the system coefficient matrix. The effect of the radiated acoustic field generated by underwater structures' on thin-walled structures such as submarines and ships is inspected in this work. Numerical examples verify the accuracy of the proposed method and the efficiency improvement. [ABSTRACT FROM AUTHOR]

In practical conditions, near-field acoustic holography (NAH) requires the measurement environment to be a free sound field. If vibrating objects are located above the reflective ground, the sound field becomes non-free in the presence of a reflecting surface, and conventional NAH may not identify the sound source. In this work, two types of half-space NAH techniques based on the Helmholtz equation least-squares (HELS) method are developed to reconstruct the sound field above a reflecting plane. The techniques are devised by introducing the concept of equivalent source in HELS-method-based NAH. Two equivalent sources are tested. In one technique, spherical waves are used as the equivalent source, and the sound reflected from the reflecting surface is regarded as a linear superposition of orthogonal spherical wave functions of different orders located below the reflecting surface. In the other technique, some monopoles are considered equivalent sources, and the reflected sound is considered a series of sounds generated by simple sources distributed under the reflecting surface. The sound field is reconstructed by matching the pressure measured on the holographic surface with the orthogonal spherical wave source in the vibrating object and replacing the reflected sound with an equivalent source. Therefore, neither technique is related to the surface impedance of the reflected plane. Compared with the HELS method, both methods show higher reconstruction accuracy for a half-space sound field and are expected to broaden the application range of HELS-method-based NAH techniques. [ABSTRACT FROM AUTHOR]