Integral Equation Research Articles

Phaseless inverse scattering with a parametrized spatial spectral volume integral equation for finite scatterers in the soft x-ray regime

Soft x-ray wafer-metrology experiments are characterized by low signal-to-noise ratios and lack phase information, which both cause difficulties with the accurate three-dimensional profiling of small geometrical features of structures on a wafer. To this end, we extend an existing phase-based inverse-scattering method to demonstrate a sub-nanometer and noise-robust reconstruction of the targets by synthetic soft x-ray scatterometry experiments. The targets are modeled as three-dimensional finite dielectric scatterers embedded in a planarly layered medium, where a scatterer’s geometry and spatial permittivity distribution are described by a uniform polygonal cross section along its height. Each cross section is continuously parametrized by its vertices and homogeneous permittivity. The combination of this parametrization of the scatterers and the employed Gabor frames ensures that the underlying linear system of the spatial spectral Maxwell solver is continuously differentiable with respect to the parameters for phaseless inverse-scattering problems. In synthetic demonstrations, we demonstrate the accurate and noise-robust reconstruction of the parameters without any regularization term. Most of the vertex parameters are retrieved with an error of less than λ/13 with λ=13.5nm, when the ideal sensor model with shot noise detects at least five photons per sensor pixel. This corresponds to a signal-to-noise ratio of 3.5 dB. These vertex parameters are retrieved with an accuracy of λ/90 when the signal-to-noise ratio is increased to 10 dB, or approximately 100 photons per pixel. The material parameters are retrieved with errors ranging from 0.05% to 5% for signal-to-noise ratios between 10 dB and 3.5 dB.

The Smirnov Property for Weighted Lebesgue Spaces

We establish lower norm bounds for multivariate functions within weighted Lebesgue spaces, characterised by a summation of functions whose components solve a system of nonlinear integral equations. This problem originates in portfolio selection theory, where these equations allow one to identify mean-variance optimal portfolios, composed of standard European options on several underlying assets. We elaborate on the Smirnov property—an integrability condition for the weights that guarantees the uniqueness of solutions to the system. Sufficient conditions on weights to satisfy this property are provided, and counterexamples are constructed, where either the Smirnov property does not hold or the uniqueness of solutions fails.

A new method for converting impulsive Riemann–Liouville fractional order system into the integral equation

A new method for converting impulsive Riemann–Liouville fractional order system into the integral equation

Integral formulation of Klein–Gordon singular waveguides

AbstractWe consider the analysis of singular waveguides separating insulating phases in two‐space dimensions. The insulating domains are modeled by a massive Schrödinger equation and the singular waveguide by appropriate jump conditions along the one‐dimensional interface separating the insulators. We present an integral formulation of the problem and analyze its mathematical properties. We also implement a fast multipole and sweeping‐accelerated iterative algorithm for solving the integral equations, and demonstrate numerically the fast convergence of this method. Several numerical examples of solutions and scattering effects illustrate our theory.

Unified framework for stochastic dynamic responses and system reliability analysis of long-span cable-stayed bridges under near-fault ground motions

Long-span cable-stayed bridges are complex systems with numerous components, and prone to failures of multiple components under near-fault ground motions with long-period velocity pulses. The evaluation of dynamic system reliability of cable-stayed bridges is an important challenging issue. In this work, a unified framework for simultaneously determining both the stochastic dynamic responses and system reliabilities of long-span cable-stayed bridges under near-fault stochastic ground motions is proposed based on direct probability integral method (DPIM) with adaptive strategy. Firstly, the 3-dimensional finite element model of typical cable-stayed bridge and the stochastic synthesis model of near-fault ground motions are established. Based on probability density integral equation, new formulas in input probability space of time-variant moments of response are derived. Then, DPIM with adaptive strategy is suggested to calculate the mean value and standard deviation of response via new formulas. Dynamic system reliability of cable-stayed bridge under near-fault stochastic ground motions is evaluated by establishing failure criteria of extreme value mappings of multiple performance functions. Finally, numerical results indicate that the long-period velocity pulse of near-fault motions imposes an important effect, resulting in large mean responses and failure of cables. Failure probability of cables is significantly increased with the occurrence of velocity pulses, which causes the decrease of system reliability of cable-stayed bridge. Thus, the randomness and velocity pulse effect of near-fault ground motions should be seriously considered for reliability-based design and safety assessment of cable-stayed bridges.

A novel numerical method for solving the Kirchhoff-Helmholtz integral equation based on the sound field representation using the reproducing kernel in the frequency domain

The paper introduces a unique numerical solution method for the Kirchhoff-Helmholtz integral equation. This method is based on the representation of the sound field using the reproducing kernel in the solution space of the homogeneous Helmholtz equation. When compared to the conventional boundary element method, the proposed method holds numerous advantages. This paper presents the analysis theory of the proposed method and demonstrates its validity by calculating bounded and unbounded sound fields. The accuracy of the calculation is assessed by computing the sound field in a rectangular room under various conditions. Furthermore, although some theoretical issues remain, the proposed method effectively avoids the non-uniqueness of solutions in unbounded domain problems.

Estimation of the surface impedance based on the discrete complex image source method and array processing

This paper introduces a method for estimating the sound absorption coefficient of a large panel of absorbing material from measurements using a microphone array. The method relies upon describing the pressure field as the set of monopoles obtained when the reflection coefficient is represented as the Laplace Transform of an image source distribution. This formulation results in a well-behaved integral equation, which is discretized using Gaussian quadrature to yield an inverse problem. It is possible to determine the regularized solution to the inverse problem and reconstruct the surface impedance of the sample by conducting measurements of the sound pressure at multiple points of observation. This sound field representation is adequate for sound sources in the close range of the material and the array. The paper presents simulated and experimental investigations to delineate the efficacy of the proposed measurement technique against the well-known image source method, thereby highlighting its significant potential to advance acoustic characterization methodologies.

Modeling of seismic wave scattering in poroelastic media

Understanding wave scattering in the earth is considered fundamental in describing seismic wave propagation and providing information on structural features of the earth’s interior. Petrophysical parameters (especially porosity and permeability) affect the reflection coefficients of subsurface interfaces, which can better explain the field data and infer the subsurface structure. However, the numerical solutions to the scattering problem for efficient modeling of wave propagation in poroelastic earth structures have limitations. We develop a numerical algorithm for solving the poroelastic scattering integral equations. Specifically, applying perturbation theory to Biot’s equations, the solutions are expressed by the Lippman-Schwinger integral equations, which can express the displacement and pressure fields. We derive the contrast-source integral equations of the decoupled poroelastic wave equations. We apply a conjugate-gradient fast Fourier transform method for fast solutions of the integral equations. We show that despite the complexity of the geologic structure, the numerical method enables the modeling of the displacement and pressure fields in the frequency and time domains. We determine that the wave scattering problem for the Biot model provides a good description to understand the earth’s interior.

Analysis of efficient discretization technique for nonlinear integral equations of Hammerstein type

Purpose This study focuses on investigating the numerical solution of second-kind nonlinear Volterra–Fredholm–Hammerstein integral equations (NVFHIEs) by discretization technique. The purpose of this paper is to develop an efficient and accurate method for solving NVFHIEs, which are crucial for modeling systems with memory and cumulative effects, integrating past and present influences with nonlinear interactions. They are widely applied in control theory, population dynamics and physics. These equations are essential for solving complex real-world problems. Design/methodology/approach Demonstrating the solution’s existence and uniqueness in the equation is accomplished by using the Picard iterative method as a key technique. Using the trapezoidal discretization method is the chosen approach for numerically approximating the solution, yielding a nonlinear system of algebraic equations. The trapezoidal method (TM) exhibits quadratic convergence to the solution, supported by the application of a discrete Grönwall inequality. A novel Grönwall inequality is introduced to demonstrate the convergence of the considered method. This approach enables a detailed analysis of the equation’s behavior and facilitates the development of a robust solution method. Findings The numerical results conclusively show that the proposed method is highly efficacious in solving NVFHIEs, significantly reducing computational effort. Numerical examples and comparisons underscore the method’s practicality, effectiveness and reliability, confirming its outstanding performance compared to the referenced method. Originality/value Unlike existing approaches that rely on a combination of methods to tackle different aspects of the complex problems, especially nonlinear integral equations, the current approach presents a significant single-method solution, providing a comprehensive approach to solving the entire problem. Furthermore, the present work introduces the first numerical approaches for the considered integral equation, which has not been previously explored in the existing literature. To the best of the authors’ knowledge, the work is the first to address this equation, providing a foundational contribution for future research and applications. This innovative strategy not only simplifies the computational process but also offers a more comprehensive understanding of the problem’s dynamics.

Q-Learning-Based Dumbo Octopus Algorithm for Parameter Tuning of Fractional-Order PID Controller for AVR Systems

The tuning of fractional-order proportional-integral-derivative (FOPID) controllers for automatic voltage regulator (AVR) systems presents a complex challenge, necessitating the solution of real-order integral and differential equations. This study introduces the Dumbo Octopus Algorithm (DOA), a novel metaheuristic inspired by machine learning with animal behaviors, as an innovative approach to address this issue. For the first time, the DOA is invented and employed to optimize FOPID parameters, and its performance is rigorously evaluated against 11 existing metaheuristic algorithms using 23 classical benchmark functions and CEC2019 test sets. The evaluation includes a comprehensive quantitative analysis and qualitative analysis. Statistical significance was assessed using the Friedman’s test, highlighting the superior performance of the DOA compared to competing algorithms. To further validate its effectiveness, the DOA was applied to the FOPID parameter tuning of an AVR system, demonstrating exceptional performance in practical engineering applications. The results indicate that the DOA outperforms other algorithms in terms of convergence accuracy, robustness, and practical problem-solving capability. This establishes the DOA as a superior and promising solution for complex optimization problems, offering significant advancements in the tuning of FOPID for AVR systems.

Exploring the effects of finite size and indenter shape on the contact behavior of functionally graded thermoelectric materials

The performance enhancement of functionally graded thermoelectric (FGTE) devices is significantly influenced by contact studies of the FGTE materials. It is unclear how the finite thickness and the punch geometry influence the FGTE materials’ contact behaviors. This paper investigates the frictionless contact problem between three types of rigid punches (flat, triangular, and cylindrical) and the FGTE strip with finite thickness. The electric-thermo-elastic parameters of the FGTE strip vary in the thickness direction according to an exponential function. Based on the Fourier integral transform and the transformation matrix method, the problem is transformed into the numerical solution of three sets of singular integral equations. The presence of singular features on either side of the punch demands the adoption of specific collocation strategies. The distribution of the normal current density, the normal energy flux, and the normal contact stress is obtained by adjusting multiple electric-thermo-elastic parameters. The contact stresses in the case of punches with varying shapes can be effectively controlled by modulating the coefficient of thermal expansion and the strip thickness, whereas the effect of the electrical conductivity, the shear modulus, and the thermoelectric load on these stresses depends on whether they are increased or decreased.

Discretization of Continuous Spaces using Barycentric Subdivision Method with Metric Space Constraints on the Nearest Neighbour Edges

The Rao-Wilton-Glisson (RWG) is a commonly used basis function in the numerical solution of the electric field integral equation (EFIE) using the Method of Moments (MoM) and Galerkin approach. This method relies on triangular patches to approximate the surface current. Traditionally, barycentric subdivision of a primary triangle into n-sub-triangles has been used with RWG basis function to solve the EFIE using MoM. This paper presents a method of approximating a surface using triangular patches by sub-dividing a primary triangle into (2n-1) subtriangles. which creates a denser mesh than the widely used nine (9) points quadrature method. The structure is approximated by small square patches, which are further sub-divided into two primary triangles. By applying barycentric sub-division, the primary triangles are decomposed into sub-triangles to create a mesh over the surface of the structure. Using graph theory, the triangular meshes are defined by the function, G (V, E), where V and E are the vertices and edges of the triangles in the mesh space, respectively. The connectivity matrix of shared edges is found by imposing a constraint on the edge length using metric spaces. This method approximates a square patch with 32 scalene triangles and shows that it can be used to reconstruct equilateral patches and doubly split double ring into mesh structures.

Existence of Solutions for a Coupled Hadamard Fractional System of Integral Equations in Local Generalized Morrey Spaces

Existence of Solutions for a Coupled Hadamard Fractional System of Integral Equations in Local Generalized Morrey Spaces

ON A BOUNDARY VALUE PROBLEM FOR HIGH-ORDER HYPERBOLIC EQUATION WITH IMPULSE DISCRETE MEMORY

The investigation focuses on a boundary value problem for a high-order hyperbolic equation with impulse discrete memory in a rectangular domain. By introducing new functions, the problem is transformed into a set of boundary value problems for a first-order differential equation with impulse discrete memory, which depends on unknown functions and integral relations. D.S. Dzhumabaev's parametrization method is applied to this equivalent problem. The domain is divided according to the time variable, and functional parameters representing discrete memory values are introduced within the interior domains. As a result, the family of boundary value problems for the first-order differential equation with impulse discrete memory and unknown functions is converted into an equivalent family of integral-multipoint boundary value problems involving functional parameters and unknown functions. These equivalent problems include initial value problems for first-order differential equations related to the new functions. The solutions to the initial value problems are expressed using Volterra integral equations. By substituting these solutions into the boundary and impulse conditions, a system of linear functional equations concerning the functional parameters is derived. An algorithm is developed to solve the equivalent problem, and sufficient conditions for the unique solvability of the family of integral-multipoint boundary value problems with functional parameters and unknown functions are provided. Additionally, sufficient conditions for the unique solvability of the original boundary value problem for the high-order hyperbolic equation with impulse discrete memory are established based on the initial data.

Discretization methods and their extrapolations for two-dimensional nonlinear Volterra-Urysohn integral equations

In this paper, a class of nonlinear two-dimensional (2D) integral equations of Volterra type, i.e. Volterra-Urysohn integral equations, is studied. Following the ideas of [24], and assuming that the kernels of the integral equation are Lipschitz functions with respect to the dependent variable, the existence and uniqueness of a solution to the integral equation is shown by a technique based on the Picard iterative method. Then, the Euler and trapezoidal discretization methods are used to reduce the solution of the integral equation to the solution of a system of nonlinear algebraic equations. It is proved that the solution of the Euler method has first order convergence to the exact solution of the integral equation while the solution of the trapezoidal method has quadratic convergence. To prove the convergence of the trapezoidal method, a new Gronwall inequality is developed. Some numerical examples are given which confirm our theoretical results.

Influence of specularity factor on heat transport in nanoribbons of different sizes

In this paper, the phonon heat transfer in two-dimensional conductors with different types of phonon reflection from the boundaries is examined. Heat conductors with arbitrary ratios of width W to length L are studied assuming that the mean free path between phonon-phonon collisions is infinite. The integral equations for the angular distribution functions of the incident and reflected phonons at a specific point on the conductor boundary were proposed. To solve these integral equations a new iterative method is proposed. The proposed iterative approach formally corresponds to taking into account subsequent collisions of phonons with the edges of the conductor. It ensures the convergence of the desired solution for any W/L ratio and for any value of the specularity coefficient p. Interpolation formulae are found to describe with sufficient accuracy the solution of the system of integral equations in the entire region of the specularity coefficient p from 0 to 1, and the W/L ratios ranging from 10 to 0.01. These formulae allow the construction of the isolines of the thermal conductance coefficient values, from which it is possible to determine the necessary values of W/L and parameter p to get the desired value of the thermal conductance.

A bidirectional fifth-order partial differential equation: Lax representation and soliton solutions

In this paper, we study the bidirectional fifth-order partial differential equation uttt−uxxxxt−4(uxut)xx−4(uxuxt)x=0, which was proposed as a new integrable equation by Wazwaz [Phys. Scr. 83, 015012 (2011)]. By means of the prolongation structure method of Wahlquist and Estabrook [J. Math. Phys. 16, 1–7 (1975)], we construct a Lax representation for this equation. This enables us to confirm its integrability and identify it as the potential second-order flow in a Gelfand-Dickey hierarchy. We also use the Darboux transformation and present the multi-soliton solutions in the Wronskian form. Finally, we comment on a more general bidirectional partial differential equation.

Finite Genus Solutions to a Hierarchy of Integrable Semi-Discrete Equations

Finite Genus Solutions to a Hierarchy of Integrable Semi-Discrete Equations

Depletion forces beyond the diluted limit

The determination of the effective interactions in many-body systems still possesses a tremendous challenge in Condensed Matter Physics. The problem mainly resides in the fact that, on the one hand, there is not a unique route to obtain the effective forces between the ”observable” constituents and, on the other hand, one typically considers that the diluted limit of those components always corresponds to the effective forces at all particle concentrations. In the particular case of colloidal dispersions, it is very important to have experimental, theoretical, and computational tools that allow us to determine, accurately and without using significant approximations, the effective interactions between colloids. In this contribution, we report on a detailed study of depletion potentials in colloidal mixtures at finite concentration, namely, binary and ternary mixtures of disks (in two dimensions or 2D) and spheres (in three dimensions or 3D). To this end, the depletion forces between the large colloidal species are determined through a recently developed scheme based on the so-called contraction of the description, which has been extended and built at the level of the bare forces, i.e., we basically consider that all particles interact through the bare potentials and the net force acting on a given particle is determined by the second’s Newton law. This scheme can be easily adapted to Molecular Dynamics simulations. To verify the physical consistency of the formalism, we explicitly show that in the diluted limit of large particles, the resulting depletion potential reproduces correctly the AO-Vrij limit. As further proof of the accuracy of the results, a comparison of the structure of the large colloids in the whole mixture with the one that results using exclusively the depletion potential is carried out. In 3D, the results are explicitly compared with the potential of mean force and those obtained with the integral equations formalism. We also report a new colloidal stability mechanism based on the use of two different species of depletant agents at low and equal concentrations. Although we highlight the fact that the depletion potential depends on the concentration of the large species, we show that this dependence can be eliminated when the colloidal dispersion is open to a reservoir of small particles, i.e., when the chemical potential of small particles is fixed. Finally, we report the effects of polydispersity on the depletion interaction between large colloids.

Accurate computation of scattering poles of acoustic obstacles with impedance boundary conditions

We propose a computation method for scattering poles of impedance obstacles. Boundary integral equations are used to formulate the problem. It is shown that the scattering poles are the eigenvalues of some integral operator. Then we employ the Nyström method to discretize the integral operator and obtain a nonlinear matrix eigenvalue problem. The eigenvalues are computed using a multistep parallel spectral indicator method. Numerical examples demonstrate the high accuracy of the proposed method and can serve as the benchmarks. Our study provides a practical approach and can be extended to other scattering problems. This paper continues our previous study on the computation method for scattering poles of sound-soft obstacles.