Kernel Of Integral Equation Research Articles

An equivalent formulation of Sonine condition

Sonine kernel is characterized by the Sonine condition (denoted by SC) and is an important class of kernels in nonlocal differential equations and integral equations. This work proposes a SC with a more general form (denoted by gSC), which is more convenient than SC to accommodate complex kernels and equations. A typical kernel is given, and the first-kind Volterra integral equation under gSC is accordingly transformed and then analyzed. Based on these results, it is finally proved that the gSC is indeed equivalent to the original SC, which indicates that the Sonine kernel may be essentially characterized by the behavior of its convolution with the associated kernel at the starting point.

SOLUTIONS OF SOME INTEGRAL EQUATIONS OF THE SECOND KIND

The article examines the solutions of some integral equations of the second kind. Such equations arise when using Levi's method to construct a fundamental solution of the Cauchy problem for a degenerate equation of the Kolmogorov type. The equation may also contain a degeneracy on the initial hyperplane. The coefficients of this equation are bounded in the group of principal terms and ones are increasing functions in the group of lowest terms. The considered classes of kernels of integral equations make it possible to preserve the function that determines the growth of the coefficients of the parabolic equation when evaluating the resolvent. In the evalutions of the kernels of integral equations, there are evaluation functions that arise when constructing the corresponding fundamental solution, and fundamental solution of the Cauchy problem of the model equation with constant coefficients.

On one method of constructing the cauchy kernel of the singular integral equation for clarifie the stress-strain state of the wedge system under anti-plane deformation

An analytical approach to the construction of a singular integral equation (SIE) with Cauchy kernel is proposed, which makes it possible to clarify the stress-strain state of multi-wedge system with loaded radial cracks. This approach is based on the use of the theory of residues and the clarification of the solutions periodicity of the multi-wedge system characteristic equation. It is illustrated on the example of the construction of a SIE with Cauchy kernel for a two-wedge system with a loaded interfacial crack under the conditions of anti-plane deformation. For particular cases of geometric and mechanical parameters of the two-wedge system, the values necessary for constructing such an equation were calculated.

A fast solver framework for acoustic hybrid integral equations

Reliable modeling of the scattering by acoustically large and geometrically complex objects can be achieved by means of subdomain-dependent problem formulation and a numerically rigorous solution. While the objects’ inhomogeneity has driven the development of differential equation formulations and solvers, integral equation formulations, where the object’s background is modeled via a Green’s function, are advantageous for unbounded domains. In the hybrid integral equations approach (Usner et al., 2006), the interaction of separate subdomains with external fields is described by pertinent integral equations. Their Galerkin discretization leads to a dense blocked stiffness matrix. The development of compressed representations of the matrix, which are necessary for the treatment of large systems, becomes non-trivial due to the multitude of integral equation kernels and the different geometrical and physical characteristic of the subdomains. As part of the development of a fast hybrid integral equation solver framework, we consider the case of objects composed of large inhomogeneous volumes, modeled as incompressible fluids, and of simplified solids, modeled via surface integral equations. A hybrid integral equation formulation is derived and solved numerically. The iterative solution is accelerated by employing the butterfly-compressed hierarchical representation of the stiffness matrix, recently used for acoustic volume integral equations (Kaplan, 2022).

Mathematical Modeling of Battery Energy Storage System Operating Modes using Volterra Integral Equations

The integration of powerful photovoltaic systems and wind farms into energy systems significantly affects the operating modes due to frequent power fluctuations, thereby justifying the use of energy storage devices. The improvement of mathematical models describing operating modes of power facilities is a consequence of structure and process complication occurring in energy systems. The article presents a method for modelling the operating parameters of battery energy storage systems, based on the Volterra integral equations. The sequence of the solution is described, as well as its applicability in energy systems research. To determine the operating modes of battery energy storage systems, a non-standard class of Volterra integral equations of the first kind is used in the research. This makes it possible to calculate the operating modes of storage batteries in terms of the inverse problem, when the generation and power imbalance functions are known. To calculate the operating modes of several types of storage devices operating simultaneously within the same power system, the form of such equations can be specified in the form of piecewise constant functions. This method has a number of advantages, since it allows one to take into account nonlinear processes in the kernels of integral equations. Thus, the processes of degradation of energy storage devices that affect the efficiency of their operation are taken into account. The use of regularizing algorithms makes it possible to reduce the error. As an example, an autonomous photovoltaic system with batteries was considered. The obtained results and their visualization show the viability of applying the approach.

A numerical algorithm for a class of nonlinear fractional Volterra integral equations via modified hat functions

A numerical algorithm via a modified hat functions (MHFs) has been proposed to solve a class of nonlinear fractional Volterra integral equations of the second kind. A fractional-order operational matrix of integration is introduced. In a new methodology, the operational matrices of MHFs and the powers of weakly singular kernels of integral equations are used as a structure for transforming the main problem into a number of systems consisting of two equations for two unknowns. Relative errors for the approximated solutions are investigated. Convergence analysis of the proposed method is evaluated and convergence rate is addressed. Finally, the extraordinary accuracy of the utilized approach is illustrated by a few examples. The results, absolute and relative errors are illustrated in some tables and diagrams. In addition, a comparison is made between the absolute errors obtained by the proposed method and two other methods; one uses a hybrid approach and the other applies second Chebyshev wavelet.

Mikhlin’s Integral Equation and the Integral Equation with the Generalized Neumann Kernel on Simply Connected Domains

Mikhlin’s integral equation is a classical integral equation for solving boundary value problems for Laplace’s equation. The kernel of the integral equation is known as the Neumann kernel. Recently, an integral equation for solving the Riemann–Hilbert problem was derived. The kernel of the new integral equation is a generalization of the Neumann kernel, and hence, it is called the generalized Neumann kernel. The objective of this paper is to present a detailed comparison between these two integral equations with emphasis on their similarities and differences. This comparison is done through applying both equations to solve Laplace’s equation with Dirichlet boundary conditions in simply connected domains with smooth and piecewise smooth boundaries.

Separate retrieval of microphysical characteristics in aerosol fractions from laser sensing data

A procedure is proposed for deriving the aerosol microphysical characteristics of the spherical particle, the complex refractivity index m = mreal + i*mimag and the bimodal size distribution function U(r), using laser sensing data at the wavelengths within 355-1064 nm. A possibility of a separate fraction-wise assessment of m + U(r) is studied for weakly absorbing particles for mimag < 0.015, when mfine ≠ mcoarse holds. The reconstruction of the mode-averaged mmean is used in the preliminary stage and provides information on the mmean + U(r, mmean) parameters and the mode difference of mreal in the current observation session. A subsequent transformation of the scattering efficiency factors (integral equation kernels) divides the refractive index with respect to the fractions and gives a correct estimation of mcoarse at different sought-for values of mrealfine and mrealcoarse. A change of the kernels and the region of allowed values of the sought-for U(r) ensures its admissible estimation for both fractions. The algorithms are tested for one mfine (1.50+i*0.01) and nine mcoarse (mreal = 1.40, 1.50, 1.60; mimag = 0.0001, 0.001, 0.01). In order to include the influence of the contribution from the modal particles into the total concentration, 462 empirical models of U(r) are used.

ON A TOPOLOGICAL METHOD FOR CALCULATING THE INDEX OF QUASI-SINGULAR INTEGRAL EQUATION

A reduced spectral problem is considered for an elliptic type operator, in particular, for the Cauchy - Riemann operator with regular boundary value conditions to a quasi-singular integral equation with continuous kernel. Structure of the kernel of the quasi-singular integral equation is studied in explicit form. Index is calculated and condition for Noetherity of the investigated quasi-singular integral equation is established by the topological method on the complex plane. In this case, the spectral parameters are described under which the nonhomogeneous boundary value problem with shift for the Cauchy-Riemann equations is solvable everywhere in the class of continuous functions on the unit circle. The given problem is a nonlocal nature, and similar problems for the Cauchy-Riemann operation are described by M. Otelbaev in 1982. The solution of some singular integral equations with kernels depending on the difference and the sum of the arguments were studied in the works of I.I. Kalmushevsky. This interest is explained both by the theoretical significance of the results obtained and by the possibilities of important applications. Similar boundary value problems arise in the mathematical modeling of gas dynamics problems, soil moisture prediction, radiation transfer, population genetics, etc. The simplest examples of these boundary conditions were formulated by V.A. Steklov 1922. In this paper, the fundamental difference from the above work is the topological approach to calculating the index and establishing the Noether condition on the complexs plane.

A Mathematical Analysis Method for Bending Problem of Clamped Shallow Spherical Shell on Elastic Foundation

A mathematical analysis method is employed to solve the bending problem of slip clamped shallow spherical shell on elastic foundation. Using the slip clamped boundary conditions, the differential equations of the problem are simplified to a biharmonic equation. Using the [Formula: see text]-function, the fundamental solution and the boundary equation of the biharmonic equation, a function is established. This function satisfies the homogeneous boundary condition of the biharmonic equation. The biharmonic equation of the slip clamped shallow spherical shell bending problem on elastic foundation is transformed to Fredholm integral equation of the second kind by using Green’s formula. The vector expression of the integral equation kernel is derived. Choosing a suitable form of the normalized boundary equation, the singularity of the integral equation kernel is overcome. To obtain the numerical results, the discretization of the integral equation of the bending problem is conducted. The treatment of singular term in the discretization equation is to use the integration by parts. Numerical results of rectangular, trapezoidal, pentagonal, L-shaped and concave shape shallow spherical shells show high accuracy of the proposed method. The numerical results show fine agreement with the ANSYS finite element method (FEM) solution, which shows the proposed method is an effective mathematical analysis method.

Asymptotic regularisation of the solution to the problem of electromagnetic field scattering from a set of small impedance particles

The solution to the problem of electromagnetic (EM) wave scattering from a set of small-size impedance particles of arbitrary shape is derived by the asymptotic approach. Particles are located in a homogeneous domain with constant ɛ0 and μ0. The solution is derived under the condition b → 0, where b is the characteristic size of the particle; further, the number M(b) of particles tends to infinity at a specific rate. The regularising procedure consists of the derivation of the explicit form of a solution that excludes the necessity to solve the respective integral equation for determination of the fields at the surface of particles and thus avoids integrating the Green function derivatives, which are in the kernel of this boundary integral equation. Practical application of this approach yields an ability to create media with the desired inhomogeneous distribution of effective refractive index n and magnetic permeability μ(x). Explicit analytical formulas are derived for these physical parameters and supported by computations.

Периодические контактные и смешанные задачи теории упругости (обзор)

Results are reviewed collected in the investigations of periodic contact and mixed problems of the plane, axially symmetric and spatial elasticity theory. Among mixed problems, cut (crack) problems are focused integral equations of which are connected with those for contact problems. The periodic contact problems stimulate research of the discrete contact of rough (wavy) surfaces. Together with classical elastic domains (half-plane, half-space, plane and full space), we consider periodic problems for cylinder, layer, cone and spatial wedge. Most publications including fun-damental ones by Westergaard and Shtaerman deals with plane periodic problems of the elasticity theory. Here, one can mention approaches based on complex variable functions, Fourier series, Green’s functions and potential func-tions. A fracture mechanics approach to the plane periodic contact problem was developed. Methods and approaches are considered which allow us to take friction forces, adhesion and wear into account in the periodic contact. For spatial periodic and doubly periodic contact and properly mixed problems, we describe such methods as the localiza-tion method, the asymptotic methods, the method of nonlinear boundary integral equations, the fast Fourier trans-form. The half-space is the simplest model for elastic solids. But for the simplest straight-line periodic punch system, some three-dimensional contact problems (normal contact or tangential contact for shifted cohesive coatings) turn out to be incorrect because their integral equations contain divergent series. Considering three-dimensional periodic problems, I.G. Goryacheva disposes circular punches in special way (circular orbits, polar coordinated are used for centers of the punches), in this case one can prove convergence of the series in the integral equation (it is important that the punches are circular). For the periodic problems for an elastic layer, V.M. Aleksandrov has shown that the series in integral equations converge but the kernels become more complicated. In the present paper, we demonstrate that for the straight-line periodic punch system of arbitrary form the contact problem for a half-space turns out to be correct in case of more complicated boundary conditions. Namely, it can be sliding support or rigid fixation of a half-plane on the half-space boundary, the half-plane boundary should be parallel to the straight-line (the punch system axis) for arbitrary finite distance between the parallel lines. On this way, for sliding support, the kernel of the period-ic problem integral equation kernel is free of integrals, it consists of single convergent series (normal contact, the kernel is given in two equivalent forms). We consider classical percolation (how neighboring contact domains pene-trate one to another, investigated by K.L. Johnson, V.A. Yastrebov with co-authors) for the three-dimensional periodic contact amplification as well as percolation for the straight-line punch system. A similar approach is suggested for the case of periodic tangential contact (coatings system cohesive with a half-space boundary shifted along its axis or perpendicular to it). Here, one can separate out unique solutions of auxiliary problems because the line of changing boundary conditions on the half-space boundary can provoke non-uniqueness. The method proposed opens possibility to consider more complicated three-dimensional periodic contact problems for straight-line punch systems with changing boundary conditions inside the period.

Classification and Applications of Randomized Functional Numerical Algorithms for the Solution of Second-Kind Fredholm Integral Equations

Systematization of numerical randomized functional algorithms for approximation of solutions to second-kind Fredholm integral equation is performed in this paper. Three types of algorithms are emphasized: projection algorithms, grid algorithms, and projection-grid algorithms. Disadvantages of grid algorithms that require calculation of values of the kernel of integral equations at fixed points are revealed (practically, the kernels of equation have integrable singularities and this calculation is impossible). Thus, for applied problems related to the solution of second-kind Fredholm integral equations, projection or projection-grid randomized algorithms are more convenient.

A Ground Penetrating Radar Imaging Approach for a Heterogeneous Subsoil With a Vertical Permittivity Gradient

The article presents an imaging approach for ground penetrating radar prospecting over a heterogeneous background soil characterized by a dielectric permittivity variation along depth. The proposed approach is based on a linear model of the electromagnetic scattering phenomenon, where the soil inhomogeneity in the reference scenario is accounted for by introducing an equivalent permittivity function. Such a function notably simplifies the computation of the kernel of the relevant linear integral equation to be inverted. Then, the data inversion is carried out in a regularized way by resorting to the truncated singular value decomposition scheme. A procedure for estimating the equivalent permittivity function, typically unknown in practical situations, is also proposed. Reconstruction results achieved from full-wave simulated data and real data collected in a controlled test site are provided to demonstrate the effectiveness and the reliability of the proposed imaging approach.

Improving the use of the randomized singular value decomposition for the inversion of gravity and magnetic data

The focusing inversion of gravity and magnetic potential-field data using the randomized singular value decomposition (RSVD) method is considered. This approach facilitates tackling the computational challenge that arises in the solution of the inversion problem that uses the standard and accurate approximation of the integral equation kernel. We have developed a comprehensive comparison of the developed methodology for the inversion of magnetic and gravity data. The results verify that there is an important difference between the application of the methodology for gravity and magnetic inversion problems. Specifically, RSVD is dependent on the generation of a rank [Formula: see text] approximation to the underlying model matrix, and the results demonstrate that [Formula: see text] needs to be larger, for equivalent problem sizes, for the magnetic problem compared to the gravity problem. Without a relatively large [Formula: see text], the dominant singular values of the magnetic model matrix are not well approximated. We determine that this is due to the spectral properties of the matrix. The comparison also shows us how the use of the power iteration embedded within the randomized algorithm improves the quality of the resulting dominant subspace approximation, especially in magnetic inversion, yielding acceptable approximations for smaller choices of [Formula: see text]. Further, we evaluate how the differences in spectral properties of the magnetic and gravity input matrices also affect the values that are automatically estimated for the regularization parameter. The algorithm is applied and verified for the inversion of magnetic data obtained over a portion of the Wuskwatim Lake region in Manitoba, Canada.

Three-boson bound states in Minkowski space with contact interactions

The structure of the three-boson bound state in Minkowski space is studied for a model with contact interaction. The Faddeev-Bethe-Salpeter equation is solved both in Minkowski and Euclidean spaces. The results are in fair agreement for comparable quantities, like the transverse amplitude obtained when the longitudinal constituent momenta of the light-front valence wave function are integrated out. The Minkowski space solution is obtained numerically by using a recently proposed method based on the direct integration over the singularities of the propagators and interaction kernel of the four-dimensional integral equation. The complex singular structure of the Faddeev components of the Bethe-Salpeter vertex function for space and time-like momenta in an example of a Borromean system is investigated in detail. Furthermore, the transverse amplitude is studied as a mean to access the double-parton transverse momentum distribution. Following that, we show that the two-body short-range correlation contained in the valence wave function is evidenced when the pair has a large relative momentum in a back-to-back configuration, where one of the Faddeev components of the Bethe-Salpeter amplitude dominates over the others. In this situation a power-law behavior is derived and confirmed numerically.

Determining the Vertical Profile of the Greenhouse-Gas Concentration in the Atmosphere up to 80 km on the Satellite-to-Earth Radio Translucence

This article discusses methods for determining trace gases in the Earth’s atmosphere based on radio translucence of the satellite–Earth or Earth–satellite–Earth route using monochromatic radiation at a given frequency set. The substantiation of a new method using the difference of differential signals on different slopes of the absorption line of the measured gas is carried out, which allows one to increase the spatial selectivity of the measurements and reduce the influence of clouds, precipitation, and third-party gases. The results of calculating the kernels of integral equations when measuring the profile of water vapor in the absorption band of 22.23 GHz are presented. The selectivity of the kernels makes it possible to restore the water-vapor profile both in the lower troposphere layer to heights of 8 km and in the 30–80 km layer. The advantage of the proposed methods is the differential nature of the measurements, which does not require absolute calibration of sources and long-term stability of the equipment. In addition, it becomes possible to register small-scale and rapidly varying processes occurring during the revolution of the satellite in orbit on a spatial scale of 2–10 km. It is precisely this scale of spatiotemporal changes in the concentration of greenhouse gases that can be expected during technological accidents, missile launches, and other anthropogenic influences.

Differential Radiometric Methods for Measurements of the Vertical Profile of Water Vapor in the Troposphere and Stratosphere of the Earth

Differential methods for determining the height profile of one of the most important greenhouse gases in the Earth’s atmosphere—water vapor—based on ground and satellite radio thermal microwave measurements in the absorption band of 22 GHz, are considered. The substantiation of a fundamentally new difference-differential method is carried out using the difference of differential signals on different slopes of the absorption line of the measured gas. This method makes it possible to increase the spatial selectivity of measurements and reduce the influence of clouds, precipitation, and third-party gases, as well as carry out measurements above the water surface. The results of the calculation of the kernels of integral equations are given for measurements in the absorption band of water vapor of 22 GHz. The selectivity of the kernels makes it possible to restore the water-vapor profile both in the lower troposphere layer to heights of 8 km and in the stratospheric–mesospheric layer of 30–80 km. The advantage of the proposed methods is the differential nature of the measurements, which does not require the absolute calibration of sources and long-term stability of the equipment.

Boundary element method for fracture mechanics analysis of 3d non-planar cracks in anisotropic solids

Among the numerical approaches to fracture mechanics analysis of cracked anisotropic solids, the boundary element method is notable for high accuracy and performance due to its semi-analytical nature and the use of only boundary mesh. Various boundary element techniques were proposed for 3D fracture mechanics analysis. However, the main problem of these approaches is the treatment of singular and hypersingular integrals, which can demand analytic evaluation of coefficient at kernel singularity at singular point in local curvilinear coordinates, which produce cumbersome equations in the case of non-planar geometries. Therefore, the paper presents novel formulation of the boundary element method for 3D fracture mechanics analysis of anisotropic solids with non-planar cracks. Pan’s single domain boundary element formulation is extended with several novelties, which allow accurate analysis of non-planar geometries. These are modified Kutt’s numerical quadratures with Chebyshev nodes for accurate evaluation of singular and hypersingular integrals; polynomial mappings for smoothing the integrand at the crack front line; and special shape functions, which account for a square-root stress singularity at crack front and allow accurate determination of stress intensity factors. The kernels of boundary integral equations are evaluated using the exponentially convergent quadrature, which allows derivation of fast boundary element technique. The procedure for accurate numerical determination of stress intensity factors at arbitrary point of the crack front is also developed. Numerical examples are presented, which show high accuracy of the proposed boundary element method. It is shown that non-planar cracks exhibit shearing mode opening along with normal one due to its geometry and direction of the applied loading. Present approach can be combined with Sih’s strain energy density criterion to study 3D cracks propagation in anisotropic elastic solids under fatigue loading, which is the direction of future research.

Exact solution to the Lippmann-Schwinger equation for an elliptical billiard

We present an exact solution to the Lippmann-Schwinger equation for a two-dimensional elliptic billiard modeled as a boundary-wall. Using a property of the Green function — which is a zeroth order Hankel function of the first kind — we managed to separate the integral equation Kernel. This approach allows us to study the problem analytically for four representative potential coupling strength γ(s), namely: (i) γ(s)=γ0/cosh2μ0−cos2s; (ii) γ(s) = γ0; (iii) γ(s) being a 2π−periodic even function; and (iv) γ(s) being a 2π−periodic odd function, where γ0 is a real constant. Also, we find the condition which needs to be verified in order to the incident plane wave to produce a resonance inside the barrier.