System Of One-dimensional Integral Equations Research Articles

Determination of non-axisymmetric stresses in the bodies of revolution based on regulized intergral equations

A solution to the 3D elasticity theory problem in the form of a Fourier series expansion by an angular coordinate, the coefficients of which are determined from the system of one-dimensional integral equations, is constructed. The relation for determining the kernels of these equations for harmonics of arbitrary order is written in analytical form. The regularization of the obtained equations is carried out. For this purpose, the kernels of the equations are represented as the sum of regular functions and singular components, which are determined through the kernels of the plane problem of the elasticity theory in the Cartesian coordinate system. In the stress tensor, the components that contain logarithmic functions as multiplier, are additionally highlighted. The test calculations are performed, which confirm the reliability of the analytically obtained relations and integral equations built on their basis. Testing of the developed numerical algorithm for solving integral equations is carried out and examples of calculation of non-axisymmetric stress concentration near cavities in bodies of various shapes are resulted. The obtained results allow to expand the application of boundary elements method to study of practically important non-axisymmetric problems for the bodies of rotation.

The Radon Transform over Cones with Vertices on the Sphere and Orthogonal Axes

Recovering a function from its integrals over circular cones recently gained significance because of its relevance to novel medical imaging technologies such as emission tomography using Compton cameras. In this paper, we investigate the case where the vertices of the cones of integration are restricted to a sphere in $n$-dimensional space and where the symmetry axes are orthogonal to the sphere. We show invertibility of the considered transform and develop an inversion method based on series expansion and reduction to a system of one-dimensional integral equations of generalized Abel type. Since the arising kernels do not satisfy standard assumptions, we also develop a uniqueness result for generalized Abel integral equations where the kernel has zeros on the diagonal. Finally, we demonstrate how to efficiently implement our inversion method and present numerical results.

Motion of a spherical particle inside a liquid film

The motion of a spherical particle inside a liquid film coated on a plane wall is considered under conditions of Stokes flow in the limit of vanishing capillary number where the interfacial deformation is infinitesimal. The problem is formulated in terms of a system of one-dimensional integral equations for the velocity and traction Fourier coefficients along the trace of the interface, wall, and particle contour in a meridional plane, and the solution is found using a boundary-element method. Comprehensive results for the force and torque resistance coefficients are presented in the case of particle rotation and translation in quiescent fluids. The velocity of translation and angular velocity or rotation of a freely suspended particle in simple shear flow are computed and discussed over a broad range of conditions.

Effect of surface slip on Stokes flow past a spherical particle in infinite fluid and near a plane wall

The motion of a spherical particle in infinite linear flow and near a plane wall, subject to the slip boundary condition on both the particle surface and the wall, is studied in the limit of zero Reynolds number. In the case of infinite flow, an exact solution is derived using the singularity representation, and analytical expressions for the force, torque, and stresslet are derived in terms of slip coefficients generalizing the Stokes–Basset–Einstein law. The slip velocity reduces the drag force, torque, and the effective viscosity of a dilute suspension. In the case of wall-bounded flow, advantage is taken of the axial symmetry of the boundaries of the flow with respect to the axis that is normal to the wall and passes through the particle center to formulate the problem in terms of a system of one-dimensional integral equations for the first sine and cosine Fourier coefficients of the unknown traction and velocity along the boundary contour in a meridional plane. Numerical solutions furnish accurate predictions for (a) the force and torque exerted on a particle translating parallel to the wall in a quiescent fluid, (b) the force and torque exerted on a particle rotating about an axis that is parallel to the wall in a quiescent fluid, and (c) the translational and angular velocities of a freely suspended particle in simple shear flow parallel to the wall. For certain combinations of the wall and particle slip coefficients, a particle moving under the influence of a tangential force translates parallel to the wall without rotation, and a particle moving under the influence of a tangential torque rotates about an axis that is parallel to the wall without translation. For a particle convected in simple shear flow, minimum translational velocity is observed for no-slip surfaces. However, allowing for slip may either increase or decrease the particle angular velocity, and the dependence on the wall and particle slip coefficients is not necessarily monotonic.

Particle motion near and inside an interface

The motion of a spherical particle near the interface between two immiscible viscous fluids undergoing simple shear flow is considered in the limit of small Reynolds and capillary numbers where the interface exhibits negligible deformation. Taking advantage of the rotational symmetry of the boundaries of the flow with respect to the axis that is normal to the interface and passes through the particle centre, the problem is formulated as a system of one-dimensional integral equations for the first Fourier coefficients of the unknown components of the traction and velocity along the particle and interface contours. The results document the particle translational and angular velocities, and reveal that the particle slips while rolling over the interface under the influence of a simple shear flow, for any viscosity ratio. In the second part of the investigation, the motion of an axisymmetric particle straddling a planar interface is considered. The results confirm a simple exact solution when a particle with top-down symmetry is immersed half-way in each fluid and translates parallel to the interface, reveal a similar simple solution for a particle that is held stationary in simple shear flow, and document the force and torque exerted on a spherical particle for more general arrangements. The onset of a non-integrable singularity of the traction at the contact line prohibits the computation of the translational and angular velocities of a freely suspended particle convected under the action of a shear flow.

Motion of a spherical particle in film flow

The motion of a spherical particle suspended in gravity-driven film flow down an inclined plane is considered in the limit of vanishing Reynolds and Bond numbers where the free-surface deformation is infinitesimal. Taking advantage of the axially symmetry of the boundaries of the flow with respect to the axis that is normal to the wall and free surface and passes through the particle centre, the problem is formulated as a system of one-dimensional integral equations for the first Fourier coefficients of the unknown traction and velocity along the boundary contours in a meridional plane. It is found that the particle translational velocity scaled by the unperturbed velocity evaluated at the particle centre increases monotonically as the particle approaches the free-surface, whereas the corresponding angular velocity of rotation scaled by the unperturbed vorticity evaluated at the particle centre reaches a maximum at a certain intermediate position. The free-surface velocity vector field and deformation are displayed, the force and torque exerted on a spherical particle adhering to the wall are tabulated, and the associated flow pattern is discussed.

Imaging three dimensional two-particle correlations for heavy-ion reaction studies

We report an extension of the source imaging method for analyzing three-dimensional sources from three-dimensional correlations. Our technique consists of expanding the correlation data and the underlying source function in spherical harmonics and inverting the resulting system of one-dimensional integral equations. With this strategy, we can image the source function quickly, even with the finely binned data sets common in three-dimensional analyses.

Three-Dimensional Imaging Analysis of Two-Particle Correlations in Heavy-Ion Reactions

We report an extension of the source imaging method for imag- ing full three-dimensional sources from three-dimensional like-pair correlations. Our technique consists of expanding the correlation data and the underlying source function in spherical harmonics and inverting the resulting system of one-dimensional integral equations. With this method of attack, we can image the source function quickly, even with the extremely large data sets common in three-dimensional analyses. We apply our method to the recently measured E859 un-Coulomb corrected data.

Solution of the elastic boundary value problems for a layer with tunnel stress raisers

A novel procedure for solving three-dimensional problems for elastic layer weakened by through-thickness tunnel cracks has been developed and is presented in this paper. This procedure reduces the given boundary value problem to an infinite system of one-dimensional singular integral equations and is based on a system of homogeneous solutions for a layer. Integral representations of single- and double-layer potentials are used for metaharmonic and harmonic functions entering in the singular integral equations. These representations provide a continuous extendibility of the stress vector while allowing a jump in the displacement vector in the transition through the cut. Expanding the potential and biharmonic solutions in the Fourier series over the thickness coordinate yields the integral representations of the displacement vector and stress tensor. The problem of reducing a denumerable set of the integral equations of the given boundary value problem to one-to-one correspondence with the set of unknown densities appearing in the Fourier’s coefficient representations has been settled efficiently. Numerical investigations show a rapid convergence of the proposed reduction procedure as applied to the solution of the infinite system of one-dimensional integral equations. Numerical examples illustrate the proposed method and demonstrate its advantages.

Depolarization of the electromagnetic field in a locally inhomogeneous earth-ionosphere waveguide

This paper presents a further development of the numerical-analytical method for the solution of three-dimensional problems in the theory of radio wave propagation. We consider a vector problem of the electromagnetic field of a vertical electric dipole in a plane Earth-ionosphere waveguide with a local large-scale irregularity on the anisotropic ionosphere wall. The possibility of lowering (elevating) of the local region of the upper waveguide wall with respect to the regular ionosphere level is taken into account. The field components on the boundary surfaces obey the Leontovich impedance conditions. The problem is reduced to a system of two-dimensional integral equations taking into account the overexcitation and depolarization of the field scattered by the irregularity. Using asymptotic (with respect to the parameter kr ≫1) integration along the direction perpendicular to the ray path, we transform this system to a system of one-dimensional integral equations. The system is solved numerically in the diagonal approximation, combining direct inversion of the Volterra integral operator and the subsequent iterations. The proposed method reduces the computer time required for solving the problem and is useful for the study of both small-scale and large-scale irregularities. We obtained estimates of the TE field components that are not excited by the source considered and originate entirely from field scattering by a three-dimensional irregularity disturbing the geometric regularity of the ionospheric waveguide wall.

Integral equation for stress concentration at the edge of a plane crack of arbitrary contour

Equations are derived for stress concentration near a crack of closed contour lying in a plane. A system of one-dimensional integral equations for the concentration factor is obtained. The right sides of the equations contain the initial approximation—a solution of the problem of a circular crack whose sides are acted upon by nonaxisymmetric loading.

Quantum lorentz gas: Effective equations and spectral analysis

We consider the quantum Lorentz gas (QLG) in the plane. We show that the spectral properties of the quantum Lorentz gas can be obtained from the study of a homogeneous system of one-dimensional integral equations. The qualitative spectral analysis is performed, and the spectrum is shown to have a band structure. The wave functions in the complete configuration space can be constructed in terms of the solutions of the obtained effective equations. We apply the obtained result to a simpler Lorentz gas model including additional symmetry.

Numerical solution of Helmholtz equation for an open boundary in space

The two-dimensional Fredholm integral equation of the first kind for the solution of Helmholtz equation for an open-boundary surface with rotational symmetry is reduced to a system of one-dimensional integral equations of the first kind. The obtained system is solved using finite elements. Furthermore the singularities of the unknown function and the kernel are treated analytically.

A numerical method for analyzing the quasistationary field above a stratified medium with an axisymmetric elevation

We consider the quasistationary electromagnetic field above a plane-parallel stratified medium with an axisymmetric elevation. The three-dimensional vector problem is reduced to scalar problems in the azimuthal halfplane for the first two field harmonics. A system of one-dimensional integral equations of the first kind is obtained.

A solution method for diffraction problems of electromagnetic waves on bodies of revolution in a stratified medium

We consider the diffraction of electromagnetic waves on bodies of revolution located in stratified media with arbitrary excitation. The problem is reduced to boundary-value problems on the azimuthal halfplane for systems of elliptical differential equations. A system of one-dimensional integral equations of the first kind is obtained.

Bound-state solutions of the Bethe-Salpeter equation in momentum space

The Bethe-Salpeter equation is investigated in the ladder approximation for a Yukawa-type coupling of three scalar fields of different masses. By means of an expansion in terms of Gegenbauer polynomials in (Euclidean) momentum space a system of one-dimensional integral equations is obtained, which turns out to be very appropriate for numerical solution. The lowest eigenvalues for bound states of angular momentum zero are given for various mass ratios of the particles involved. The convergence of the numerical approximations is examined. The method can be applied also in the case, in which a superposition of particles with different masses is exchanged. As an example the inclusion of self-energy effects in the propagator of the exchanged particle is considered for the case of equal external masses.