On Solvability of Integral Equations for Harmonic Single Layer Potential on the Boundary of a Domain with Cusp

2009 ◽  
pp. 303-314
Author(s):  
Sergei V. Poborchi
Keyword(s):  
Integral Equations ◽  
Single Layer ◽  
Single Layer Potential ◽  
Layer Potential

The motion of particles in the Hele-Shaw cell

1994 ◽  
Vol 261 ◽  
pp. 199-222 ◽  
Author(s):  
C. Pozrikidis
Keyword(s):  
Integral Equations ◽  
Stokes Flow ◽  
Single Layer ◽  
Surface Force ◽  
Spherical Particles ◽  
Fredholm Integral Equations ◽  
Single Layer Potential ◽  
Layer Potential ◽  
Single Plane ◽  
Boundary Velocity

The force and torque on a particle that translates, rotates, or is held stationary in an incident flow within a channel with parallel-sided walls, are considered in the limit of Stokes flow. Assuming that the particle has an axisymmetric shape with axis perpendicular to the channel walls, the problem is formulated in terms of a boundary integral equation that is capable of describing arbitrary three-dimensional Stokes flow in an axisymmetric domain. The method involves: (a) representing the flow in terms of a single-layer potential that is defined over the physical boundaries of the flow as well as other external surfaces, (b) decomposing the polar cylindrical components of the velocity, boundary surface force, and single-layer potential in complex Fourier series, and (c) collecting same-order Fourier coefficients to obtain a system of one-dimensional Fredholm integral equations of the first kind for the coefficients of the surface force over the traces of the natural boundaries of the flow in an azimuthal plane. In the particular case where the polar cylindrical components of the boundary velocity exhibit a first harmonic dependence on the azimuthal angle, we obtain a reduced system of three real integral equations. A numerical method of solution that is based on a standard boundary element-collocation procedure is developed and tested. For channel flow, the effect of domain truncation on the nature of the far flow is investigated with reference to plane Hagen–Poiseuille flow past a cylindrical post. Numerical results are presented for the force and torque exerted on a family of oblate spheroids located above a single plane wall or within a parallel-sided channel. The effect of particle shape on the structure of the flow is illustrated, and some novel features of the motion are discussed. The numerical computations reveal the range of accuracy of previous asymptotic solutions for small or tightly fitting spherical particles.

Two-level methods for the single layer potential in ℝ3

Computing ◽  
1998 ◽  
Vol 60 (3) ◽  
pp. 243-266 ◽  
Author(s):  
P. Mund ◽  
E. P. Stephan ◽  
J. Weiße
Keyword(s):  
Single Layer ◽  
Single Layer Potential ◽  
Layer Potential

Quadrature Formula for the Direct Value of the Normal Derivative of the Single Layer Potential

2020 ◽  
Vol 56 (9) ◽  
pp. 1237-1255
Author(s):  
P. A. Krutitskii ◽  
I. O. Reznichenko ◽  
V. V. Kolybasova
Keyword(s):  
Quadrature Formula ◽  
Single Layer ◽  
Normal Derivative ◽  
Single Layer Potential ◽  
Layer Potential

Solvability of a Dirichlet problem for a time fractional diffusion-wave equation in Lipschitz domains

2012 ◽  
Vol 15 (2) ◽  
Author(s):  
Jukka Kemppainen
Keyword(s):  
Wave Equation ◽  
Dirichlet Problem ◽  
Original Problem ◽  
Single Layer ◽  
Fractional Diffusion ◽  
Lipschitz Domains ◽  
Single Layer Potential ◽  
Layer Potential ◽  
Diffusion Wave ◽  
Sesquilinear Form

AbstractThis paper investigates a Dirichlet problem for a time fractional diffusion-wave equation (TFDWE) in Lipschitz domains. Since (TFDWE) is a reasonable interpolation of the heat equation and the wave equation, it is natural trying to adopt the techniques developed for solving the aforementioned problems. This paper continues the work done by the author for a time fractional diffusion equation in the subdiffusive case, i.e. the order of the time differentiation is 0 < α < 1. However, when compared to the subdiffusive case, the operator ∂ tα in (TFDWE) is no longer positive. Therefore we follow the approach applied to the hyperbolic counterpart for showing the existence and uniqueness of the solution.We use the Laplace transform to obtain an equivalent problem on the space-Laplace domain. Use of the jump relations for the single layer potential with density in H −1/2(Γ) allows us to define a coercive and bounded sesquilinear form. The obtained variational form of the original problem has a unique solution, which implies that the original problem has a solution as well and the solution can be represented in terms of the single layer potential.

The BPX preconditioner for the single layer potential operator

1997 ◽  
Vol 67 (3-4) ◽  
pp. 327-340 ◽  
Author(s):  
Stefan A. Funken ◽  
Ernst P. Stephan
Keyword(s):  
Single Layer ◽  
Potential Operator ◽  
Single Layer Potential ◽  
Layer Potential

scholarly journals Some spectral properties of an integral operator in potential theory

1986 ◽  
Vol 29 (3) ◽  
pp. 405-411 ◽  
Author(s):  
John F. Ahner
Keyword(s):  
Potential Theory ◽  
Spectral Properties ◽  
Three Dimensional ◽  
Single Layer ◽  
Integral Operators ◽  
Normal Derivative ◽  
Potential Operator ◽  
Single Layer Potential ◽  
Layer Potential ◽  
Integral Equation Methods

In [7] Plemelj established some fundamental results in two- and three-dimensional potential theory about the eigenvalues of both the double layer potential operator and its adjoint, the normal derivative of the single layer potential operator. In [3] Blumenfeld and Mayer established some additional results concerning the eigenvalues of these integral operators in the case of ℝ2. The spectral properties established by Plemelj [7] and by Blumenfeld and Mayer [3] have had a profound effect in the area of integral equation methods in scattering and potential theory in both ℝ2 and ℝ3.

On Invertibility of Elastic Single-Layer Potential Operator

2004 ◽  
Vol 74 (2) ◽  
pp. 147-173 ◽  
Author(s):  
R. Vodička ◽  
V. Mantič
Keyword(s):  
Single Layer ◽  
Potential Operator ◽  
Single Layer Potential ◽  
Layer Potential

Computational Methods for the Detection of the Source of Acoustical Noise

2000 ◽  
Author(s):  
Thomas DeLillo ◽  
Victor Isakov ◽  
Nicolas Valdivia ◽  
Lianju Wang
Keyword(s):  
Inverse Problem ◽  
Computational Methods ◽  
Single Layer ◽  
Singular Value ◽  
Pressure Measurements ◽  
Single Layer Potential ◽  
Interior Region ◽  
Layer Potential ◽  
Representation Of Solutions ◽  
Value Decomposition

Abstract Computational methods for the inverse problem of detecting the source of acoustical noise in an interior region from pressure measurements in the nearfield are discussed. The methods are based on a single layer potential representation of solutions to the Helmholtz equation. Regularization is peformed using the singular value decomposition and the conjugate gradient method.

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