Representation of a solution of the dirichlet problem in a planar cusp domain as a logarithmic single-layer potential

2010 ◽  
Vol 43 (3) ◽  
pp. 175-182
Author(s):  
S. V. Poborchii
Keyword(s):  
Dirichlet Problem ◽  
Single Layer ◽  
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.

True and Spurious Eigensolutions of an Elliptical Membrane by Using the Nondimensional Dynamic Influence Function Method

2014 ◽  
Vol 136 (2) ◽  
Author(s):  
Jeng-Tzong Chen ◽  
Jia-Wei Lee ◽  
Ying-Te Lee ◽  
Wen-Che Lee
Keyword(s):  
Dirichlet Problem ◽  
Influence Function ◽  
Single Layer ◽  
Boundary Integral ◽  
Double Layer Potential ◽  
Single Layer Potential ◽  
Layer Potential ◽  
Potential Formulation ◽  
The Neumann Problem ◽  
Spurious Eigenvalues

In this paper, we employ the nondimensional dynamic influence function (NDIF) method to solve the free vibration problem of an elliptical membrane. It is found that the spurious eigensolutions appear in the Dirichlet problem by using the double-layer potential approach. Besides, the spurious eigensolutions also occur in the Neumann problem if the single-layer potential approach is utilized. Owing to the appearance of spurious eigensolutions accompanied with true eigensolutions, singular value decomposition (SVD) updating techniques are employed to extract out true and spurious eigenvalues. Since the circulant property in the discrete system is broken, the analytical prediction for the spurious solution is achieved by using the indirect boundary integral formulation. To analytically study the eigenproblems containing the elliptical boundaries, the fundamental solution is expanded into a degenerate kernel by using the elliptical coordinates and the unknown coefficients are expanded by using the eigenfunction expansion. True and spurious eigenvalues are simultaneously found to be the zeros of the modified Mathieu functions of the first kind for the Dirichlet problem when using the single-layer potential formulation, while both true and spurious eigenvalues appear to be the zeros of the derivative of modified Mathieu function for the Neumann problem by using the double-layer potential formulation. By choosing only the imaginary-part kernel in the indirect boundary integral equation method (BIEM) to solve the eigenproblem of an elliptical membrane, spurious eigensolutions also appear at the same position with those of NDIF since boundary distribution can be lumped. The NDIF method can be seen as a special case of the indirect BIEM by lumping the boundary distribution. Both the analytical study and the numerical experiments match well with the same true and spurious solutions.

scholarly journals The Dirichlet Problem for the EquationΔu−k2u=0in the Exterior of Nonclosed Lipschitz Surfaces

2013 ◽  
Vol 2013 ◽  
pp. 1-4
Author(s):  
P. A. Krutitskii
Keyword(s):  
Integral Equation ◽  
Weak Solution ◽  
Dirichlet Problem ◽  
Integral Representation ◽  
Laplace Equation ◽  
Existence And Uniqueness ◽  
Single Layer ◽  
Single Layer Potential ◽  
Layer Potential

We study the Dirichlet problem for the equationΔu−k2u=0in the exterior of nonclosed Lipschitz surfaces inR3. The Dirichlet problem for the Laplace equation is a particular case of our problem. Theorems on existence and uniqueness of a weak solution of the problem are proved. The integral representation for a solution is obtained in the form of single-layer potential. The density in the potential is defined as a solution of the operator (integral) equation, which is uniquely solvable.

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

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.

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.

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