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.

scholarly journals The solution of the third problem for the Laplace equation on planar domains with smooth boundary and inside cracks and modified jump conditions on cracks

2006 ◽  
Vol 2006 ◽  
pp. 1-14
Author(s):  
Dagmar Medková
Keyword(s):  
Integral Equation ◽  
Laplace Equation ◽  
Smooth Boundary ◽  
Single Layer ◽  
Double Layer Potential ◽  
Jump Conditions ◽  
Single Layer Potential ◽  
Layer Potential ◽  
The Third ◽  
Planar Domains

This paper studies the third problem for the Laplace equation on a bounded planar domain with inside cracks. The third condition∂u/∂n+hu=fis given on the boundary of the domain. The skip of the functionu+−u−=gand the modified skip of the normal derivatives(∂u/∂n)+−(∂u/∂n)−+hu+=fare given on cracks. The solution is looked for in the form of the sum of a modified single-layer potential and a double-layer potential. The solution of the corresponding integral equation is constructed.

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.

ON THE THEORETICAL JUSTIFICATION OF POCKLINGTON'S EQUATION

2009 ◽  
Vol 19 (08) ◽  
pp. 1325-1355 ◽  
Author(s):  
XAVIER CLAEYS
Keyword(s):  
Integral Equation ◽  
Single Layer ◽  
Single Layer Potential ◽  
Theoretical Justification ◽  
Model Case ◽  
Layer Potential ◽  
Potential Equation ◽  
One Dimensional ◽  
Dimensional Integral ◽  
Dimensional Surface

Pocklington's model consists in a one-dimensional integral equation relating the current at the surface of a straight finite wire to the tangential trace of an incident electromagnetic field. It is a simplification of the more usual single layer potential equation posed on a two-dimensional surface. We are interested in estimating the error between the solution of the exact integral equation and the solution of Pocklington's model. We address this problem for the model case of acoustics in a smooth geometry using results of asymptotic analysis.

scholarly journals A Numerical Study for the Dirichlet Problem of the Helmholtz Equation

Mathematics ◽  
2021 ◽  
Vol 9 (16) ◽  
pp. 1953
Author(s):  
Yao Sun ◽  
Shijie Hao
Keyword(s):  
Integral Equation ◽  
Dirichlet Problem ◽  
Helmholtz Equation ◽  
Numerical Study ◽  
Single Layer ◽  
Boundary Integral ◽  
Tikhonov Regularization Method ◽  
Weakly Singular Kernel ◽  
Layer Potential ◽  
Value Decomposition

In this paper, an effective numerical method for the Dirichlet problem connected with the Helmholtz equation is proposed. We choose a single-layer potential approach to obtain the boundary integral equation with the density function, and then we deal with the weakly singular kernel of the integral equation via singular value decomposition and the Nystrom method. The direct problem with noisy data is solved using the Tikhonov regularization method, which is used to filter out the errors in the boundary condition data, although the problems under investigation are well-posed. Finally, a few examples are provided to demonstrate the effectiveness of the proposed method, including piecewise boundary curves with corners.

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

scholarly journals On a Nonlinear Wave Equation of Kirchhoff-Carrier Type: Linear Approximation and Asymptotic Expansion of Solution in a Small Parameter

2018 ◽  
Vol 2018 ◽  
pp. 1-18
Author(s):  
Nguyen Huu Nhan ◽  
Le Thi Phuong Ngoc ◽  
Nguyen Thanh Long
Keyword(s):  
Asymptotic Expansion ◽  
Wave Equation ◽  
Weak Solution ◽  
Dirichlet Problem ◽  
Small Parameter ◽  
Nonlinear Wave ◽  
Nonlinear Wave Equation ◽  
Existence And Uniqueness ◽  
Linearization Method ◽  
Asymptotic Expansion Of Solution

We consider the Robin-Dirichlet problem for a nonlinear wave equation of Kirchhoff-Carrier type. Using the Faedo-Galerkin method and the linearization method for nonlinear terms, the existence and uniqueness of a weak solution are proved. An asymptotic expansion of high order in a small parameter of a weak solution is also discussed.

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
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