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

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.

Download Full-text

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

Fractional Calculus and Applied Analysis ◽

10.2478/s13540-012-0014-3 ◽

2012 ◽

Vol 15 (2) ◽

Cited By ~ 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.

Download Full-text

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

International Journal of Mathematics and Mathematical Sciences ◽

10.1155/ijmms/2006/91983 ◽

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.

Download Full-text

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

International Journal of Mathematics and Mathematical Sciences ◽

10.1155/2013/302628 ◽

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.

Download Full-text

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

Vestnik St Petersburg University Mathematics ◽

10.3103/s1063454110030088 ◽

2010 ◽

Vol 43 (3) ◽

pp. 175-182

Author(s):

S. V. Poborchii

Keyword(s):

Dirichlet Problem ◽

Single Layer ◽

Single Layer Potential ◽

Layer Potential

Download Full-text

ON THE TRUE AND SPURIOUS EIGENVALUES BY USING THE REAL OR THE IMAGINARY-PART OF THE METHOD OF FUNDAMENTAL SOLUTIONS

International Journal of Computational Methods ◽

10.1142/s021987621341003x ◽

2013 ◽

Vol 10 (02) ◽

pp. 1341003 ◽

Cited By ~ 3

Author(s):

I. L. CHEN ◽

Y. T. LEE ◽

P. S. KUO ◽

J. T. CHEN

Keyword(s):

Double Layer ◽

Imaginary Part ◽

Single Layer ◽

Fundamental Solutions ◽

Method Of Fundamental Solutions ◽

Double Layer Potential ◽

Layer Potential ◽

The Common ◽

Value Decomposition ◽

Spurious Eigenvalues

In this paper, the method of fundamental solutions (MFS) of real-part or imaginary-part kernels is employed to solve two-dimensional eigenproblems. The occurring mechanism of spurious eigenvalues for circular and elliptical membranes is examined. It is found that the spurious eigensolution using the MFS depends on the location of the fictitious boundary where the sources are distributed. By employing the singular value decomposition technique, the common left unitary vectors of the true eigenvalue for the single- and double-layer potential approaches are found while the common right unitary vectors of the spurious eigenvalue are obtained. Dirichlet and Neumann eigenproblems are both considered. True eigenvalues are dependent on the boundary condition while spurious eigenvalues are different in the different approach, single-layer or double-layer potential MFS. Two examples of circular and elliptical membranes are numerically demonstrated to see the validity of the present method and the results are compared well with the theoretical prediction.

Download Full-text

Modified Representations for the Close Evaluation Problem

Mathematical and Computational Applications ◽

10.3390/mca26040069 ◽

2021 ◽

Vol 26 (4) ◽

pp. 69

Author(s):

Camille Carvalho

Keyword(s):

Single Layer ◽

Linear Partial Differential Equation ◽

Representation Formula ◽

Boundary Integral ◽

Three Dimensions ◽

Layer Potentials ◽

Single Layer Potential ◽

Layer Potential ◽

Integral Equation Methods ◽

Evaluation Problem

When using boundary integral equation methods, we represent solutions of a linear partial differential equation as layer potentials. It is well-known that the approximation of layer potentials using quadrature rules suffer from poor resolution when evaluated closed to (but not on) the boundary. To address this challenge, we provide modified representations of the problem’s solution. Similar to Gauss’s law used to modify Laplace’s double-layer potential, we use modified representations of Laplace’s single-layer potential and Helmholtz layer potentials that avoid the close evaluation problem. Some techniques have been developed in the context of the representation formula or using interpolation techniques. We provide alternative modified representations of the layer potentials directly (or when only one density is at stake). Several numerical examples illustrate the efficiency of the technique in two and three dimensions.

Download Full-text

A Numerical Study for the Dirichlet Problem of the Helmholtz Equation

Mathematics ◽

10.3390/math9161953 ◽

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.

Download Full-text