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.

scholarly journals Influence of viscosity on the scattering of an air pressure wave by a rigid body: a regular boundary integral formulation

2008 ◽  
Vol 464 (2097) ◽  
pp. 2303-2320 ◽  
Author(s):  
Dorel Homentcovschi
Keyword(s):  
Integral Equation ◽  
Rigid Body ◽  
Stress Tensor ◽  
Double Layer ◽  
Pressure Wave ◽  
Direct Proof ◽  
Single Layer ◽  
Boundary Integral ◽  
Fredholm Alternative ◽  
Layer Potential

This paper gives a regular vector boundary integral equation for solving the problem of viscous scattering of a pressure wave by a rigid body. Firstly, single-layer viscous potentials and a generalized stress tensor are introduced. Correspondingly, generalized viscous double-layer potentials are defined. By representing the scattered field as a combination of a single-layer viscous potential and a generalized viscous double-layer potential, the problem is reduced to the solution of a vectorial Fredholm integral equation of the second kind. Generally, the vector integral equation is singular. However, there is a particular stress tensor, called pseudostress, which yields a regular integral equation. In this case, the Fredholm alternative applies and permits a direct proof of the existence and uniqueness of the solution. The results presented here provide the foundation for a numerical solution procedure.

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.

scholarly journals A Numerical Method for Filtering the Noise in the Heat Conduction Problem

Mathematics ◽  
2019 ◽  
Vol 7 (6) ◽  
pp. 502
Author(s):  
Yao Sun ◽  
Xiaoliang Wei ◽  
Zibo Zhuang ◽  
Tian Luan
Keyword(s):  
Boundary Condition ◽  
Integral Equation ◽  
Heat Conduction ◽  
Numerical Method ◽  
Density Function ◽  
Heat Conduction Problem ◽  
Boundary Integral ◽  
Jump Discontinuity ◽  
Tikhonov Regularization Method ◽  
Layer Potential

In this paper, we give an effective numerical method for the heat conduction problem connected with the Laplace equation. Through the use of a single-layer potential approach to the solution, we get the boundary integral equation about the density function. In order to deal with the weakly singular kernel of the integral equation, we give the projection method to deal with this part, i.e., using the Lagrange trigonometric polynomials basis to give an approximation of the density function. Although the problems under investigation are well-posed, herein the Tikhonov regularization method is not used to regularize the aforementioned direct problem with noisy data, but to filter out the noise in the corresponding perturbed data. Finally, the effectiveness of the proposed method is demonstrated using a few examples, including a boundary condition with a jump discontinuity and a boundary condition with a corner. Whilst a comparative study with the method of fundamental solutions (MFS) is also given.

On modified Green functions in exterior problems for the Helmholtz equation

1982 ◽  
Vol 383 (1785) ◽  
pp. 313-332 ◽  
Keyword(s):  
Integral Equation ◽  
Green Function ◽  
Helmholtz Equation ◽  
Least Squares ◽  
Present Note ◽  
Boundary Integral ◽  
Green Functions ◽  
Exterior Problems ◽  
The Green Function ◽  
Definition Of

The question of non-uniqueness in boundary integral equation formu­lations of exterior problems for the Helmholtz equation has recently been resolved with the use of additional radiating multipoles in the definition of the Green function. The present note shows how this modification may be included in a rigorous formalism and presents an explicit choice of co­efficients of the added terms that is optimal in the sense of minimizing the least-squares difference between the modified and exact Green functions.

Numerical Study on the Growth and Collapse of a Cavitation Bubble inside a Rigid Cylinder with a Compliant Coating

2014 ◽  
Vol 875-877 ◽  
pp. 1194-1198
Author(s):  
Fardin Rouzbahani ◽  
M.T. Shervani-Tabar
Keyword(s):  
Integral Equation ◽  
Boundary Integral Equation ◽  
Cavitation Bubble ◽  
Numerical Study ◽  
Finite Difference Methods ◽  
Life Time ◽  
Integral Equation Method ◽  
Boundary Integral ◽  
Compliant Coating ◽  
Rigid Cylinder

In this paper, growth and collapse of a cavitation bubble inside a rigid cylinder with a compliant coating (a model of humans vessels) is studied using Boundary Integral Equation and Finite Difference Methods. The fluid flow is treated as a potential flow and Boundary Integral Equation Method is used to solve Laplaces equation for velocity potential. The compliant coating is modeled as a membrane with a spring foundation. The effects of the parameters describing the flow and the parameters describing the compliant coating on the interaction between the fluid and the cylindrical compliant coating are shown throughout the numerical results. It is shown that by increasing the compliancy of the coating, the bubble life time is decreased and the mass per unit area has an important role in bubble behavior.

scholarly journals An integral equation–based numerical method for the forced heat equation on complex domains

2020 ◽  
Vol 46 (5) ◽  
Author(s):  
Fredrik Fryklund ◽  
Mary Catherine A. Kropinski ◽  
Anna-Karin Tornberg
Keyword(s):  
Integral Equation ◽  
Heat Equation ◽  
Helmholtz Equation ◽  
Singular Integrals ◽  
Boundary Integral ◽  
Boundary Integral Method ◽  
Elliptic Pdes ◽  
Time Step ◽  
Modified Helmholtz Equation ◽  
High Regularity

Abstract Integral equation–based numerical methods are directly applicable to homogeneous elliptic PDEs and offer the ability to solve these with high accuracy and speed on complex domains. In this paper, such a method is extended to the heat equation with inhomogeneous source terms. First, the heat equation is discretised in time, then in each time step we solve a sequence of so-called modified Helmholtz equations with a parameter depending on the time step size. The modified Helmholtz equation is then split into two: a homogeneous part solved with a boundary integral method and a particular part, where the solution is obtained by evaluating a volume potential over the inhomogeneous source term over a simple domain. In this work, we introduce two components which are critical for the success of this approach: a method to efficiently compute a high-regularity extension of a function outside the domain where it is defined, and a special quadrature method to accurately evaluate singular and nearly singular integrals in the integral formulation of the modified Helmholtz equation for all time step sizes.

Integral Equation Approach for Computing Green’s Function on Doubly Connected Regions via the Generalized Neumann Kernel

2014 ◽  
Vol 71 (1) ◽  
Author(s):  
Siti Zulaiha Aspon ◽  
Ali Hassan Mohamed Murid ◽  
Mohamed M. S. Nasser ◽  
Hamisan Rahmat
Keyword(s):  
Integral Equation ◽  
Dirichlet Problem ◽  
Linear System ◽  
Green’S Function ◽  
Green's Function ◽  
Boundary Integral ◽  
Generalized Neumann Kernel ◽  
Integral Equation Approach ◽  
Gauss Elimination ◽  
Neumann Kernel

This research is about computing the Green’s function on doubly connected regions by using the method of boundary integral equation. The method depends on solving a Dirichlet problem. The Dirichlet problem is then solved using a uniquely solvable Fredholm integral equation on the boundary of the region. The kernel of this integral equation is the generalized Neumann kernel. The method for solving this integral equation is by using the Nystrӧm method with trapezoidal rule to discretize it to a linear system. The linear system is then solved by the Gauss elimination method. Mathematica plots of Green’s functions for several test regions are also presented.

scholarly journals The Dirichlet Problem for Second-Order Divergence Form Elliptic Operators with Variable Coefficients: The Simple Layer Potential Ansatz

2015 ◽  
Vol 2015 ◽  
pp. 1-11 ◽  
Author(s):  
Alberto Cialdea ◽  
Vita Leonessa ◽  
Angelica Malaspina
Keyword(s):  
Dirichlet Problem ◽  
Differential Operators ◽  
Differential Forms ◽  
Variable Coefficients ◽  
Divergence Form ◽  
Boundary Integral ◽  
Second Order ◽  
Boundary Integral Method ◽  
Layer Potential ◽  
Partial Differential Operators

We investigate the Dirichlet problem related to linear elliptic second-order partial differential operators with smooth coefficients in divergence form in bounded connected domains ofRm(m≥3) with Lyapunov boundary. In particular, we show how to represent the solution in terms of a simple layer potential. We use an indirect boundary integral method hinging on the theory of reducible operators and the theory of differential forms.

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