Problem For Laplace Equation Research Articles

The boundary value problem for the two-dimensional Laplace equation with transmission conditions given on open Lipschitz curve

We consider boundary value problem for Laplace equation in bounded two-dimensional Lipschitz domain with transmission boundary condition given upon open curve. This conditions includes itself the jump of solution of boundary value problem and the meaning of boundary value of its normal derivative. We prove the equivalence of considered boundary value problem and obtained variational problem. As a result we prove existence and uniqueness of solution of the posed problems in appropriate functional spaces. Based on the integral representation of the solution the considered boundary value problem is reduced to the system of boundary integral equation which has unique solution.

Dirihlet-Ventcel bounsdary problem for Laplace equation in an unbounded sector

We are concerned with boundary value problems for Laplace equation in an unbounded sector $s_\theta$ with vertex at the origin, the boundary conditions being of mixed type and jumping at corner. The boundary conditions are these: Dirichlet datum on one of the radial lines, while on the other the values of an Ventcel boundary condition is prescribed. We are interested in looking for solutions having a prescribed degree of smoothness up to the origin: more precisely we search for solutions of problem having all the derivatives up to the order that are square integrable with a power weight. This problem has a background in physical modeling of electrostatic or thermal imaging. Determining the geometry and the physical nature of an corrosion within a conducting medium from voltage and current measurements on the accessible boundary of the medium can be modeled as an inverse boundary value problem for the Laplace equation subject to appropriate boundary conditions on the corrosion surface. We are interesting in investigation of a regularity properties of solution to the @direct@ problem. Applying Mellin transform we pass to a finite difference equation.We use the methods of V.A.Solonnikov and E.V.Frolova just as in the case of the analogous finite difference equation obtained under the Dirichlet or the Neumann conditions indstead of the Ventcel condition in our case. We obtain the sulution of homogeneous difference equation in the form of infinite product. Then we find asymptotic formulas for this solution.Returning to nonhomogeneous differerence equation we find its solution in the form of contour integral. we define the solution of the starting problem by the help of the inverse Mellin transform. We estimate this solution in the norm of V.Kondratiev spaces $H^k_\mu(s_\theta$ under some conditions on weight $\mu$, higher order of derivatives $k$ and the opening of the angle $\theta$.

ПОДБОР ОПТИМАЛЬНОЙ СТРУКТУРЫ RBF-СЕТИ ДЛЯ АППРОКСИМАЦИИ РЕШЕНИЯ ДВУМЕРНОГО УРАВНЕНИЯ ЛАПЛАСА

An algorithm for neural network approximation of the solution of a boundary value problem for the twodimensional Laplace equation on a square domain is proposed. An approximate solution of the Laplace equation is a radial basis neural network with one hidden layer (RBF network). Optimal parameters of the RBF network are obtained from the minimization problem of the quadratic functional for the Laplace equation. In our work we also proposed an algorithm for choosing the minimum number of neurons on the hidden layer of the RBFnetwork for a given accuracy of the approximate solution. Two problems for Laplace equation with different boundary conditions were considered. In the first problem for the Laplace equation, we set discontinuous boundary conditions in the corners of the square, and in the second problem we set regular boundary conditions with error.

Numerical Estimation of a Mode III Fracture Mechanics Parameter for a Three-Dimensional V-Notch on a Steel Bold

In this work a couple of three-dimensional problems in the domain of linear elastic Fracture Mechanics are examined. These are problems of solid bodies (they could be steel bolds or rivets) with a surface crack singularity (V-notch). They are reduced to Laplace equation problems by considering a Lamé potential. The boundary singularity is numerically treated as per the singular function boundary integral method (SFBIM), which in the literature is known as one of the so-called Trefftz methods. Thus, the general solution of the governing equation, in the vicinity of the surface crack, is expressed as an asymptotic expansion, the coefficients of which are approximated by polynomials. The remaining numerical steps are followed according to this method with which very fast convergence and very high accuracy are observed. In fact, the CPU time and the numerical error recorded with this numerical technique are significantly smaller than those achieved with the finite element method (FEM) which was also used to solve the same problems. The calculated value of Mode III Fracture Mechanics parameter (FMP) indicates that there is no danger of crack propagation. Thus, the extension of the method to this category of problems is considered as a novel application of this algorithm in Fracture Mechanics. Keywords: Mode III Fracture Mechanics parameter, crack singularity, governing equation, singular

Some Remarks on Solvability of Dirichlet Problem for Laplace Equation in Non-standard Function Spaces

Some Remarks on Solvability of Dirichlet Problem for Laplace Equation in Non-standard Function Spaces

Numerical estimation of a Mode III fracture Mechanics parameter for three-dimensional elliptic boundary value problems with a crack singularity

In this study two spatial problems of solid bodies with a surface crack singularity (V-notch) are examined. These are problems in the domain of linear elastic fracture Mechanics which are reduced to Laplace equation problems by considering a Lamé potential. The boundary singularity is numerically treated by first determining the general solution of the governing equation, in the vicinity of the surface crack and by expressing it as an asymptotic expansion, the coefficients of which are approximated by polynomials. The remaining numerical steps are followed according to the singular function boundary integral method (SFBIM), which in the literature is known as one of the so-called Trefftz methods. Very fast convergence and very high accuracy are observed in implementing the method to solve a general 3-D model problem of a solid body having a cross section in the form of an ellipse and a specific spatial model problem of a steel rivet, which connects metal members of a structure. In fact, the CPU time and the numerical error recorded with this numerical technique are significantly smaller than those achieved with the finite element method (FEM) which was used to solve the same problems. The calculated value of Mode III fracture Mechanics parameter (FMP) indicates that there is no danger of crack propagation. Thus, the extension of the method to this category of problems is quite interesting since it involves a novel application of this algorithm in fracture Mechanics, a domain in which the method demonstrates its capability to treat problems with crack discontinuities and provide results more efficiently (less computing effort and greater accuracy) than the classical FEM with grid refinement. This advantage of the SFBIM can be exploited for research or design purposes, in several fields of applied Mechanics, by implementing its algorithm with subroutines embedded in engineering packages and aiming to make them more efficient in solving specific problems.

Transmutation Operator Method for Solving Heat Conduction Problem

The transmutation operator method is extended to the case of functions of two variables. The transmutation operator flattens the function, i.e. the transmutation operator replaces a function with discontinuous partial derivatives on the coordinate axes by a continuously differentiable function. The work reveal the properties of the transmutation operator, and prove the commutativity of the transmutation operator and the Laplace operator. It was found that the Cauchy problem for the Laplace equation with internal conjugations in an unbounded domain can be replaced with the model Cauchy problem for the twodimensional Laplace equation. As a result, a new analytical method for solving initial-boundary value problems for a two-dimensional heat equation has been developed. The factorization of the transmutation operator is established as a product of two one-dimensional transmutation operators. The form of the transmutation operator establishing the isomorphism of two mathematical models of heat conduction in unbounded media with different physical characteristics was found and descrfibed.

Study of approximate solution to integral equation associated with mixed boundary value problem for Laplace equation

Study of approximate solution to integral equation associated with mixed boundary value problem for Laplace equation

Locating events in a segment of a segmented HPGe detector at KRISS

Abstract A specific segment of the 24 fold segmented HPGe detector was studied for improving the interaction position resolution from several cm to mm. The segmented HPGe detector was modeled by CAD program with their electrodes which was isolated by the Boolean function. The Shockley-Ramo theorem changed the Gauss law to Laplace equation problem for calculating the induced charge on each electrode. Finite Element Method solved the Poisson and Laplace equation with their boundary conditions for Electric field and potential. The drift velocities of charge carriers were compared between both along the 100 > and one between 100 > and 110 > Ge crystal axis at a specific segment and used to simulate the pulse shapes with 1 mm position resolution. Coincidence system between a specific segment and 3 × 3 HPGe detector verified the simulated pulse shapes by comparing these with the experimental ones which are generated at the specific target region with a fixed scattering angle. The experimental and simulated ones were χ 2 fitted to find the least χ 2 value and locate the interaction positions. The interaction positions of experimental pulse shapes were analyzed statistically after the rise time correction along 100 > and 110 > Ge crystal axis. The position resolution as FWHM on x, y, and z axis is the 3.12, 4.38 and 8.69 mm with the collimator size 3, 6, and 6 mm, respectively.

An optimal quasi solution for the Cauchy problem for Laplace equation in the framework of inverse ECG

The inverse ECG problem is set as a boundary data completion for the Laplace equation: at each time the potential is measured on the torso and its normal derivative is null. One aims at reconstructing the potential on the heart. A new regularization scheme is applied to obtain an optimal regularization strategy for the boundary data completion problem. We consider the ℝn+1domain Ω. The piecewise regular boundary of Ω is defined as the union∂Ω = Γ1∪ Γ0∪ Σ, where Γ1and Γ0are disjoint, regular, andn-dimensional surfaces. Cauchy boundary data is given in Γ0, and null Dirichlet data in Σ, while no data is given in Γ1. This scheme is based on two concepts: admissible output data for an ill-posed inverse problem, and the conditionally well-posed approach of an inverse problem. An admissible data is the Cauchy data in Γ0corresponding to an harmonic function inC2(Ω) ∩H1(Ω). The methodology roughly consists of first characterizing the admissible Cauchy data, then finding the minimum distance projection in theL2-norm from the measured Cauchy data to the subset of admissible data characterized by givena prioriinformation, and finally solving the Cauchy problem with the aforementioned projection instead of the original measurement.

On Cauchy problem for Laplace equation

On Cauchy problem for Laplace equation

THE SPECTRAL DECOMPOSITION OF CAUCHY PROBLEM’S SOLUTION FOR LAPLACE EQUATION

The spectral decomposition of Cauchy problem for Laplace equation is obtained in Krein space, and is made a regularization of a problem, using the resolvent of the corresponding operator.

Determining surface heat flux for noncharacteristic Cauchy problem for Laplace equation

In this paper, the noncharacteristic Cauchy problem for the Laplace equation {wxx+wyy=0x∈(0,1),y∈R,w(0,y)=g(y)y∈R,wx(0,y)=h(y)y∈R, is investigated, where the Cauchy data is given at x=0 and the heat flux is sought in the interval 0<x≤1. This problem is severely ill-posed: the solution (if it exists) does not depend continuously on the given data. A modified regularization method is used to solve this problem. Furthermore, some error estimates for the heat flux between the regularization solution and the exact solution are given. Finally, a numerical example shows that the proposed method works well.

Numerical solution of a Cauchy problem for Laplace equation in 3-dimensional domains by integral equations

A numerical method based on integral equations is proposed and investigated for the Cauchy problem for the Laplace equation in 3-dimensional smooth bounded doubly connected domains. To numerically reconstruct a harmonic function from knowledge of the function and its normal derivative on the outer of two closed boundary surfaces, the harmonic function is represented as a single-layer potential. Matching this representation against the given data, a system of boundary integral equations is obtained to be solved for two unknown densities. This system is rewritten over the unit sphere under the assumption that each of the two boundary surfaces can be mapped smoothly and one-to-one to the unit sphere. For the discretization of this system, Weinert’s method (PhD, Göttingen, 1990) is employed, which generates a Galerkin type procedure for the numerical solution, and the densities in the system of integral equations are expressed in terms of spherical harmonics. Tikhonov regularization is incorporated, and numerical results are included showing the efficiency of the proposed procedure.

The Singular Function Boundary Integral Method for the 2-D and 3-D Laplace Equation Problems in Mechanics, with a Boundary Singularity

In this study the Singular Function Boundary Integral Method (SFBIM) is implemented in the case of a planar elliptic boundary value problem in Mechanics, with a point boundary singularity. The method is also extended in the case of a typical problem of Solid Mechanics, concerning the Laplace equation problem in three dimensions, defined in a domain with a straight edge singularity on the surface boundary. In both the 2-D and 3-D cases, the general solution of the Laplace equation is approximated by the leading terms (which contain the singular functions) of the local asymptotic solution expansion. The singular functions are used to weight the governing equation in the Galerkin sense. For the 2-D Laplacian model problem of this study, which is defined over a domain with a re-entrant corner, the resulting discretized equations are reduced to boundary integrals by means of Green’s second identity. For the 3-D model problem of this work, the volume integrals of the discretized equations are reduced to surface integrals by implementing Gauss’ divergence theorem. The Dirichlet boundary conditions are then weakly enforced by means of Lagrange multipliers. The values of the latter are calculated together with the singular coefficients, in the 2-D case or the Edge Flux Intensity Functions (EFIFs), in the 3-D model problem, which appear in the local solution expansion. For the planar problem, the numerical results are favorably compared with the analytic solution. Especially for the extension of the method in three dimensions, the preliminary numerical results compare favorably with available post-processed finite element results.

Optimal Stable Approximation for the Cauchy Problem for Laplace Equation

Cauchy problem for Laplace equation in a strip is considered. The optimal error bounds between the exact solution and its regularized approximation are given, which depend on the noise level either in a Hölder continuous way or in a logarithmic continuous way. We also provide two special regularization methods, that is, the generalized Tikhonov regularization and the generalized singular value decomposition, which realize the optimal error bounds.

The necessary and sufficient condition for the existence and uniqueness of a system of Fredholm integral equations of the first kind

In this paper, we consider the existence and uniqueness of a system of Fredholm integral equations of the first kind, which arises from the Dirichlet problem for Laplace equation on 2D multiply-connected domains. We introduce a computable index γ N and prove that the necessary and sufficient condition for the existence and uniqueness of solution of this Fredholm integral equations of the first kind is γ N ≠ 0. The numerical examples demonstrate the validity of our theoretic results.

Решение смешанной краевой задачи для уравнения Лапласа в многомерном бесконечном слое

Решение смешанной краевой задачи для уравнения Лапласа в многомерном бесконечном слое

A New Version of Fast Convergent GMRES(m) Algorithm

In this paper, a kind of modified GMRES(m) algorithm is proposed and its convergence theory is well established. By properly varying the restart parameter m in the GMRES(m) algorithm, the computational efficiency and precision can be significantly improved. Because the parameter m can be very small, the storage and computations are greatly saved. In addition, the new algorithm is successfully used to solve a heat conduction problem and a Laplace equation problem. Numerical results show that it is not only highly effective and accurate, but also can avoid slow convergence or failure behaviour of the GMRES(m) algorithm. The superiorities will be much more remarkable for the solution of large scale problems. So it has extensive prospect in scientific computing and engineering applications.

On the use of an integral equation approach for the numerical solution of a Cauchy problem for Laplace equation in a doubly connected planar domain

We consider the Cauchy problem for the Laplace equation, i.e. the reconstruction of a harmonic function from knowledge of the value of the function and its normal derivative given on a part of the boundary of the solution domain. The solution domain considered is a bounded smooth doubly connected planar domain bounded by two simple disjoint closed curves. Since the Cauchy problem is ill-posed, as a regularizing method we generalize the novel direct integral equation approach in [1], originally proposed for a circular outer boundary curve, to a more general simply connected domain. The solution is represented as a sum of two single-layer potentials defined on each of the two boundary curves and in which both densities are unknown. To identify these densities, the representation is matched with the given Cauchy data to generate a system to solve for the densities. It is shown that the operator corresponding to this system is injective and has dense range, thus Tikhonov regularization is applied to solve it in a stable way. For the discretisation, the Nyström method is employed generating a linear system to solve, and via Tikhonov regularization a stable discrete approximation to these densities are obtained. Using these one can then find an approximation to the solution of the Cauchy problem. A numerical example is included and we compare with other regularizing methods as well (implemented via integral methods). These results show that the proposed direct method gives accurate reconstructions with little computational effort (the computational time is of order of seconds). Moreover, the obtained approximation can be used as an initial guess in more involved regularizing methods to further improve the accuracy.