Problem For Laplace Equation Research Articles

Computational techniques in the first course on electromagnetism

It is widely acknowledged that the introductory engineering electromagnetics (EM) course is in trouble because its concepts and techniques are novel and difficult, the time allowed is insufficient, and the range of topics is great and expanding. The introduction of computer-numerical solutions and the advantages and disadvantages of various approaches are discussed. It is suggested that the use of spreadsheet capabilities is pedagogically advantageous and time-effective for solving Laplace's equation problems by the method of finite differences. >

On an exterior Laplace equation problem with Robin boundary condition

An exterior problem for Laplace's equation in R 3 with a Robin boundary condition is investigated. The particular form of the assumed boundary condition can be considered to be a perturbation of a Dirichlet boundary condition. The convergence of the solution to the Robin problem to that of the corresponding Dirichlet problem is proven for sufficiently smooth prescribed boundary functions. The proof uses an integral equation method combined with results dealing with singular perturbation problems. In addition, the Robin problem is reformulated as a new boundary integral equation, which is shown to be uniquely solvable. A constructive method is established for obtaining the solution to this integral equation. An example is supplied to illustrate the method given.

Solution of a Boundary-Value Problem associated with Diffraction of Water Waves by a Nearly Vertical Barrier

An exact solution is derived for a boundary-value problem for Laplace's equation which is a generalization of the one occurring in the course of solution of the problem of diffraction of surface water waves by a nearly vertical submerged barrier. The method of solution involves the use of complex function theory, the Schwarz reflection principle, and reduction to a system of two uncoupled Riemann-Hilbert problems. Known results, representing the reflection and transmission coefficients of the water wave problem involving a nearly vertical barrier, are derived in terms of the shape function.

On the treatment of corners in the boundary element method

A new technique for treating surface discontinuities within boundary element calculations is proposed. Multiple nodes are used to represent the geometry, and the necessary additional equations are obtained by differentiating the usual boundary element integral equation. In deriving expressions for the resulting singular integrals, it is found that constraints must be placed on the functional approximation at the discontinuity. An algorithm for incorporating these constraints is developed and numerical tests for an exactly solvable three-dimensional Laplace equation problem are presented.

The CVBEM for multiply connected domains using a linear trial function

The objective of this paper is to present a modelling technique that approximates boundary value problems of the Laplace equation over two-dimensional multiply connected regions. By using this method, two-dimensional Laplace equation problems can be solved by use of analytic functions. The flexibility of this technique is demonstrated on problems with multiply connected domains, dissimilar materials, and many types of boundary conditions that have previously been difficult to handle.

The convergence of a direct BEM for the plane mixed boundary value problem of the Laplacian

In this paper the convergence analysis of a direct boundary elecment method for the mixed boundary value problem for Laplace equation in a smooth plane domain is given. The method under consideration is based on the collocation solution by constant elements of the corresponding system of boundary integral equations. We prove the convergence of this method, provide asymptotic error estimates for the BEM-solution and give some numerical examples.

Some developments of the CVBEM-application to the mixed boundary value problem for the Laplace equation

The main developments of the CVBEM given in the paper consists in removing the need for approximating the boundary curve by a poligonal line and the introduction of the singular boundary elements for the case of re-entrant corners or collision points on the boundary. The paper gives also explicit formulae for solving the mixed Dirichlet-Neumann boundary value problem for two-dimensional Laplace equation by CVBEM. The Fortran program MIXED coded by this idea is applied to some test problems.

A useful potential solution V(x,y,z) for symmetric rectangular stepwise constant-potential boundaries at y = ±Y

Three-dimensional Laplace-equation problems are generally quite difficult. In a number of electron-optics devices, a rectangular structure of stepwise constant potentials V mn at y = ±Y form symmetric boundaries. It is shown that useful solutions of the potential V(x, y, z) in the interior of such devices can be developed systematically.

On a certain boundary problem for Laplace equation

On a certain boundary problem for Laplace equation