The Use of the Singular Value Decomposition Method in the Synthesis of the Neumann's Boundary Problem

Abstract

If you would like to write for this, or any other Emerald publication, then please use our Emerald for Authors service information about how to choose which publication to write for and submission guidelines are available for all. Please visit www.emeraldinsight.com/authors for more information. About Emerald www.emeraldinsight.comEmerald is a global publisher linking research and practice to the benefit of society. The company manages a portfolio of more than 290 journals and over 2,350 books and book series volumes, as well as providing an extensive range of online products and additional customer resources and services.Emerald is both COUNTER 4 and TRANSFER compliant. The organization is a partner of the Committee on Publication Ethics (COPE) and also works with Portico and the LOCKSS initiative for digital archive preservation.ABSTRACT: This paper discusses an electrostatic, homogeneous field in a uniform two-dimensional domain with Neumann's boundary conditions. The boundary conditions are known only at some segments of the boundary. The synthesis is understood as the computation of the remaining boundary conditions which would ensure the required potential distribution in some subdomains within the boundary.The introduction of a single-layer potential leads to Fredholm's equation of the second order. Stepwise approximation of the source distribution along the boundary rearranges Fredholm's equation and the requirements concerning the single layer potential distribution. It leads to a matrix equation with a rectangular coefficient matrix.In order to solve approximately this equation, in the sense of the least squares minimization, the singular value decomposition (SVD) method is used. The choice of subdomains with determined potential distribution influences significantly the conditioning of the equation. Easy selection of an acceptable solution among all possible solutions proves the suitability of the SVD method in the above problem. The numerical experiments reported in the paper are a good illustration of this.