Boundary Value Integral Equations Research Articles

Combination of the Laguerre Transform with the Boundary-element Method for the Solution of Integral Equations with Retarded Kernel

We apply the Laguerre transform with respect to time to a time-dependent boundary-value integral equation encountered in the solution of three-dimensional Dirichlet initial-boundary-value problems for the homogeneous wave equation with homogeneous initial conditions by using the retarded potential of single layer. The obtained system of boundary integral equations is reduced to a sequence of Fredholm integral equations of the first kind that differ solely by the recursively dependent right-hand sides. To find their numerical solution, we use the boundary-element method. We establish an asymptotic estimate of the error of numerical solution and present the results of numerical simulations aimed at finding the solutions of retarded-potential integral equations for model examples.

Analysis of Concentrated Colloidal Dispersions

A long-standing area of research in colloid science concerns the determination of electrostatic, magnetic, and elastostatic fields in dispersions comprising densely packed particles whose material properties differ markedly from that of the background. As new materials with complex microstructures are being developed, there is a corresponding need for analytic and numerical methods to predict and understand their properties. Dense systems with large numbers of close-to-touching particles require very fine discretization which causes the linear system of equations that arise to be highly ill-conditioned. A framework for solving the many-body interface problems is provided by a mathematical transformation of the boundary value integral equation. The transformation avoids the use of asymptotics and involves no uncontrolled approximations which distinguishes this work from previous methods. Pointwise analysis of local charge densities in the bulk and at interfaces is now possible.

A fast integral equation approach to acoustic scattering from three-dimensional objects using a natural basis set

Boundary value integral equations and the moment method solution technique provide a succinct and elegant strategy for the solution of acoustic surface scattering problems. This paper explores the application of a subdomain basis set, based on the phase of the incident radiation, which is used to solve for the unknown normal pressure gradient on the surface of the scatterer using the moment method. The advantage of this basis set lies in the fact that it can accurately represent the fast phase character of the pressure gradient on the surface of the scatterer. This allows for far fewer discretizations/basis elements in the moment method formulation and therefore significantly reduces the density of unknowns that must be numerically computed. The main contribution of this paper is to show how to efficiently employ this natural basis and to explore the physical justification for using this approach. The method is compared with other techniques including the Kirchhoff solution technique, high frequency single scattering techniques, the Nussenzveig–Fock technique, and the parabolic equation approximation. Unlike these methods no approximation to the physical system need be introduced.