An alternating boundary integral based method for a Cauchy problem for the Laplace equation in semi-infinite regions

We consider a Cauchy problem for the Laplace equation in a two-dimensional semi-infinite region with a bounded inclusion, i.e. the region is the intersection between a half-plane and the exterior of a bounded closed curve contained in the half-plane. The Cauchy data are given on the unbounded part of the boundary of the region and the aim is to construct the solution on the boundary of the inclusion. In 1989, Kozlov and Maz'ya [10] proposed an alternating iterative method for solving Cauchy problems for general strongly elliptic and formally self-adjoint systems in bounded domains. We extend their approach to our setting and in each iteration step mixed boundary value problems for the Laplace equation in the semi-infinite region are solved. Well-posedness of these mixed problems are investigated and convergence of the alternating procedure is examined. For the numerical implementation an efficient boundary integral equation method is proposed, based on the indirect variant of the boundary integral equation approach. The mixed problems are reduced to integral equations over the (bounded) boundary of the inclusion. Numerical examples are included showing the feasibility of the proposed method.

An alternating potential based approach for a Cauchy problem for the Laplace equation in a planar domain with a cut

We consider a Cauchy problem for the Laplace equation in a bounded region containing a cut, where the region is formed by removing a sufficiently smooth arc (the cut) from a bounded simply connected domain D. The aim is to reconstruct the solution on the cut from the values of the solution and its normal derivative on the boundary of the domain D. We propose an alternating iterative method which involves solving direct mixed problems for the Laplace operator in the same region. These mixed problems have either a Dirichlet or a Neumann boundary condition imposed on the cut and are solved by a potential approach. Each of these mixed problems is reduced to a system of integral equations of the first kind with logarithmic and hypersingular kernels and at most a square root singularity in the densities at the endpoints of the cut. The full discretization of the direct problems is realized by a trigonometric quadrature method which has super-algebraic convergence. The numerical examples presented illustrate the feasibility of the proposed method.

Simulation in the supply chain context:matching the simulation tool to the problem

The supply chain can be a source of competitive advantage for the firm. Simulation is an effective tool for investigating supply chain problems. The three main simulation approaches in the supply chain context are System Dynamics (SD), Discrete Event Simulation (DES) and Agent Based Modelling (ABM). A sample from the literature suggests that whilst SD and ABM have been used to address strategic and planning problems, DES has mainly been used on planning and operational problems., A review of received wisdom suggests that historically, driven by custom and practice, certain simulation techniques have been focused on certain problem types. A theoretical review of the techniques, however, suggests that the scope of their application should be much wider and that supply chain practitioners could benefit from applying them in this broader way.

Exact solution of the Bragg-diffraction problem in sillenites

A method for the exact solution of the Bragg-difrraction problem for a photorefractive grating in sillenite crystals based on Pauli matrices is proposed. For the two main optical configurations explicit analytical expressions are found for the diffraction efficiency and the polarization of the scattered wave. The exact solution is applied to a detailed analysis of a number of particular cases. For the known limiting cases there is agreement with the published results.

Numerical solution of the Dirichlet initial boundary value problem for the heat equation in exterior 3-dimensional domains using integral equations

A numerical method for the Dirichlet initial boundary value problem for the heat equation in the exterior and unbounded region of a smooth closed simply connected 3-dimensional domain is proposed and investigated. This method is based on a combination of a Laguerre transformation with respect to the time variable and an integral equation approach in the spatial variables. Using the Laguerre transformation in time reduces the parabolic problem to a sequence of stationary elliptic problems which are solved by a boundary layer approach giving a sequence of boundary integral equations of the first kind to solve. Under the assumption that the boundary surface of the solution domain has a one-to-one mapping onto the unit sphere, these integral equations are transformed and rewritten over this sphere. The numerical discretisation and solution are obtained by a discrete projection method involving spherical harmonic functions. Numerical results are included.