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We present a fast algorithm for the evaluation of the exact nonreflecting boundary conditions for the SchrSdinger equation in one dimension. The exact nonrefleeting boundary condition contains a nonloeal term which is a convolution integral in time, with a kernel proportional to 1/v~. The key observation is that this integral can be split into two parts: a(More)
A second kind integral equation formulation is presented for the Dirichlet problem for the Laplace equation in two dimensions, with the boundary conditions specified on a collection of open curves. The performance of the obtained apparatus is illustrated with several numerical examples. The formulation is a simplification of the equation previously(More)
A detailed analysis is presented of all pseudo-differential operators of orders up to 2 encountered in classical potential theory in two dimensions. Each of the operators under investigation turns out to be a sum of one or more of standard operators (second derivative, derivative of the Hilbert transform, etc.), and an integral operator with smooth kernel.(More)
We present an integral equation formulation for the unsteady Stokes equations in two dimensions. This problem is of interest in its own right, as a model for slow viscous flow, but perhaps more importantly, as an ingredient in the solution of the full, incompressible Navier-Stokes equations. Using the unsteady Green's function, the velocity evolves(More)
We present a fast and accurate algorithm for the evaluation of nonlocal (long-range) Coulomb and dipole-dipole interactions in free space. The governing potential is simply the convolution of an interaction kernel U (x) and a density function ρ(x) = |ψ(x)| 2 for some complex-valued wave function ψ(x), permitting the formal use of Fourier methods. These are(More)
We present a fast algorithm for the evaluation of exact nonreflecting boundary conditions for the time-dependent Schrödinger equation in two dimensions on the unit circle. After separation of variables, the exact outgoing condition for each Fourier mode contains a nonlocal term which is a convolution integral in time. The kernel for that convolution is the(More)