Abstract This expository paper is concerned with the direct integral formulations for boundary value problems of the Helmholtz equation. We discuss unique solvability for the corresponding boundary integral equations and its relations to the interior eigenvalue value problems of the Laplacian. Based on the integral representations, we study the asymptotic behaviors of the solutions to the bound...

Harmonic functions u : R → R are equivalent to integral manifolds of an exterior differential system with independence condition (M, I, ω). To this system one associates the space of conservation laws C. They provide necessary conditions for g : Sn−1 →M to be the boundary of an integral submanifold. We show that in a local sense these conditions are also sufficient to guarantee the existence of...

The payoffs of the barrier options depend on the time path of the underlying price as opposed to just the price at expiry. It implies that both the boundary conditions and the initial condition are imposed on the Black-Scholes partial differential equation. Therefore, the valuation of the barrier options is a boundary value problem. Since the Black-Scholes equation can be converted into a homog...

This article reviews several fast algorithms for boundary integral equations. After a brief introduction of the boundary integral equations for the Laplace and Helmholtz equations, we discuss in order the fast multipole method and its kernel independent variant, the hierarchical matrix framework, the wavelet based method, the high frequency fast multipole method, and the recently proposed multi...

We study boundary quantum Knizhnik-Zamolodchikov equations associated with quantum affine symmetry Uq(ŝln). We consider the case where the deformation parameter q satisfies |q| = 1, and the difference parameter λ > 0. We give an integral representation of boundary quantum Knizhnik-Zamolodchikov equations. Our approach is based on the free field realizations of the vertex operators and the bound...

We study the coercivity properties and the norm dependence on the wavenumber k of certain regularized combined field boundary integral operators that we recently introduced for the solution of two and three-dimensional acoustic scattering problems with Neumann boundary conditions. We show that in the case of circular and spherical boundaries, our regularized combined field boundary integral ope...

Keywords: Rotating Navier–Stokes equations Exterior domain Boundary integral method Finite element approximation a b s t r a c t In this paper, we apply the boundary integral method to the linearized rotating Navier– Stokes equations in exterior domain. Introducing some open ball which decomposes the exterior domain into a finite domain and an infinite domain, we obtain a coupled problem by the...

We study the short time behavior of the early exercise boundary for American style put options in the Black–Scholes theory. We develop an asymptotic expansion which shows that the simple lower bound of Barles et al. is a more accurate approximation to the actual boundary than the more complex upper bound. Our expansion is obtained through iteration using a boundary integral equation. This integ...

This paper deals with the second order boundary value problem with integral boundary conditions on a half-line: (p(t)x′(t))′ + g(t)f(t, x(t)) = 0, a.e. in (0,∞), x(0) = ∫ ∞ 0 x(s)g(s)ds, lim t→∞ p(t)x′(t) = p(0)x′(0). A new result on the existence of positive solutions is obtained. The interesting points are: firstly, the boundary value problem involved in the integral boundary condition on unb...

Abstract: Numerically solving the Poisson-Boltzmann equation is a challenging task due to the existence of the dielectric interface, singular partial charges representing the biomolecule, discontinuity of the electrostatic field, infinite simulation domains, etc. Boundary integral formulation of the Poisson-Boltzmann equation can circumvent these numerical challenges and meanwhile conveniently ...