A Source Problem for the Helmholtz Equation via a Dirichlet-to-Neumann Map

In this paper, we consider a source problem for a time harmonic acoustic wave in two-dimensional space. Based on the boundary integral equation method, a Dirichlet-to-Neumann map in terms of boundary integral operators on the boundary of the source is constructed to transform this problem into two boundary value problems for the Helmholtz equation.

Self-Adjoint Dirac Operators on Domains in $$\mathbb {R}^3$$

In this paper, the spectral and scattering properties of a family of self-adjoint Dirac operators in $$L^2(\Omega ; \mathbb {C}^4)$$ L 2 ( Ω ; C 4 ) , where $$\Omega \subset \mathbb {R}^3$$ Ω ⊂ R 3 is either a bounded or an unbounded domain with a compact $$C^2$$ C 2 -smooth boundary, are studied in a systematic way. These operators can be viewed as the natural relativistic counterpart of Laplacians with boundary conditions as of Robin type. Our approach is based on abstract boundary triple techniques from extension theory of symmetric operators and a thorough study of certain classes of (boundary) integral operators, that appear in a Krein-type resolvent formula. The analysis of the perturbation term in this formula leads to a description of the spectrum and a Birman–Schwinger principle, a qualitative understanding of the scattering properties in the case that $$\Omega $$ Ω is an exterior domain, and corresponding trace formulas.

Multiple traces formulation and semi-implicit scheme for modelling biological cells under electrical stimulation

We model the electrical behavior of several biological cells under external stimuli by extending and computationally improving the multiple traces formulation introduced in Henríquez et al. [Numer. Math. 136 (2016) 101–145]. Therein, the electric potential and current for a single cell are retrieved through the coupling of boundary integral operators and non-linear ordinary differential systems of equations. Yet, the low-order discretization scheme presented becomes impractical when accounting for interactions among multiple cells. In this note, we consider multi-cellular systems and show existence and uniqueness of the resulting non-linear evolution problem in finite time. Our main tools are analytic semigroup theory along with mapping properties of boundary integral operators in Sobolev spaces. Thanks to the smoothness of cellular shapes, solutions are highly regular at a given time. Hence, spectral spatial discretization can be employed, thereby largely reducing the number of unknowns. Time-space coupling is achieved via a semi-implicit time-stepping scheme shown to be stable and second order convergent. Numerical results in two dimensions validate our claims and match observed biological behavior for the Hodgkin–Huxley dynamical model.