We model pacemaker effects of an algebraically localized heterogeneity in a 1 dimensional array of oscillators with nonlocal coupling. We assume the oscillators obey simple phase dynamics and that the array is large enough so that it can be approximated by a continuous nonlocal evolution equation. We concentrate on the case of heterogeneities with positive average and show that steady solutions...

We study integrability of a system of nonlinear partial differential equations consisting of the nonlinear d’Alembert equation 2u = F (u) and nonlinear eikonal equation uxμuxμ = G(u) in the complex Minkowski space R(1, 3). A method suggested makes it possible to establish necessary and sufficient compatibility conditions and construct a general solution of the d’Alembert-eikonal system for all ...

We investigate how a catastrophic event (modeled as a temporary fall of the reproduction rate) increases the extinction probability of an isolated self-regulated stochastic population. Using a variant of the Verhulst logistic model as an example, we combine the probability generating function technique with an eikonal approximation to evaluate the exponentially large increase in the extinction ...

Fast Marching represents a very efficient technique for solving front propagation problems, which can be formulated as partial differential equations with Dirichlet boundary conditions, called Eikonal equation:

We present a method of ray tracing that is based on a system of differential equations equivalent to the eikonal equation, but formulated in the ray coordinate system. We use a first-order discretization scheme that is interpreted very simply in terms of the Huygens’ principle. The method has proved to be a robust alternative to conventional ray tracing, while being faster and having a better a...

As Planck’s constant ~ (treated as a free parameter) tends to zero, the solution to the eikonal equation |∇S(X)| = f(X) can be increasingly closely approximated by the solution to the corresponding Schrödinger equation. When the forcing function f(X) is set to one, we get the Euclidean distance function problem. We show that the corresponding Schrödinger equation has a closed form solution whic...

The Eikonal equation is well established for modelling wave propagation in the field of cardiac electrophysiology[?Franzone]. It offers highly tempting computation speeds in contrast to the bidomain or monodomain models. Consequently, developing effective algorithmic solution techniques to the Eikonal equation is an immediate and valuable task. In this paper, we build on the methods existing in...

Traveltime calculations amount to solving the nonlinear eikonal equation for a given source location. We analyze the relationship between the eikonal solution and its perturbations with respect to the source location and develop a partial differential equation that relates the traveltime field for one source location to that for a nearby source. This linear first-order equation in one form depe...

We apply the level-set methodology to compute multivalued solutions of the paraxial eikonal equation in both isotropic and anisotropic metrics. This paraxial equation is obtained from 2D stationary eikonal equations by using one of the spatial directions as the artificial evolution direction. The advection velocity field used to move level sets is obtained by the method of characteristics; ther...

A new concept called a boat-sail distance is introduced on the surface of water with flow, and it is used to define a generalized Voronoi diagram, in such a way that the water surface is partitioned into regions belonging to the nearest harbors with respect to this distance. The problem of computing this Voronoi diagram is reduced to a boundary value problem of a partial differential equation, ...