We study a two-species reaction-diffusion problem described by a system consisting of a semilinear parabolic equation and a first order ordinary differential equation, endowed with suitable conditions. We prove the existing of a unique traveling wave profile and give necessary conditions and sufficient conditions for the occurrence of penetration and conversion fronts.

This note is concerned with the damped wave equation ε∂ttu+ ∂tu− Δu+ f (u) = g depending on a small parameter ε and with the corresponding parabolic equation ∂tu− Δu+ f (u) = g obtained in the singular limit ε → 0. The existence of a family Mε of exponential attractors which is Hölder continuous with respect to ε is proved.

The time-dependent form of Tappert’s range refraction parabolic equation is derived and proposed as an artificial boundary condition for the wave equation in a waveguide. The numerical comparison with Higdon’s absorbing boundary conditions shows sufficiently good quality of the new boundary condition at low computational cost.

Our aim in this article is to construct exponential attractors for singularly perturbed damped wave equations that are continuous with respect to the perturbation parameter. The main difficulty comes from the fact that the phase spaces for the perturbed and unperturbed equations are not the same; indeed, the limit equation is a (parabolic) reaction-diffusion equation. Therefore, previous constr...

The Parabolic Equation Method ( P E M ) is the commonly used method to deal with wave propagation in random media. This method basically requires the small ratio of wave length to correlation length (small angle scattering). Recently, de Wolf/2/ and Rino/3/ extended the PEM by including the backscattered wave, retaining the small angle scattering. In his dissertation, GroBe/4/ generalized the P...

We develop an efficient algorithm for simulating wave propagation over long distances with both weak and strong scatterers. In domains with weak heterogeneities the wave field is decomposed into forward propagating and back scattered modes using two coupled parabolic equations. In the region near strong scatterers, the Helmholtz equation is used to capture the strong scattering events. The key ...

Numerical methods are presented for simulating stochastic-in-time current pulses for an electrodiffusion model of the biological channel, with a fixed applied voltage across the channel. The electrodiffusion model consists of the parabolic advection-diffusion equation coupled either to Gauss' law or Poisson's equation, depending on the choice of boundary conditions. The TRBDF2 method is employe...