4 research outputs found
Stability of equilibrium solutions of a double power reaction diffusion equation with a Dirac interaction
In this paper we provide detailed information about the instability of
equilibrium solutions of a nonlinear family of localized reaction-difussion
equations in dimensione one. Beyond we provide explicit formulas to the
equilibrium solutions, via perturbation method and we calculate the exact
number of positive eigenvalues of the linear operator associated to the
stability problem, which allow us to compute the dimension of the unstable
manifold.Comment: 16 pages, 3 figure
A Nonlinear Heat Equation Arising from Automated-Vehicle Traffic Flow Models
In this paper, a new nonlinear heat equation is studied that arises as a
model of the collective behavior of automated vehicles. The properties of the
solutions of this equation are studied by introducing the appropriate notion of
a weak solution that requires certain entropy-like conditions. To obtain an
approximation of the solution of the nonlinear heat equation, a new
conservative first-order finite difference scheme is proposed that respects the
corresponding entropy conditions, and certain links between the weak solution
and the numerical scheme are provided. Finally, a traffic simulation scenario
and a comparison with the Lighthill-Witham-Richards (LWR) model are provided,
illustrating the benefits of the use of automated vehicles.Comment: 36 pages, 9 figure
Differentiable positive definite kernels on two-point homogeneous spaces
In this work we study continuous kernels on compact two-point homogeneous spaces which are positive definite and zonal (isotropic). Such kernels were characterized by R. Gangolli some forty years ago and are very useful for solving scattered data interpolation problems on the spaces. In the case the space is the d-dimensional unit sphere, J. Ziegel showed in 2013 that the radial part of a continuous positive definite and zonal kernel is continuously differentiable up to order ⌊(d−1)/2⌋ in the interior of its domain. The main issue here is to obtain a similar result for all the other compact two-point homogeneous spaces.CNPq (grant 141908/2015-7)FAPESP (grant 2014/00277-5