49,063 research outputs found
The Effect of Diffusion on the Particle Spectra in Pulsar Wind Nebulae
A possible way to calculate particle spectra as a function of position in
pulsar wind nebulae is to solve a Fokker-Planck transport equation. This paper
presents numerical solutions to the transport equation with the processes of
convection, diffusion, adiabatic losses, and synchrotron radiation included. In
the first part of the paper the steady-state version of the transport equation
is solved as a function of position and energy. This is done to distinguish the
various effects of the aforementioned processes on the solutions to the
transport equation. The second part of the paper deals with a time-dependent
solution to the transport equation, specifically taking into account the effect
of a moving outer boundary. The paper highlights the fact that diffusion can
play a significant role in reducing the amount of synchrotron losses, leading
to a modification in the expected particle spectra. These modified spectra can
explain the change in the photon index of the synchrotron emission as a
function of position. The solutions presented in this paper are not limited to
pulsar wind nebulae, but can be applied to any similar central source system,
e.g. globular clusters
Mathematical Modelling of Turning Delays in Swarm Robotics
We investigate the effect of turning delays on the behaviour of groups of
differential wheeled robots and show that the group-level behaviour can be
described by a transport equation with a suitably incorporated delay. The
results of our mathematical analysis are supported by numerical simulations and
experiments with e-puck robots. The experimental quantity we compare to our
revised model is the mean time for robots to find the target area in an unknown
environment. The transport equation with delay better predicts the mean time to
find the target than the standard transport equation without delay.Comment: Submitted to the IMA Journal of Applied Mathematic
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