64 research outputs found
A time discretization scheme for a nonlocal degenerate problem modelling resistance spot welding
This is the author's PDF version of an article published in Mathematical Modelling of Natural Phenomena© 2015. The definitive version is available at http://www.mmnp-journal.org/articles/mmnp/abs/2015/06/mmnp2015106p90/mmnp2015106p90.htmlIn the current work we construct a nonlocal mathematical model describing the phase transition occurs during the resistance spot welding process in the industry of metallurgy. We then consider a time discretization scheme for solving the resulting nonlocal moving boundary problem. The scheme consists of solving at each time step a linear elliptic partial differential equation and then making a correction to account for the nonlinearity. The stability and error estimates of the developed scheme are investigated. Finally some numerical results are presented confirming the efficiency of the developed numerical algorithm
Estimate of blow-up and relaxation time for self-gravitating Brownian particles and bacterial populations
We determine an asymptotic expression of the blow-up time t_coll for
self-gravitating Brownian particles or bacterial populations (chemotaxis) close
to the critical point. We show that t_coll=t_{*}(eta-eta_c)^{-1/2} with
t_{*}=0.91767702..., where eta represents the inverse temperature (for Brownian
particles) or the mass (for bacterial colonies), and eta_c is the critical
value of eta above which the system blows up. This result is in perfect
agreement with the numerical solution of the Smoluchowski-Poisson system. We
also determine the asymptotic expression of the relaxation time close but above
the critical temperature and derive a large time asymptotic expansion for the
density profile exactly at the critical point
Particle approximation of the one dimensional Keller-Segel equation, stability and rigidity of the blow-up
We investigate a particle system which is a discrete and deterministic
approximation of the one-dimensional Keller-Segel equation with a logarithmic
potential. The particle system is derived from the gradient flow of the
homogeneous free energy written in Lagrangian coordinates. We focus on the
description of the blow-up of the particle system, namely: the number of
particles involved in the first aggregate, and the limiting profile of the
rescaled system. We exhibit basins of stability for which the number of
particles is critical, and we prove a weak rigidity result concerning the
rescaled dynamics. This work is complemented with a detailed analysis of the
case where only three particles interact
Hyperbolic quenching problem with damping in the micro-electro mechanical system device
[[abstract]]We study the initial boundary value problem for the damped hyperbolic
equation arising in the micro-electro mechanical system device with
local or nonlocal singular nonlinearity. For both cases, we provide some criteria
for quenching and global existence of the solution. We also derive the
existence of the quenching curve for the corresponding Cauchy problem with
local source[[notice]]補正完畢[[journaltype]]國外[[incitationindex]]SCI[[ispeerreviewed]]Y[[countrycodes]]US
The dual integral equation method in hydromechanical systems
Some hydromechanical systems are investigated by applying the dual
integral equation method. In developing this method we suggest
from elementary appropriate solutions of Laplace's equation, in
the domain under consideration, the introduction of a potential
function which provides useful combinations in cylindrical and
spherical coordinates systems. Since the mixed boundary conditions
and the form of the potential function are quite
general, we obtain integral equations with mth-order Hankel kernels. We then discuss a kind of approximate practicable
solutions. We note also that the method has important applications
in situations which arise in the determination of the temperature
distribution in steady-state heat-conduction problems
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