# American Institute of Mathematical Sciences

## Journals

NHM
Networks & Heterogeneous Media 2014, 9(4): 635-654 doi: 10.3934/nhm.2014.9.635
The subject of the present paper is the derivation and asymptotic analysis of a mathematical model for the formation of a mushy region during sulphation of calcium carbonate. The model is derived by averaging, with the use of the multiple scales method, applied on microscopic moving - boundary problems. The latter problems describe the transformation of calcium carbonate into gypsum on the microscopic scale. The derived macroscopic model is solved numerically with the use of a finite element method. The results of some simulations and a relevant discussion are also presented.
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PROC
Conference Publications 2007, 2007(Special): 768-778 doi: 10.3934/proc.2007.2007.768
We consider the following non-local elliptic boundary value problem:

− $w''(x) = \lambda (f(w(x)))/((\eq^1_(-1) f(w(z)) dz)^2) \all x \in$ (−1, 1),
$w'(1) + \alpha w(1) = 0$, $w'$(−1) − $\alpha w$(−1)$=$0,

where $\alpha$ and $\lambda$ are positive constants and $f$ is a function satisfying $f(s)$ > 0, $f'(s) < 0, f''(s) > 0$ for $s > 0, \eq^\infty_0 f(s)ds < \infty.$ The solution of the equation represents the steady state of a thermistor device. The problem has a unique solution for a critical value $\lambda$* of the parameter $\lambda$, at least two solutions for $\lambda < \lambda$* and has no solution for $\lambda > \lambda$*. We apply a finite element and a finite volume method in order to find a numerical approximation of the solution of the problem from the space of continuous piecewise quadratic functions, for the case that $\lambda < \lambda$* and for the stable branch of the bifurcation diagram. A comparison of these two methods is made regarding their order of convergence for $f(s) = e^( - s)$ and $f(s) = (1 + s)^( - 2)$. Also, for the same equation but with Dirichlet boundary conditions, a situation where the solution is unique for $\lambda < \lambda$*, a similar comparison of the finite element and the finite volume method is presented.

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DCDS
Discrete & Continuous Dynamical Systems - A 2015, 35(3): 1009-1037 doi: 10.3934/dcds.2015.35.1009
In this work we study the semilinear wave equation of the form $u_{tt}=u_{xx} + {\lambda}/{ (1-u)^2},$ with homogeneous Dirichlet boundary conditions and suitable initial conditions, which, under appropriate circumstances, serves as a model of an idealized electrostatically actuated MEMS device. First we establish local existence of the solutions of the problem for any $\lambda>0.$ Then we focus on the singular behaviour of the solution, which occurs through finite-time quenching, i.e. when $||u(\cdot,t)||_{\infty}\to 1$ as $t\to t^*- < \infty$, investigating both conditions for quenching and the quenching profile of $u.$ To this end, the non-existence of a regular similarity solution near a quenching point is first shown and then a formal asymptotic expansion is used to determine the local form of the solution. Finally, using a finite difference scheme, we solve the problem numerically, illustrating the preceding results.
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