This paper concerns the study of the numerical approximation for the following boundary value problem: $$\left\{\begin{array}{cccc}{u}_t(x,t)-u_{xx}(x,t)=-u^{-p}(x,t),\hfill & 0<x<1,t>0,u_x(0,t)=0,\hfill & u(1,t)=1,t>0,u(x,0)=u_0\left(x\right)>0,& 0\le x\le 1,\end{array}\right.$$ where $p>0$. We obtain some conditions under which the solution of a semidiscrete form of the above problem quenches in a finite time and estimate its semidiscrete quenching time. We also establish the convergence of the semidiscrete quenching time. Finally, we give some numerical experiments to illustrate our analysis.

In this paper, we consider the following initial-boundary value problem $$\left\}\begin{array}{c}{u}_{tt}(x,t)=\epsilon Lu(x,t)+f\left(u(x,t)\right)\text{in}\Omega \times (0,T),\hfill \\ u(x,t)=0\text{on}\partial \Omega \times (0,T),\hfill \\ u(x,0)=0\text{in}\Omega ,\hfill \\ u_t(x,0)=0\text{in}\Omega ,\hfill \end{array}\right.$$ where $\Omega$ is a bounded domain in ${ℝ}^N$ with smooth boundary $\partial \Omega$, $L$ is an elliptic operator, $\epsilon$ is a positive parameter, $f\left(s\right)$ is a positive, increasing, convex function for $s\in (-\infty ,b)$, ${lim}_{s\to b}f\left(s\right)=\infty$ and ${\int }_0^b\frac{ds}{f\left(s\right)}<\infty$ with $b=const>0$. Under some assumptions, we show that the solution of the above problem quenches in a finite time and its quenching time goes to that of the solution of the following differential equation $$\left\}\begin{array}{cc}{\alpha }^{\text{'}\text{'}}\left(t\right)=f\left(\alpha \left(t\right)\right),\hfill & t>0,\hfill \\ \alpha \left(0\right)=0,{\alpha }^{\text{'}}\left(0\right)=0,\hfill \end{array}\right.$$ as $\epsilon$ goes to zero. We also show that the above result remains...

The dimensionality and the amount of data that need to be processed when intensive data streams are observed grow rapidly together with the development of sensors arrays, CCD and CMOS cameras and other devices. The aim of this paper is to propose an approach to dimensionality reduction as a first stage of training RBF nets. As a vehicle for presenting the ideas, the problem of estimating multivariate probability densities is chosen. The linear projection method is briefly surveyed. Using random...

In this paper, we consider a 2nd order semilinear parabolic initial boundary value problem (IBVP) on a bounded domain $\Omega \subset {ℝ}^N$, with nonstandard boundary conditions (BCs). More precisely, at some part of the boundary we impose a Neumann BC containing an unknown additive space-constant $\alpha \left(t\right)$, accompanied with a nonlocal (integral) Dirichlet side condition. We design a numerical scheme for the approximation of a weak solution to the IBVP and derive error estimates for the approximation of the solution $u$ and...

The model order reduction methodology of reduced basis (RB) techniques offers efficient treatment of parametrized partial differential equations (P2DEs) by providing both approximate solution procedures and efficient error estimates. RB-methods have so far mainly been applied to finite element schemes for elliptic and parabolic problems. In the current study we extend the methodology to general linear evolution schemes such as finite volume schemes for parabolic and hyperbolic evolution equations....

In this contribution, we present a solution to the stochastic Galerkin (SG) matrix equations coming from the Darcy flow problem with uncertain material coefficients in the separable form. The SG system of equations is kept in the compressed tensor form and its solution is a very challenging task. Here, we present the reduced basis (RB) method as a solver which looks for a low-rank representation of the solution. The construction of the RB consists of iterative expanding of the basis using Monte...

A method for reducing controllers for systems described by partial differential equations (PDEs) is applied to Burgers' equation with periodic boundary conditions. This approach differs from the typical approach of reducing the model and then designing the controller, and has developed over the past several years into its current form. In earlier work it was shown that functional gains for the feedback control law served well as a dataset for reduced order basis generation via the proper orthogonal...

A phase field approach for structural topology optimization which allows for topology changes and multiple materials is analyzed. First order optimality conditions are rigorously derived and it is shown via formally matched asymptotic expansions that these conditions converge to classical first order conditions obtained in the context of shape calculus. We also discuss how to deal with triple junctions where e.g. two materials and the void meet. Finally, we present several numerical results for...

In this work, we propose a general framework for the construction of pressure law for phase transition. These equations of state are particularly suitable for a use in a relaxation finite volume scheme. The approach is based on a constrained convex optimization problem on the mixture entropy. It is valid for both miscible and immiscible mixtures. We also propose a rough pressure law for modelling a super-critical fluid.

We show that it is possible to construct a class of entropic schemes for the multicomponent Euler system describing a gas or fluid homogeneous mixture at thermodynamic equilibrium by applying a relaxation technique. A first order Chapman–Enskog expansion shows that the relaxed system formally converges when the relaxation frequencies go to the infinity toward a multicomponent Navier–Stokes system with the classical Fick and Newton laws, with a thermal diffusion which can be assimilated to a Soret...

We show that it is possible to construct a class of entropic schemes for the multicomponent Euler system describing a gas or fluid homogeneous mixture at thermodynamic equilibrium by applying a relaxation technique. A first order Chapman–Enskog expansion shows that the relaxed system formally converges when the relaxation frequencies go to the infinity toward a multicomponent Navier–Stokes system with the classical Fick and Newton laws, with a thermal diffusion which can be assimilated to a Soret...

We investigate biological processes, particularly the propagation of malaria. Both the continuous and the numerical models on some fixed mesh should preserve the basic qualitative properties of the original phenomenon. Our main goal is to give the conditions for the discrete (numerical) models of the malaria phenomena under which they possess some given qualitative property, namely, to be between zero and one. The conditions which guarantee this requirement are related to the time-discretization...