We rigorously derive energy estimates for the second order vector wave equation with gauge condition for the electric field with non-constant electric permittivity function. This equation is used in the stabilized Domain Decomposition Finite Element/Finite Difference approach for time-dependent Maxwell’s system. Our numerical experiments illustrate efficiency of the modified hybrid scheme in two and three space dimensions when the method is applied for generation of backscattering data in the reconstruction...

We show that while Runge-Kutta methods cannot preserve polynomial invariants in general, they can preserve polynomials that are the energy invariant of canonical Hamiltonian systems.

Les méthodes sans maillage emploient une interpolation associée à un ensemble de particules : aucune information concernant la connectivité ne doit être fournie. Un des atouts de ces méthodes est que la discrétisation peut être enrichie d’une façon très simple, soit en augmentant le nombre de particules (analogue à la stratégie de raffinement $h$), soit en augmentant l’ordre de consistance (analogue à la stratégie de raffinement $p$). Néanmoins, le coût du calcul des fonctions d’interpolation est très...

Les méthodes sans maillage emploient une interpolation associée à un ensemble de particules : aucune information concernant la connectivité ne doit être fournie. Un des atouts de ces méthodes est que la discrétisation peut être enrichie d'une façon très simple, soit en augmentant le nombre de particules (analogue à la stratégie de raffinement h), soit en augmentant l'ordre de consistance (analogue à la stratégie de raffinement p). Néanmoins, le coût du calcul des fonctions d'interpolation est...

We prove the existence of solutions to nonlinear parabolic problems of the following type: $$\left\{\begin{array}{cc}{\displaystyle \frac{\partial b\left(u\right)}{\partial t}}+A\left(u\right)=f+\mathrm{div}\left(\Theta (x;t;u)\right)\hfill & \text{in}Q,\hfill \\ u(x;t)=0\hfill & \text{on}\partial \Omega \times [0;T],\hfill \\ b\left(u\right)(t=0)=b\left(u_0\right)\hfill & \text{on}\Omega ,\hfill \end{array}\right.$$ where $b:ℝ\to ℝ$ is a strictly increasing function of class ${𝒞}^1$, the term $$A\left(u\right)=-\mathrm{div}\left(a\right(x,t,u,\nabla u\left)\right)$$ is an operator of Leray-Lions type which satisfies the classical Leray-Lions assumptions of Musielak type, $\Theta :\Omega \times [0;T]\times ℝ\to ℝ$ is a Carathéodory, noncoercive function which satisfies the following condition: ${sup}_{\left|s\right|\le k}\left|\Theta (·,·,s)\right|\in E_{\psi }\left(Q\right)$ for all $k>0$, where $\psi$ is the Musielak complementary function of $\Theta$, and the second term $f$ belongs to $L^1\left(Q\right)$.

Basic models suitable to explain the epidemiology of dengue fever have previously shown the possibility of deterministically chaotic attractors, which might explain the observed fluctuations found in empiric outbreak data. However, the region of bifurcations and chaos require strong enhanced infectivity on secondary infection, motivated by experimental findings of antibody-dependent-enhancement. Including temporary cross-immunity in such models, which is common knowledge among field researchers...

For contractive interval functions $\left[g\right]$ we show that $\left[g\right]\left({\left[x\right]}_{ϵ}^{k_0}\right)\subseteq \int \left({\left[x\right]}_{ϵ}^{k_0}\right)$ results from the iterative process ${\left[x\right]}^{k+1}:=\left[g\right]\left({\left[x\right]}_{ϵ}^k\right)$ after finitely many iterations if one uses the epsilon-inflated vector ${\left[x\right]}_{ϵ}^k$ as input for $\left[g\right]$ instead of the original output vector ${\left[x\right]}^k$. Applying Brouwer’s fixed point theorem, zeros of various mathematical problems can be verified in this way.

We investigate functional equations $f\left(p\right(x\left)\right)=q\left(f\right(x\left)\right)$ where $p$ and $q$ are given real functions defined on the set $ℝ$ of all real numbers. For these investigations, we can use methods for constructions of homomorphisms of mono-unary algebras. Our considerations will be confined to functions $p,q$ which are strictly increasing and continuous on $ℝ$. In this case, there is a simple characterization for the existence of a solution of the above equation. First, we give such a characterization. Further, we present a construction...