Mathematical Models and Methods in Applied Sciences

In this paper we study the local-in-time Cauchy problem for the Schrödinger–Debye equations. This model occurs in nonlinear optics and describes the non-resonant delayed interaction of an electromagnetic wave with a medium. We extend the study to nonphysical cases such as the three-dimensional case or more general nonlinearities.

This paper is devoted to the analysis of two Helmholtz equations in ℝ² coupled via quasiperiodic transmission conditions on a set of piecewise smooth interfaces. The solution of this system is quasiperiodic in one direction and satisfies outgoing wave conditions with respect to the other direction. It is shown that Maxwell's equations for the diffraction of a time-harmonic oblique incident plane wave by periodic interfaces can be reduced to problems of this kind. The analysis is based on a strongly elliptic variational formulation of the differential problem in a bounded periodic cell involving nonlocal boundary operators. We obtain existence and uniqueness results for solutions corresponding to electromagnetic fields with locally finite energy. Special attention is paid to the regularity and leading asymptotics of solutions near the edges of the interface.

We present two modifications of continuum models used in large eddy simulation. The first modification is a closure approximation which better attenuates small eddies. The second modification is a change in the boundary conditions for the large eddy model from strict adherence to slip with resistance. For model consistency, the resistance coefficient is a function of the averaging radius so that the model's boundary conditions reduce to a no-slip as the averaging radius decreases to zero.

The aim of this paper is the study of the relaxation limit of the 3-D bipolar hydrodynamic model for semiconductors. We prove the convergence for the weak solutions to the bipolar Euler–Poisson system towards the solutions to the bipolar drifthyphen;diffusion system, as the relaxation time tends to zero.

A problem of conjugation of hyperbolic and parabolic equations in domain with moving boundaries is considered. Existence and uniqueness of a strong solution of the given problem are proved. A priori estimate for operator-difference scheme with non-self-adjoint spatial operator is obtain. Homogeneous difference scheme with constant weights for the conjugation problem is constructed. Moreover, consistency conditions are approximated with the second-order of accuracy with respect to spatial variables. Stability and convergence of the suggested scheme are investigated.

This paper proposes a multicell model to describe the evolution of tumour growth from the avascular stage to the vascular one through the angiogenic process. The model is able to predict the formation of necrotic regions, the control of mitosis by the presence of an inhibitory factor, the angiogenesis process with proliferation of capillaries just outside the tumour surface with penetration of capillary sprouts inside the tumour, the regression of the capillary network induced by the tumour when angiogenesis is controlled or inhibited, say as an effect of angiostatins, and finally the regression of the tumour size. The three-dimensional model is deduced both in a continuum mechanics framework and by a lattice scheme in order to put in evidence the relation between microscopic phenomena and macroscopic parameters. The evolution problem can be written as a free-boundary problem of mixed hyperbolic–parabolic type coupled with an initial-boundary value problem in a fixed domain.

Here we consider a magnetized plasma composed of electrons of low mean density and of one species of multicharged ions. Starting with the Vlasov–Fokker–Planck equations for both particles, we derive a diffusion model for the electrons, with explicit transport coefficients which are related to the ions. Finally, we study the kinetic boundary layers in the case of a bounded domain in space variables. For that purpose, we are led to study a Milne problem and to introduce a generalized extrapolation length.

This paper introduces a novel construction of wavelets on the unit interval. With this construction, explicit upper bounds for the length of the modified border wavelets filters can be given. This insures a good localization of the border wavelets when a triangular biorthogonalization scheme is employed. The resulting wavelet bases are then well-suited for the adaptive solution of partial differential equations.