Mathematical Models and Methods in Applied Sciences

The Cauchy problem for an infinite plate with transverse shear deformation is studied by means of an area potential and some special initial potentials. This is a fundamental step in the construction of the potential theory for dynamic problems for plates, since such results make it possible to reduce various initial-boundary value problems for the equations of motion to analogous ones for the homogeneous system, which can then be solved by means of retarded potentials.

We describe a model related to nuclear collisions using Boltzmann operators. An asymptotic analysis is performed concerning the gain operator for the outgoing particles. Some numerical methods related to this model are also described and numerical results are given.

An integral representation of electrograms is used for large scale simulations in an anisotropic model of the whole left ventricle. Numerical artifacts, like spurious oscillations or peaks, may appear if the computational grid is not fine enough. To avoid an excessive increase in the number of elements and nodes, we present a numerical procedure based on a combination of: a special refinement of the grid, a sub-element technique, a nonlinear interpolation and a split form of the integral. The numerical simulations show that in this way it is possible to suppress the numerical artifacts thus allowing an accurate computation of electrograms in any point inside or outside the myocardial wall.

When charged particles are submitted to a large external magnetic field, their movement in first approximation occurs along the magnetic field lines and obeys a one-dimensional Vlasov equation along these field lines. However, when observing the particles on a sufficiently long time scale, a drift phenomenon perpendicular to the magnetic field lines superposes to this first movement. In this paper, we present a rigorous asymptotic analysis of the two-dimensional Vlasov equation when the magnetic field tends to infinity, the observation time scale increases accordingly. Techniques based on the two-scale convergence and the introduction of a second problem enable us to find an equation verified by the weak limit of the distribution function.

We examine the stability properties of a sandwich beam consisting of two outer layers and a thin core. The outer layers are modeled as Euler Bernoulli beams and the inner core provides both elastic and viscous resistance to shearing. We show for both clamped and hinged boundary conditions that (i) if rotational inertia terms are neglected, the model is described by an analytic semigroup, and (ii) if rotational inertia is retained in the outer layers, the model is uniformly exponentially stable.

By way of examples we show that the initial value problem for transport equations, with coefficients that involve a nonlocal (in space) functional of the solution, may have multiple solutions. We try to expose the essential features of the examples, thus motivating certain assumptions in the theory of physiologically structured population models.

We present a nonconforming mixed finite element scheme for the approximate solution of the time-harmonic Maxwell's equations in a three-dimensional, bounded domain with absorbing boundary conditions on artificial boundaries. The numerical procedures are employed to solve the direct problem in magnetotellurics consisting in determining a scattered electromagnetic field in a model of the earth having bounded conductivity anomalies of arbitrary shapes. A domain-decomposition iterative algorithm which is naturally parallelizable and is based on a hybridization of the mixed method allows the solution of large three-dimensional models. Convergence of the approximation by the mixed method is proved, as well as the convergence of the iteration.

We analyze the solution of the time-harmonic Maxwell equations with vanishing electric permittivity in bounded domains and subject to absorbing boundary conditions. The problem arises naturally in magnetotellurics when considering the propagation of electromagnetic waves within the earth's interior. Existence and uniqueness are shown under the assumption that the source functions are square integrable. In this case, the electric and magnetic fields belong to H(curl; Ω). If, in addition, the divergences of the source functions are square integrable and the coefficients are Lipschitz-continuous, a stronger regularity result is obtained. A decomposition of the space of square integrable vector functions and a new compact imbedding result are exploited.