Mathematical Models and Methods in Applied Sciences

The aim of this paper is to study the reflection-transmission of geometrical optic rays described by semi-linear symmetric hyperbolic systems such as the Maxwell–Lorentz equations with the anharmonic model of polarisation. The framework is both that of Donnat and Williams since we consider dispersive media and profiles with hyperbolic (imaginary) phases and elliptic phases (complex with non-null real part). We first give hypothesis close to the Maxwell equation. Then we introduce a decomposition for both profile into boundary (tangential) and normal part and we solve the so-called "microscopic" equation of the small scales for each boundary frequency. Then we show that the non-linearities generate harmonics which interact at the boundary and generate new resonant profiles with harmonic tangential frequency. Lastly we make a WKB expansion at any order and give a precise description of the correctors.

We prove a global in time abstract existence and uniqueness result for a general parabolic problem of reconstruction of a convolution kernel. The result is, in particular, applicable to the theory of heat conduction for materials with memory.

We consider fully discrete schemes for the one-dimensional linear Schrödinger equation and analyze whether the classical dispersive properties of the continuous model are presented in these approximations. In particular, Strichartz estimates and the local smoothing of the numerical solutions are analyzed. Using a backward Euler approximation of the linear semigroup we introduce a convergent scheme for the nonlinear Schrödinger equation with nonlinearities which cannot be treated by energy methods.

Let ϒ be a three-dimensional Lipschitz polyhedron, and assume that the matrix function μ is piecewise constant on a polyhedral partition of ϒ. Based on regularity results for solutions to two-dimensional anisotropic transmission problems near corner points we obtain conditions on μ and the intersection angles between interfaces and ∂ϒ ensuring that the operator -∇ · μ∇ maps the Sobolev space isomorphically onto W^-1,q(ϒ) for some q > 3.

We consider here the "generalized von Kármán equations", which constitute a mathematical model for a nonlinearly elastic plate subjected to boundary conditions "of von Kármán type" only on a portion of its lateral face, the remaining portion being free. As already shown elsewhere, solving these equations amounts to solving a "cubic" operator equation, which generalizes an equation introduced by Berger and Fife. Two noticeable features of this equation, which are not encountered in the "classical" von Kármán equations are the lack of strict positivity of its cubic part and the lack of an associated functional. We establish here the convergence of a conforming finite element approximation to these equations. The proof relies in particular on a compactness method due to J. L. Lions and on Brouwer's fixed point theorem. This convergence proof provides in addition an existence proof for the original problem.

Thermoacoustic computed tomography (TACT) is an emerging hybrid imaging method for non-invasive medical diagnosis and fully three-dimensional visualization of biological probes. Within this modality electromagnetic illumination is used to induce acoustic waves inside an object of interest. In this paper, we assume that a cylindrical array of line detectors is used to record the acoustical data. This leads to the mathematical problem of inverting the circular Radon transform.

The circular Radon transform arises in several other up-to-date imaging modalities, such as RADAR imaging or ultrasound tomography. In this paper we prove a novel stable formula for recovering a planar function from its circular Radon transform. We apply this formula to obtain an exact three-dimensional imaging algorithm for TACT. Numerical reconstructions from real and synthetic data demonstrate the potential and robustness of our algorithm.