117 research outputs found

    Global Well-Posedness and Exponential Stability for Heterogeneous Anisotropic Maxwell's Equations under a Nonlinear Boundary Feedback with Delay

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    We consider an initial-boundary value problem for the Maxwell's system in a bounded domain with a linear inhomogeneous anisotropic instantaneous material law subject to a nonlinear Silver-Muller-type boundary feedback mechanism incorporating both an instantaneous damping and a time-localized delay effect. By proving the maximal monotonicity property of the underlying nonlinear generator, we establish the global well-posedness in an appropriate Hilbert space. Further, under suitable assumptions and geometric conditions, we show the system is exponentially stable.Comment: updated and improved versio

    Fractional equations and diffusive systems: an overview

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    The aim of this discussion is to give a broad view of the links between fractional differential equations (FDEs) or fractional partial differential equations (FPDEs) and so-called diffusive representations (DR). Many aspects will be investigated: theory and numerics, continuous time and discrete time, linear and nonlinear equations, causal and anti-causal operators, optimal diffusive representations, fractional Laplacian. Many applications will be given, in acoustics, continuum mechanics, electromagnetism, identification, ..

    Mathematical models for dispersive electromagnetic waves: an overview

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    In this work, we investigate mathematical models for electromagnetic wave propagation in dispersive isotropic media. We emphasize the link between physical requirements and mathematical properties of the models. A particular attention is devoted to the notion of non-dissipativity and passivity. We consider successively the case of so-called local media and general passive media. The models are studied through energy techniques, spectral theory and dispersion analysis of plane waves. For making the article self-contained, we provide in appendix some useful mathematical background.Comment: 46 pages, 16 figure

    Diffusive approximation of a time-fractional Burger's equation in nonlinear acoustics

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    A fractional time derivative is introduced into the Burger's equation to model losses of nonlinear waves. This term amounts to a time convolution product, which greatly penalizes the numerical modeling. A diffusive representation of the fractional derivative is adopted here, replacing this nonlocal operator by a continuum of memory variables that satisfy local-in-time ordinary differential equations. Then a quadrature formula yields a system of local partial differential equations, well-suited to numerical integration. The determination of the quadrature coefficients is crucial to ensure both the well-posedness of the system and the computational efficiency of the diffusive approximation. For this purpose, optimization with constraint is shown to be a very efficient strategy. Strang splitting is used to solve successively the hyperbolic part by a shock-capturing scheme, and the diffusive part exactly. Numerical experiments are proposed to assess the efficiency of the numerical modeling, and to illustrate the effect of the fractional attenuation on the wave propagation.Comment: submitted to Siam SIA

    Computational Electromagnetism and Acoustics

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    It is a moot point to stress the significance of accurate and fast numerical methods for the simulation of electromagnetic fields and sound propagation for modern technology. This has triggered a surge of research in mathematical modeling and numerical analysis aimed to devise and improve methods for computational electromagnetism and acoustics. Numerical techniques for solving the initial boundary value problems underlying both computational electromagnetics and acoustics comprise a wide array of different approaches ranging from integral equation methods to finite differences. Their development faces a few typical challenges: highly oscillatory solutions, control of numerical dispersion, infinite computational domains, ill-conditioned discrete operators, lack of strong ellipticity, hysteresis phenomena, to name only a few. Profound mathematical analysis is indispensable for tackling these issues. Many outstanding contributions at this Oberwolfach conference on Computational Electromagnetism and Acoustics strikingly confirmed the immense recent progress made in the field. To name only a few highlights: there have been breakthroughs in the application and understanding of phase modulation and extraction approaches for the discretization of boundary integral equations at high frequencies. Much has been achieved in the development and analysis of discontinuous Galerkin methods. New insight have been gained into the construction and relationships of absorbing boundary conditions also for periodic media. Considerable progress has been made in the design of stable and space-time adaptive discretization techniques for wave propagation. New ideas have emerged for the fast and robust iterative solution for discrete quasi-static electromagnetic boundary value problems

    Decay rates for solutions of semilinear wave equations with a memory condition at the boundary

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    In this paper, we study the stability of solutions for semilinear wave equations whoseboundary condition includes a integral that represents the memory effect. We show that the dissipation is strong enough to produce exponential decay of the solution, provided the relaxation function also decays exponentially. When the relaxation function decays polinomially, we show that the solution decays polynomially and with the same rate

    On numerical methods for direct and inverse problems in electromagnetism

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    This thesis is devoted to the study of processes in the propagation of electromagnetic fields. We do not aim at one particular problem, actually very different kinds of topics are analyzed here. We deal with direct problems as well as with inverse ones, low frequency electromagnetism is discussed and consequently the wave propagation problem in high frequency domain is studied. Study of electromagnetic materials and their behavior is of a huge interest for the technological world. Its importance originates from the increasing requirements for high performance devices as motors, transformers, radars,... To improve character of electromagnets, new models, accurate numerical schemes and their rigorous analysis are needed to be worked out. In the first part we aimed at a time dependent eddy current equation of the magnetic field when a non-perfect contact of two different materials is considered on the boundary. The second part concerning the problems in the high-frequency domain includes the stuyd of elctromagnetic waves and propagation of energy through matter. These problems are mostly posed on an unbounded domain. To solve these problems we use the method of Artificial Boundary Condition (ABC) - widely used for wave problems since 1977. The third part of the thesis is devoted to the inverse problems in low-frequency electromagnetism. For a design of the material characteristics of the magnetic circuit, such as the electromagnetic loss P, the permeability µ and the electrical conductivity sigma, is essential. In this thesis, the missing material constant turns out to be a measure for the electromagnetic losses. The end of each part is devoted to numerical examples confirming the theoretical results. All the computations are performed using The Finite Element Toolbox ALBERT

    Nonlinear Evolution Equations: Analysis and Numerics

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    The workshop was devoted to the analytical and numerical investigation of nonlinear evolution equations. The main aim was to stimulate a closer interaction between experts in analytical and numerical methods for areas such as wave and Schrödinger equations or the Navier–Stokes equations and fluid dynamics
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