Electromagnetism of Continuous Media Mathematical Modelling and Applications Oxford Mathematical Monographs Series

For graduate students and researchers, this self contained text provides a carefully structured, coherent, and comprehensive treatment of the mathematical modelling in electromagnetism of continuous media. The authors provide a systematic review of known subjects along with many original results. Part I reviews basic notions and approaches in electromagnetism (Maxwell's equations, Green's functions, harmonic fields, dispersive effects) and emphasizes the physical motivation for the modelling of non-conventional materials. The frequency-dependent properties (such as conductivity, polarizability, and magnetizability), which enter wave diffraction and dispersion, are shown, and these lead to a discussion of models of materials with fading memory in the time domain. Part II develops the thermodynamics of electromagnetic and thermoelectromagnetic materials with memory and provides a systematic account of thermodynamic restrictions. Existence, uniqueness and stability problems are investigated. Also, variational formulations and wave propagation solution are established. Part III is devoted to more involved models which are motivated by the interest in materials and structures with non-conventional properties. The mathematical modelling deals with non-linearity, non-locality and hysteresis. In non-linear materials attention is focussed on the generation of harmonics and in discontinuity waves. Non-locality is examined in a general way and hence is applied to superconductivity. Hysteresis is developed for magnetism. A review of known schemes is given along with new results about the modelling of hysteresis loops. The wide application of technologies in new mechanical, electronic and biomedical systems calls for materials and structures with non-conventional properties (e.g materials with 'memory'). Of equal importance is the understanding of the physical behaviour of these materials and consequently developing mathematical modelling techniques for prediction. Includes appendices that include some properties of Bessel functions, Fourier transforms and Sobolev spaces, compact operators and eigenfunctions, differential operators in curvilinear coordinates, and finite formulation of electromagnetism.

Preface, Part. Part 1: Basic Notions in Electromagnetism.: Electromagnetic fields, 2: Green's functions and retarded potentials, 3: Time-harmonic fields, 4: Models of materials with memory, Part. Part 2: Thermodynamics and Mathematical Problems.: Thermodynamics of simple electromagnetic systems, 6: Thermoelectromagnetic systems, 7: Existence and uniqueness, 8: Wave propagation, 9: Extremum principles, Part. Part 3: Nonlinearity and Nonlocality.0: Problems in nonlinear electromagnetism, 11: Nonlocal electromagnetism and superconductivity, 12: Magnetic hysteresis, AppendicesA: Some properties of Bessel functions, B: Fourier transforms and Sobolev spaces, C: Compact operators and eigenfunctions, D: Differential operators in curvilinear coordinates, E: Finite formulation of electromagnetism