# publichpfem/esco2012-boa

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 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 \title{Maxwell Equations with New Dissipative Memory Boundary Conditions }\tocauthor{M. Urev} \author{} \institute{}\maketitle\begin{center}{\large Mikhail Urev}\\ Institute of Computational Mathematics and Mathematical Geophysics SB RAS\\{\tt mih.urev2010@yandex.ru}\end{center}\section*{Abstract}The paper considers the initial-boundary value problem for Maxwellequations in a bounded domain and finite time interval.Dissipative memory boundary conditions that are discussed in thepaper are different from those used at present [1], [2]. Thisboundary condition was obtained in author's paper [3]$$\mathbf{E}_\tau(\mathbf{x},t) = \frac{1}{2\pi}\sqrt{\frac{\mu}{\sigma} } \frac{\partial}{\partial t}\int_{t_0}^t\frac{\mathbf{H} (\mathbf{x},\xi) \times \mathbf{n}(\mathbf{x}) d\xi }{\sqrt{t-\xi} } \equiv C_\sigma (D^{1/2}_{0+,t} \mathbf{H})(\mathbf{x},t) \times \mathbf{n}(\mathbf {x}),$$where $D^{1/2}_{0+,t}$ denotes the operator of fractionaldifferentiation with respect to $t$ of Riemann-Liouville order$1/2$. In case of a harmonic time dependency of theelectromagnetic field this boundary condition becomes equal to theclassical Leontovich impedance condition. Maxwell operator withthese boundary conditions has been studied in suitable functionspaces and theorem of existence and uniqueness of thecorresponding initial-boundary value problem has been proven.\bibliographystyle{plain}\begin{thebibliography}{10}\bibitem{1urev}{\sc M. Fabrizio and A. Morro }. {A boundary condition with memory in electromagnetism}. Arch. Rational Mech. Anal. 136, 359-381 (1996).\bibitem{2urev}{\sc S. Nicaise and C. Pignotti}. { Energy decay rates for solutions of Maxwell's system with a memory boundary condition}. Collectanea Mathematica, North America, 58, 1-16 (2007).\bibitem{3urev}{\sc M.V. Urev }. {Boundary conditions for Maxwell equations with arbitrary time dependence}. Comput. Math. Math. Phys., 37:12, 1444-1451 (1997).\end{thebibliography}
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