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Problem of Determining the Anisotropic Conductivity in Electrodynamic Equations
Doklady Mathematics ( IF 0.5 ) Pub Date : 2021-03-04 , DOI: 10.1134/s1064562421010099
V. G. Romanov

Abstract

For a system of electrodynamic equations, the inverse problem of determining an anisotropic conductivity is considered. It is supposed that the conductivity is described by a diagonal matrix σ(x) = \({\text{diag}}({{\sigma }_{1}}(x),{{\sigma }_{2}}(x)\), σ3(x)) with \(\sigma (x) = 0\) outside of the domain Ω = \(\{ x \in {{\mathbb{R}}^{3}}|\left| x \right| < R\} \), \(R > 0\), and the permittivity ε and the permeability μ of the medium are positive constants everywhere in \({{\mathbb{R}}^{3}}\). Plane waves coming from infinity and impinging on an inhomogeneity localized in Ω are considered. For the determination of the unknown functions \({{\sigma }_{1}}(x)\), \({{\sigma }_{2}}(x)\), and \({{\sigma }_{3}}(x)\), information related to the vector of electric intensity is given on the boundary S of the domain Ω. It is shown that this information reduces the inverse problem to three identical problems of X-ray tomography.



中文翻译:

确定电动方程中各向异性电导率的问题

摘要

对于一个电动方程组,要考虑确定各向异性电导率的反问题。假设电导率由对角矩阵σ(x)= \({\ text {diag}}({{\ sigma} _ {1}}(x),{{\ sigma} _ {2} }(X)\),σ 3X)),与\(\西格马(X)= 0 \)域Ω之外= \(\ {X \在{{\ mathbb {R}} ^ {3} } | \ left | x \ right | <R \} \)\(R> 0 \)和介质的介电常数ε和磁导率μ在\({{\ mathbb {R}} ^ {3}} \)。考虑了来自无穷大并撞击在Ω中的不均匀性的平面波。为了确定未知函数\({{\ sigma __ {1}}(x)\)\({{\ sigma} _ {2}}(x)\)\({{\ sigma } _ {3}}(x)\),在域Ω的边界S上给出了与电强度矢量有关的信息。结果表明,该信息将反问题简化为X射线断层扫描的三个相同问题。

更新日期:2021-03-05
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