Inverse source problems in electrodynamics

Guanghui Hu; Peijun Li; Xiaodong Liu; Yue Zhao

doi:10.3934/ipi.2018059

Article Contents

2018, Volume 12, Issue 6: 1411-1428. Doi: 10.3934/ipi.2018059

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Inverse source problems in electrodynamics

1.
Beijing Computational Science Research Center, Building 9, East Zone, ZPark Ⅱ, No.10 Xibeiwang East Road, Haidian District, Beijing 100193, China
2.
Department of Mathematics, Purdue University, West Lafayette, Indiana 47907, USA
3.
Institute of Applied Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China
4.
School of Mathematics and Statistics, Central China Normal University, Wuhan 430079, China

* Corresponding author: Yue Zhao
* Corresponding author: Yue Zhao

Received: January 2018

Revised: July 2018

Published: December 2018

The research of PL was supported in part by the NSF grant DMS-1151308. The work of GH is supported by the NSFC grant (No. 11671028) and NSAF grant (No. U1530401). The research of XL is supported by the NNSF of China under grant 11571355 and the Youth Innovation Promotion Association, CAS.

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Abstract

This paper concerns inverse source problems for the time-dependent Maxwell equations. The electric current density is assumed to be the product of a spatial function and a temporal function. We prove uniqueness and stability in determining the spatial or temporal function from the electric field, which is measured on a sphere or at a point over a finite time interval.

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Mathematics Subject Classification: Primary: 35R30; Secondary: 35Q61.

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Figure 1. (left): A Gaussian-modulated sinusoidal pulse function $\chi$ with $\omega = 6$, $\sigma = 1.6$, $\tau = 3$; (right): Fourier spectrum of $\chi$

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