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The original boundary value problem for Maxwell\u2019s equations is reduced to a system of integro-differential equations. An integral formulation of the vector inverse diffraction problem is proposed and the uniqueness of the solution of the first-kind integro-differential equation in special function classes is established. A two-step method for solving the vector inverse diffraction problem on the hemisphere is developed. Unlike traditional approaches, the two-step method for solving the inverse problem is non-iterative and does not require knowledge of the exact initial approximation. Consequently, there are no issues related to the convergence of the numerical method. The results of calculations of approximate solutions to the inverse problem are presented. It is shown that the two-step method is an efficient approach to solving vector problems in near-field tomography.<\/jats:p>","DOI":"10.3390\/computation12110213","type":"journal-article","created":{"date-parts":[[2024,10,22]],"date-time":"2024-10-22T06:11:25Z","timestamp":1729577485000},"page":"213","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Solution of the Vector Three-Dimensional Inverse Problem on an Inhomogeneous Dielectric Hemisphere Using a Two-Step Method"],"prefix":"10.3390","volume":"12","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-0168-0282","authenticated-orcid":false,"given":"Eugen","family":"Smolkin","sequence":"first","affiliation":[{"name":"Academy of Technology and Environment, University of G\u00e4vle, 801 76 G\u00e4vle, Sweden"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9040-628X","authenticated-orcid":false,"given":"Yury","family":"Smirnov","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Supercomputing, Penza State University, 440026 Penza, Russia"}]},{"given":"Maxim","family":"Snegur","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Supercomputing, Penza State University, 440026 Penza, Russia"}]}],"member":"1968","published-online":{"date-parts":[[2024,10,22]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Paulsen, K. 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