{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,9]],"date-time":"2026-04-09T04:42:52Z","timestamp":1775709772646,"version":"3.50.1"},"reference-count":24,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2022,6,11]],"date-time":"2022-06-11T00:00:00Z","timestamp":1654905600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"the National Natural Science Foundation of China","award":["11961044"],"award-info":[{"award-number":["11961044"]}]},{"name":"the National Natural Science Foundation of China","award":["21JR7RA214"],"award-info":[{"award-number":["21JR7RA214"]}]},{"name":"the Doctor Fund of Lan Zhou University of Technology","award":["11961044"],"award-info":[{"award-number":["11961044"]}]},{"name":"the Doctor Fund of Lan Zhou University of Technology","award":["21JR7RA214"],"award-info":[{"award-number":["21JR7RA214"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In this manuscript, the Cauchy problem of the modified Helmholtz equation is researched. This inverse problem is a serious ill-posed problem. The classical Landweber iterative regularization method is designed to find the regularized solution of this inverse problem. The error estimations between the exact solution and the regularization solution are all obtained under the a priori and the a posteriori regularization parameter selection rule. The Landweber iterative regularization method can also be applied to solve the Cauchy problem of the modified Helmholtz equation on the spherically symmetric and cylindrically symmetric regions.<\/jats:p>","DOI":"10.3390\/sym14061209","type":"journal-article","created":{"date-parts":[[2022,6,13]],"date-time":"2022-06-13T06:31:59Z","timestamp":1655101919000},"page":"1209","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["The Landweber Iterative Regularization Method for Solving the Cauchy Problem of the Modified Helmholtz Equation"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-7597-3371","authenticated-orcid":false,"given":"Yong-Gang","family":"Chen","sequence":"first","affiliation":[{"name":"School of Science, China University of Petroleum, Qindao 266580, China"}]},{"given":"Fan","family":"Yang","sequence":"additional","affiliation":[{"name":"School of Science, Lanzhou University of Technology, Lanzhou 730050, China"}]},{"given":"Qian","family":"Ding","sequence":"additional","affiliation":[{"name":"School of Science, Lanzhou University of Technology, Lanzhou 730050, China"}]}],"member":"1968","published-online":{"date-parts":[[2022,6,11]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"616","DOI":"10.1016\/j.jcp.2005.06.006","article-title":"An adaptive fast solver for the modified Helmholtz equation in two dimensions","volume":"211","author":"Cheng","year":"2006","journal-title":"J. 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