{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T01:21:19Z","timestamp":1760059279965,"version":"build-2065373602"},"reference-count":28,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2025,6,6]],"date-time":"2025-06-06T00:00:00Z","timestamp":1749168000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>This paper identifies a source term depending on spatial variable in a heat equation from just part of the boundary conditions. The measurement data are specified at an internal moment of time. The ill-posedness of the problem is higher than most of the previous source identification problems. This is because the problem becomes a noncharacteristic Cauchy problem for the heat equation if the source term is given, which is known as severely ill-posed. The method of fundamental solutions (MFS) in conjunction with the classical Tikhonov regularization method is proposed to reconstruct a stable approximation. The fundamental solutions for the heat equation are spherically symmetric in spatial variable and satisfy the equation automatically, and thus only the boundary conditions need to be satisfied. This characteristic allows the discretization to be performed only on boundary-like geometry and improve the computational efficiency. In this paper, several numerical examples are listed to show the feasibility and effectiveness of the suggested method.<\/jats:p>","DOI":"10.3390\/sym17060894","type":"journal-article","created":{"date-parts":[[2025,6,6]],"date-time":"2025-06-06T09:02:03Z","timestamp":1749200523000},"page":"894","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Identification of Source Term from Part of the Boundary Conditions"],"prefix":"10.3390","volume":"17","author":[{"given":"Yunjie","family":"Ma","sequence":"first","affiliation":[{"name":"School of Mathematics and Informational Science, Yantai University, Yantai 264005, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2025,6,6]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"127151","DOI":"10.1016\/j.ijheatmasstransfer.2025.127151","article-title":"The general solution of the inverse heat source calibration problem in weld modelling","volume":"249","author":"Rissaki","year":"2025","journal-title":"Int. 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