{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,18]],"date-time":"2026-05-18T13:35:11Z","timestamp":1779111311823,"version":"3.51.4"},"reference-count":52,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2024,2,6]],"date-time":"2024-02-06T00:00:00Z","timestamp":1707177600000},"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>In the paper, we develop three new methods for estimating unknown initial temperature in a backward time-fractional diffusion problem, which is transformed to a space-dependent inverse source problem for a new variable in the first method. Then, the initial temperature can be recovered by solving a second-order boundary value problem. The boundary functions and a unique zero element constitute a group symmetry. We derive energetic boundary functions in the symmetry group as the bases to retrieve the source term as an unknown function of space and time. In the second method, the solution bases are energetic boundary functions, and then by collocating the governing equation we obtain the expansion coefficients for retrieving the entire solution and initial temperature. For the first two methods, boundary fluxes are over-specified to retrieve the initial condition. In the third method, we give two boundary conditions and a final time temperature to construct the bases in another symmetry group; the governing equation is collocated to a linear system to obtain the whole solution (initial temperature involved). These three methods are assessed and compared by numerical experiments.<\/jats:p>","DOI":"10.3390\/sym16020191","type":"journal-article","created":{"date-parts":[[2024,2,6]],"date-time":"2024-02-06T05:36:43Z","timestamp":1707197803000},"page":"191","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["A Symmetry of Boundary Functions Method for Solving the Backward Time-Fractional Diffusion Problems"],"prefix":"10.3390","volume":"16","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-6366-3539","authenticated-orcid":false,"given":"Chein-Shan","family":"Liu","sequence":"first","affiliation":[{"name":"Center of Excellence for Ocean Engineering, National Taiwan Ocean University, Keelung 202301, Taiwan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Chung-Lun","family":"Kuo","sequence":"additional","affiliation":[{"name":"Center of Excellence for Ocean Engineering, National Taiwan Ocean University, Keelung 202301, Taiwan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9846-0694","authenticated-orcid":false,"given":"Chih-Wen","family":"Chang","sequence":"additional","affiliation":[{"name":"Department of Mechanical Engineering, National United University, Miaoli 360302, Taiwan"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2024,2,6]]},"reference":[{"key":"ref_1","first-page":"303","article-title":"An advanced implicit meshless approach for the non-linear anomalous subdiffusion equation","volume":"56","author":"Gu","year":"2010","journal-title":"Comput. 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