{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,8]],"date-time":"2026-02-08T12:30:59Z","timestamp":1770553859906,"version":"3.49.0"},"reference-count":21,"publisher":"Wiley","license":[{"start":{"date-parts":[[2014,1,1]],"date-time":"2014-01-01T00:00:00Z","timestamp":1388534400000},"content-version":"unspecified","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by\/3.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["51190093"],"award-info":[{"award-number":["51190093"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["51179151"],"award-info":[{"award-number":["51179151"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["51190093"],"award-info":[{"award-number":["51190093"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["51179151"],"award-info":[{"award-number":["51179151"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Journal of Applied Mathematics"],"published-print":{"date-parts":[[2014]]},"abstract":"<jats:p>We investigate the inverse problem in the nonhomogeneous heat equation involving the recovery of the initial temperature from measurements of the final temperature. This problem is known as the backward heat problem and is severely ill-posed. We show that this problem can be converted into the first Fredholm integral equation, and an algorithm of inversion is given using Tikhonov's regularization method. The genetic algorithm for obtaining the regularization parameter is presented. We also present numerical computations that verify the accuracy of our approximation.<\/jats:p>","DOI":"10.1155\/2014\/529618","type":"journal-article","created":{"date-parts":[[2014,3,3]],"date-time":"2014-03-03T07:58:23Z","timestamp":1393833503000},"page":"1-7","source":"Crossref","is-referenced-by-count":2,"title":["Inverse Estimates for Nonhomogeneous Backward Heat Problems"],"prefix":"10.1155","volume":"2014","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-0813-8122","authenticated-orcid":true,"given":"Tao","family":"Min","sequence":"first","affiliation":[{"name":"School of Science, Xi'an University of Technology, Xi'an, Shaanxi 710054, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Weimin","family":"Fu","sequence":"additional","affiliation":[{"name":"School of Science, Xi'an University of Technology, Xi'an, Shaanxi 710054, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Qiang","family":"Huang","sequence":"additional","affiliation":[{"name":"State Key Laboratory of Eco-Hydraulic Engineering in Shaanxi, Xi'an University of Technology, Xi'an, Shaanxi 710048, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"311","reference":[{"key":"1","doi-asserted-by":"publisher","DOI":"10.1016\/j.na.2005.09.030"},{"key":"2","doi-asserted-by":"publisher","DOI":"10.1016\/j.amc.2005.08.040"},{"key":"3","doi-asserted-by":"publisher","DOI":"10.1016\/j.ijthermalsci.2008.11.019"},{"key":"4","doi-asserted-by":"crossref","first-page":"265","DOI":"10.1080\/00036810701858193","volume":"87","year":"2008","journal-title":"Applicable Analysis"},{"key":"5","doi-asserted-by":"publisher","DOI":"10.1016\/j.amc.2006.11.062"},{"key":"7","year":"1967"},{"issue":"3","key":"8","doi-asserted-by":"crossref","first-page":"784","DOI":"10.1016\/0022-247X(72)90083-2","volume":"38","year":"1972","journal-title":"Journal of Mathematical Analysis and Applications"},{"key":"9","doi-asserted-by":"publisher","DOI":"10.1088\/0266-5611\/15\/5\/310"},{"issue":"23","key":"10","doi-asserted-by":"crossref","first-page":"3731","DOI":"10.1016\/S0017-9310(98)00102-1","volume":"41","year":"1998","journal-title":"International Journal of Heat and Mass Transfer"},{"key":"11","year":"1985"},{"key":"12","year":"1991"},{"key":"13","first-page":"1","volume":"2006","year":"2006","journal-title":"Electronic Journal of Differential Equations"},{"issue":"2","key":"14","doi-asserted-by":"crossref","first-page":"231","DOI":"10.4171\/zaa\/1321","volume":"26","year":"2007","journal-title":"Zeitschrift fur Analysis und ihre Anwendung"},{"key":"15","doi-asserted-by":"publisher","DOI":"10.1163\/156939406778247615"},{"key":"16","doi-asserted-by":"publisher","DOI":"10.1088\/0266-5611\/17\/2\/308"},{"key":"17","doi-asserted-by":"publisher","DOI":"10.1080\/17415970600850785"},{"key":"18","first-page":"1000","volume":"8","year":"1967","journal-title":"Soviet Mathematics: Doklady"},{"key":"19","year":"1984"},{"key":"20","year":"1989"},{"issue":"7","key":"21","doi-asserted-by":"crossref","first-page":"1205","DOI":"10.2514\/3.25195","volume":"28","year":"1990","journal-title":"AIAA Journal"},{"key":"22","doi-asserted-by":"publisher","DOI":"10.1007\/BF01743964"}],"container-title":["Journal of Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/downloads.hindawi.com\/journals\/jam\/2014\/529618.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/downloads.hindawi.com\/journals\/jam\/2014\/529618.xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/downloads.hindawi.com\/journals\/jam\/2014\/529618.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,5,24]],"date-time":"2024-05-24T18:35:28Z","timestamp":1716575728000},"score":1,"resource":{"primary":{"URL":"http:\/\/www.hindawi.com\/journals\/jam\/2014\/529618\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2014]]},"references-count":21,"alternative-id":["529618","529618"],"URL":"https:\/\/doi.org\/10.1155\/2014\/529618","relation":{},"ISSN":["1110-757X","1687-0042"],"issn-type":[{"value":"1110-757X","type":"print"},{"value":"1687-0042","type":"electronic"}],"subject":[],"published":{"date-parts":[[2014]]}}}