{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,2,21]],"date-time":"2025-02-21T14:36:04Z","timestamp":1740148564921,"version":"3.37.3"},"reference-count":34,"publisher":"Wiley","license":[{"start":{"date-parts":[[2022,8,26]],"date-time":"2022-08-26T00:00:00Z","timestamp":1661472000000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100007614","name":"Al Jouf University","doi-asserted-by":"publisher","award":["DSR-2021-03-0342"],"award-info":[{"award-number":["DSR-2021-03-0342"]}],"id":[{"id":"10.13039\/501100007614","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Journal of Sensors"],"published-print":{"date-parts":[[2022,8,26]]},"abstract":"<jats:p>In this article, we will determine the source term of the fractional diffusion equation (FDE). Our contribution to this work is the generalization of the common inverse diffusion equation issues and the inverse diffusion equation problems for fractional diffusion equations with energy source and using Caputo fractional derivatives in time and space. The problem is reformulated in a least-squares framework, which leads to a nonconvex minimization problem, which is solved using a Tikhonov regularization. By considering the direct problem with an implicit finite difference scheme (IFDS), the numerical inversions are performed for the source term in several approximate spaces. The inversion algorithm (IA) uniqueness is obtained. Furthermore, the effect of fractional order and regularization parameter on the inversion algorithm is carried out and shows that the inversion algorithm is effective. The order of fractional derivatives expresses the global property of the direct problem and also shows the badly posed nature of the inverted problem in question.<\/jats:p>","DOI":"10.1155\/2022\/7984688","type":"journal-article","created":{"date-parts":[[2022,8,26]],"date-time":"2022-08-26T22:35:14Z","timestamp":1661553314000},"page":"1-9","source":"Crossref","is-referenced-by-count":3,"title":["Determination of an Energy Source Term for Fractional Diffusion Equation"],"prefix":"10.1155","volume":"2022","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-6891-7849","authenticated-orcid":true,"given":"Sid Ahmed","family":"Ould Ahmed Mahmoud","sequence":"first","affiliation":[{"name":"Mathematics Department, College of Science, Jouf University, P.O. Box: 2014, Sakaka, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0379-0162","authenticated-orcid":true,"given":"Hamed","family":"Ould Sidi","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Sciences University of Nouakchott Al Aasriya, Nouakchott, Mauritania"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0034-5661","authenticated-orcid":true,"given":"Maawiya","family":"Ould Sidi","sequence":"additional","affiliation":[{"name":"Mathematics Department, College of Science, Jouf University, P.O. Box: 2014, Sakaka, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"311","reference":[{"key":"1","doi-asserted-by":"publisher","DOI":"10.1029\/92WR01757"},{"key":"2","doi-asserted-by":"publisher","DOI":"10.1029\/98WR00214"},{"key":"3","doi-asserted-by":"publisher","DOI":"10.1016\/0378-4371(92)90566-9"},{"key":"4","doi-asserted-by":"publisher","DOI":"10.1029\/1999WR900295"},{"key":"5","doi-asserted-by":"publisher","DOI":"10.2136\/sssaj2003.1079"},{"key":"6","doi-asserted-by":"publisher","DOI":"10.1016\/j.jconhyd.2006.03.001"},{"key":"7","doi-asserted-by":"publisher","DOI":"10.1002\/mma.7804"},{"key":"8","doi-asserted-by":"publisher","DOI":"10.1016\/j.aej.2021.10.030"},{"key":"9","doi-asserted-by":"publisher","DOI":"10.12693\/APhysPolA.140.265"},{"issue":"2","key":"10","first-page":"163","article-title":"On solvability of linear fractional differential equations in Banach spaces","volume":"2","author":"R. Gorenflo","year":"1999","journal-title":"Fractional Calculus and Applied Analysis"},{"key":"11","doi-asserted-by":"publisher","DOI":"10.1016\/S0378-4371(99)00503-8"},{"key":"12","doi-asserted-by":"publisher","DOI":"10.1016\/j.cam.2004.01.033"},{"key":"13","doi-asserted-by":"publisher","DOI":"10.1016\/j.amc.2006.08.162"},{"volume-title":"Fractional Differential Equations","year":"1999","author":"I. Podlubnv","key":"14"},{"key":"15","doi-asserted-by":"publisher","DOI":"10.1088\/0266-5611\/25\/11\/115002"},{"key":"16","doi-asserted-by":"publisher","DOI":"10.1515\/jip-2011-0012"},{"key":"17","doi-asserted-by":"publisher","DOI":"10.1088\/0266-5611\/27\/3\/035010"},{"key":"18","doi-asserted-by":"publisher","DOI":"10.1137\/20M1334267"},{"key":"19","doi-asserted-by":"publisher","DOI":"10.1016\/j.jde.2016.09.036"},{"key":"20","doi-asserted-by":"publisher","DOI":"10.1364\/JOSAA.36.000826"},{"key":"21","doi-asserted-by":"publisher","DOI":"10.1186\/s13662-020-02657-2"},{"key":"22","doi-asserted-by":"publisher","DOI":"10.1016\/j.aml.2018.11.025"},{"key":"23","doi-asserted-by":"publisher","DOI":"10.1137\/19M1253605"},{"key":"24","doi-asserted-by":"publisher","DOI":"10.1016\/j.aml.2018.11.012"},{"key":"25","doi-asserted-by":"publisher","DOI":"10.1007\/s10915-021-01501-3"},{"key":"26","doi-asserted-by":"publisher","DOI":"10.1016\/j.matpur.2019.06.006"},{"key":"27","doi-asserted-by":"publisher","DOI":"10.1088\/1361-6420\/ab762c"},{"key":"28","doi-asserted-by":"publisher","DOI":"10.11948\/20180279"},{"key":"29","doi-asserted-by":"publisher","DOI":"10.1016\/j.aml.2020.106554"},{"key":"30","doi-asserted-by":"publisher","DOI":"10.1016\/j.aml.2020.106213"},{"key":"31","doi-asserted-by":"publisher","DOI":"10.1080\/00036811.2018.1469011"},{"key":"32","doi-asserted-by":"publisher","DOI":"10.1007\/s40314-020-1103-2"},{"key":"33","doi-asserted-by":"publisher","DOI":"10.1007\/s10915-021-01704-8"},{"key":"34","doi-asserted-by":"publisher","DOI":"10.1016\/j.cnsns.2022.106590"}],"container-title":["Journal of Sensors"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/downloads.hindawi.com\/journals\/js\/2022\/7984688.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/downloads.hindawi.com\/journals\/js\/2022\/7984688.xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/downloads.hindawi.com\/journals\/js\/2022\/7984688.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,8,26]],"date-time":"2022-08-26T22:35:18Z","timestamp":1661553318000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.hindawi.com\/journals\/js\/2022\/7984688\/"}},"subtitle":[],"editor":[{"given":"Mohamed","family":"Louzazni","sequence":"additional","affiliation":[],"role":[{"role":"editor","vocabulary":"crossref"}]}],"short-title":[],"issued":{"date-parts":[[2022,8,26]]},"references-count":34,"alternative-id":["7984688","7984688"],"URL":"https:\/\/doi.org\/10.1155\/2022\/7984688","relation":{},"ISSN":["1687-7268","1687-725X"],"issn-type":[{"type":"electronic","value":"1687-7268"},{"type":"print","value":"1687-725X"}],"subject":[],"published":{"date-parts":[[2022,8,26]]}}}