{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,21]],"date-time":"2026-01-21T16:02:18Z","timestamp":1769011338353,"version":"3.49.0"},"reference-count":42,"publisher":"Springer Science and Business Media LLC","issue":"6","license":[{"start":{"date-parts":[[2025,5,14]],"date-time":"2025-05-14T00:00:00Z","timestamp":1747180800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2025,5,14]],"date-time":"2025-05-14T00:00:00Z","timestamp":1747180800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Comp. Appl. Math."],"published-print":{"date-parts":[[2025,9]]},"DOI":"10.1007\/s40314-025-03246-3","type":"journal-article","created":{"date-parts":[[2025,5,13]],"date-time":"2025-05-13T21:31:56Z","timestamp":1747171916000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["Analysis of a time-dependent source function for the heat equation with nonlocal boundary conditions through a local meshless procedure"],"prefix":"10.1007","volume":"44","author":[{"given":"Elyas","family":"Shivanian","sequence":"first","affiliation":[]},{"given":"Ahmad","family":"Jafarabadi","sequence":"additional","affiliation":[]},{"given":"Tannaz G.","family":"Chegini","sequence":"additional","affiliation":[]},{"given":"Abdollah","family":"Dinmohammadi","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2025,5,14]]},"reference":[{"issue":"7","key":"3246_CR1","doi-asserted-by":"publisher","first-page":"5465","DOI":"10.1007\/s00521-022-07988-7","volume":"35","author":"M Aller","year":"2023","unstructured":"Aller M, Mera D, Cotos JM, Villaroya S (2023) Study and comparison of different machine learning-based approaches to solve the inverse problem in electrical impedance tomographies. Neural Comput Appl 35(7):5465\u20135477","journal-title":"Neural Comput Appl"},{"key":"3246_CR2","doi-asserted-by":"publisher","first-page":"278","DOI":"10.1016\/j.enganabound.2015.12.016","volume":"64","author":"M Amirfakhrian","year":"2016","unstructured":"Amirfakhrian M, Arghand M, Kansa EJ (2016) A new approximate method for an inverse time-dependent heat source problem using fundamental solutions and rbfs. Eng Anal Bound Elem 64:278\u2013289","journal-title":"Eng Anal Bound Elem"},{"key":"3246_CR3","doi-asserted-by":"publisher","DOI":"10.1016\/j.measurement.2024.116599","volume":"246","author":"A Aouali","year":"2025","unstructured":"Aouali A, Messal O, Benabou A, Sommier A, Saleh L, Bastsale J-C (2025) Iron loss evaluation using infrared thermography imaging and inverse problem resolution. Measurement 246:116599","journal-title":"Measurement"},{"key":"3246_CR4","doi-asserted-by":"publisher","DOI":"10.1201\/b16760","volume-title":"Modeling and inverse problems in the presence of uncertainty","author":"HT Banks","year":"2014","unstructured":"Banks HT, Hu S, Thompson WC (2014) Modeling and inverse problems in the presence of uncertainty. CRC Press, Boca Raton"},{"key":"3246_CR5","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1017\/S0962492900000015","volume":"9","author":"MD Buhmann","year":"2000","unstructured":"Buhmann MD (2000) Radial basis functions. Acta Numer 9:1\u201338","journal-title":"Acta Numer"},{"key":"3246_CR6","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9781139086967","volume-title":"The one-dimensional heat equation","author":"JR Cannon","year":"1984","unstructured":"Cannon JR (1984) The one-dimensional heat equation, 23rd edn. Cambridge University Press, Cambridge","edition":"23"},{"issue":"3","key":"3246_CR7","doi-asserted-by":"publisher","first-page":"535","DOI":"10.1088\/0266-5611\/14\/3\/010","volume":"14","author":"JR Cannon","year":"1998","unstructured":"Cannon JR, DuChateau P (1998) Structural identification of an unknown source term in a heat equation. Inverse Prob 14(3):535","journal-title":"Inverse Prob"},{"issue":"2","key":"3246_CR8","doi-asserted-by":"publisher","first-page":"517","DOI":"10.1016\/0022-247X(86)90012-0","volume":"115","author":"JR Cannon","year":"1986","unstructured":"Cannon JR, van der Hoek J (1986) Diffusion subject to the specification of mass. J Math Anal Appl 115(2):517\u2013529","journal-title":"J Math Anal Appl"},{"issue":"4","key":"3246_CR9","doi-asserted-by":"publisher","first-page":"772","DOI":"10.1016\/j.cpc.2009.12.010","volume":"181","author":"M Dehghan","year":"2010","unstructured":"Dehghan M, Ghesmati A (2010) Numerical simulation of two-dimensional sine-gordon solitons via a local weak meshless technique based on the radial point interpolation method (rpim). Comput Phys Commun 181(4):772\u2013786","journal-title":"Comput Phys Commun"},{"issue":"6","key":"3246_CR10","doi-asserted-by":"publisher","first-page":"284","DOI":"10.1007\/s40314-023-02414-7","volume":"42","author":"A Dinmohammadi","year":"2023","unstructured":"Dinmohammadi A, Jafarabadi A (2023) Inverse heat conduction problem with a nonlinear source term by a local strong form of meshless technique based on radial point interpolation method. Comput Appl Math 42(6):284","journal-title":"Comput Appl Math"},{"issue":"1","key":"3246_CR11","doi-asserted-by":"publisher","first-page":"37","DOI":"10.1007\/s10851-023-01162-x","volume":"66","author":"M Duff","year":"2024","unstructured":"Duff M, Campbell NDF, Ehrhardt MJ (2024) Regularising inverse problems with generative machine learning models. J Math Imaging Vis 66(1):37\u201356","journal-title":"J Math Imaging Vis"},{"issue":"1","key":"3246_CR12","first-page":"9655008","volume":"2024","author":"D Gemechu","year":"2024","unstructured":"Gemechu D (2024) Sparse regularization based on orthogonal tensor dictionary learning for inverse problems. Math Probl Eng 2024(1):9655008","journal-title":"Math Probl Eng"},{"key":"3246_CR13","unstructured":"Hansen PC (1999) The l-curve and its use in the numerical treatment of inverse problems"},{"key":"3246_CR14","doi-asserted-by":"publisher","DOI":"10.1137\/1.9780898718836","volume-title":"Discrete inverse problems: insight and algorithms","author":"Hansen Per Christian","year":"2010","unstructured":"Hansen Per Christian (2010) Discrete inverse problems: insight and algorithms. SIAM, Singapore"},{"issue":"2","key":"3246_CR15","doi-asserted-by":"publisher","first-page":"189","DOI":"10.1007\/s11075-007-9136-9","volume":"46","author":"PC Hansen","year":"2007","unstructured":"Hansen PC (2007) Regularization tools version 4.0 for matlab 7.3. Numer Algor 46(2):189\u2013194","journal-title":"Numer Algor"},{"key":"3246_CR16","doi-asserted-by":"publisher","first-page":"13","DOI":"10.1016\/j.apnum.2013.02.004","volume":"69","author":"A Hazanee","year":"2013","unstructured":"Hazanee A, Ismailov MI, Lesnic D, Kerimov NB (2013) An inverse time-dependent source problem for the heat equation. Appl Numer Math 69:13\u201333","journal-title":"Appl Numer Math"},{"issue":"20","key":"3246_CR17","doi-asserted-by":"publisher","first-page":"6258","DOI":"10.1016\/j.apm.2015.01.058","volume":"39","author":"A Hazanee","year":"2015","unstructured":"Hazanee A, Lesnic D, Ismailov MI, Kerimov NB (2015) An inverse time-dependent source problem for the heat equation with a non-classical boundary condition. Appl Math Model 39(20):6258\u20136272","journal-title":"Appl Math Model"},{"key":"3246_CR18","first-page":"800","volume":"346","author":"A Hazanee","year":"2019","unstructured":"Hazanee A, Lesnic D, Ismailov MI, Kerimov NB (2019) Inverse time-dependent source problems for the heat equation with nonlocal boundary conditions. Appl Math Comput 346:800\u2013815","journal-title":"Appl Math Comput"},{"issue":"5","key":"3246_CR19","doi-asserted-by":"publisher","first-page":"489","DOI":"10.1016\/S0955-7997(03)00102-4","volume":"28","author":"YC Hon","year":"2004","unstructured":"Hon YC, Wei T (2004) A fundamental solution method for inverse heat conduction problem. Eng Anal Bound Elem 28(5):489\u2013495","journal-title":"Eng Anal Bound Elem"},{"issue":"2","key":"3246_CR20","first-page":"294","volume":"13","author":"NI Ionkin","year":"1977","unstructured":"Ionkin NI (1977) The solution of a certain boundary value problem of the theory of heat conduction with a nonclassical boundary condition. Differ Uravn 13(2):294\u2013304","journal-title":"Differ Uravn"},{"issue":"8","key":"3246_CR21","first-page":"4138","volume":"218","author":"MI Ismailov","year":"2011","unstructured":"Ismailov MI, Kanca F, Lesnic D (2011) Determination of a time-dependent heat source under nonlocal boundary and integral overdetermination conditions. Appl Math Comput 218(8):4138\u20134146","journal-title":"Appl Math Comput"},{"key":"3246_CR22","volume-title":"Statistical and computational inverse problems","author":"J Kaipio","year":"2006","unstructured":"Kaipio J, Somersalo E (2006) Statistical and computational inverse problems, vol 160. Springer Science & Business Media, Berlin"},{"key":"3246_CR23","doi-asserted-by":"publisher","DOI":"10.7717\/peerj-cs.951","volume":"8","author":"S Kamyab","year":"2022","unstructured":"Kamyab S, Azimifar Z, Sabzi R, Fieguth P (2022) Deep learning methods for inverse problems. PeerJ Comput Sci 8:e951","journal-title":"PeerJ Comput Sci"},{"key":"3246_CR24","doi-asserted-by":"publisher","first-page":"2303","DOI":"10.1016\/j.ijheatmasstransfer.2017.01.086","volume":"108","author":"SG Kandlikar","year":"2017","unstructured":"Kandlikar SG, Perez-Raya I, Raghupathi PA, Gonzalez-Hernandez J-L, Dabydeen D, Medeiros L, Phatak P (2017) Infrared imaging technology for breast cancer detection-current status, protocols and new directions. Int J Heat Mass Transf 108:2303\u20132320","journal-title":"Int J Heat Mass Transf"},{"issue":"12","key":"3246_CR25","doi-asserted-by":"publisher","first-page":"894","DOI":"10.1190\/tle37120894.1","volume":"37","author":"Y Kim","year":"2018","unstructured":"Kim Y, Nakata N (2018) Geophysical inversion versus machine learning in inverse problems. Lead Edge 37(12):894\u2013901","journal-title":"Lead Edge"},{"issue":"2","key":"3246_CR26","doi-asserted-by":"publisher","first-page":"157","DOI":"10.2478\/v10006-007-0014-3","volume":"17","author":"D Krawczyk-Sta\u0144Do","year":"2007","unstructured":"Krawczyk-Sta\u0144Do D, Rudnicki M (2007) Regularization parameter selection in discrete ill-posed problems-the use of the U-curve. Int J Appl Math Comput Sci 17(2):157\u2013164","journal-title":"Int J Appl Math Comput Sci"},{"issue":"6","key":"3246_CR27","doi-asserted-by":"publisher","first-page":"713","DOI":"10.12989\/sem.2002.14.6.713","volume":"14","author":"GR Liu","year":"2002","unstructured":"Liu GR, Yan L, Wang JG, Gu YT (2002) Point interpolation method based on local residual formulation using radial basis functions. Struct Eng Mech 14(6):713\u2013732","journal-title":"Struct Eng Mech"},{"key":"3246_CR28","doi-asserted-by":"crossref","unstructured":"Maass P (2019) Deep learning for trivial inverse problems. In: Compressed sensing and its applications: Third International MATHEON Conference 2017, pages 195\u2013209. Springer","DOI":"10.1007\/978-3-319-73074-5_6"},{"key":"3246_CR29","unstructured":"Morozov VA (1966) On the solution of functional equations by the method of regularization. In Doklady Akademii Nauk, volume 167, pages 510\u2013512. Russian Academy of Sciences"},{"key":"3246_CR30","unstructured":"Nakhushev AM (1995) Equations of mathematical biology. Vysshaya Shkola, Moscow 1:995"},{"key":"3246_CR31","doi-asserted-by":"crossref","unstructured":"Salim C, Mariyanto M, Bohal YM, Sihombing HL (2020) Application of tikhonov regularization for 1d geothermal heat flux ill-posed inverse problem: A case study on chad sedimentary basin. In: IOP Conference Series: Earth and Environmental Science, volume 417, page 012003. IOP Publishing","DOI":"10.1088\/1755-1315\/417\/1\/012003"},{"issue":"3","key":"3246_CR32","doi-asserted-by":"publisher","first-page":"431","DOI":"10.1007\/s00366-016-0482-x","volume":"33","author":"E Shivanian","year":"2017","unstructured":"Shivanian E, Jafarabadi A (2017) Inverse Cauchy problem of annulus domains in the framework of spectral meshless radial point interpolation. Eng Comput 33(3):431\u2013442","journal-title":"Eng Comput"},{"issue":"12","key":"3246_CR33","doi-asserted-by":"publisher","first-page":"1743","DOI":"10.1080\/17415977.2017.1289194","volume":"25","author":"E Shivanian","year":"2017","unstructured":"Shivanian E, Jafarabadi A (2017) Numerical solution of two-dimensional inverse force function in the wave equation with nonlocal boundary conditions. Inverse Probl Sci Eng 25(12):1743\u20131767","journal-title":"Inverse Probl Sci Eng"},{"key":"3246_CR34","doi-asserted-by":"publisher","first-page":"50","DOI":"10.1016\/j.enganabound.2018.03.006","volume":"91","author":"E Shivanian","year":"2018","unstructured":"Shivanian E, Jafarabadi A (2018) The numerical solution for the time-fractional inverse problem of diffusion equation. Eng Anal Bound Elem 91:50\u201359","journal-title":"Eng Anal Bound Elem"},{"key":"3246_CR35","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1016\/j.apnum.2018.02.008","volume":"129","author":"E Shivanian","year":"2018","unstructured":"Shivanian E, Jafarabadi A (2018) The spectral meshless radial point interpolation method for solving an inverse source problem of the time-fractional diffusion equation. Appl Numer Math 129:1\u201325","journal-title":"Appl Numer Math"},{"issue":"12","key":"3246_CR36","doi-asserted-by":"publisher","first-page":"2264","DOI":"10.3390\/cancers16122264","volume":"16","author":"N Sritharan","year":"2024","unstructured":"Sritharan N, Gutierrez C, Perez-Raya I, Gonzalez-Hernandez J-L, Owens A, Dabydeen D, Medeiros L, Kandlikar S, Phatak P (2024) Breast cancer screening using inverse modeling of surface temperatures and steady-state thermal imaging. Cancers 16(12):2264","journal-title":"Cancers"},{"issue":"2","key":"3246_CR37","doi-asserted-by":"publisher","first-page":"266","DOI":"10.1137\/1021044","volume":"21","author":"RA Willoughby","year":"1979","unstructured":"Willoughby RA (1979) Solutions of ill-posed problems (an tikhonov and vy arsenin). SIAM Rev 21(2):266","journal-title":"SIAM Rev"},{"issue":"3","key":"3246_CR38","doi-asserted-by":"publisher","first-page":"216","DOI":"10.1016\/j.enganabound.2007.08.002","volume":"32","author":"L Yan","year":"2008","unstructured":"Yan L, Chu-Li F, Yang F-L (2008) The method of fundamental solutions for the inverse heat source problem. Eng Anal Bound Elem 32(3):216\u2013222","journal-title":"Eng Anal Bound Elem"},{"key":"3246_CR39","doi-asserted-by":"publisher","first-page":"185","DOI":"10.1016\/j.ijheatmasstransfer.2019.01.002","volume":"134","author":"H Yang Zeng","year":"2019","unstructured":"Yang Zeng H, Wang SZ, Cai Y, Li E (2019) A novel adaptive approximate bayesian computation method for inverse heat conduction problem. Int J Heat Mass Transf 134:185\u2013197","journal-title":"Int J Heat Mass Transf"},{"issue":"5\u20136","key":"3246_CR40","doi-asserted-by":"publisher","first-page":"701","DOI":"10.1515\/jip-2012-0020","volume":"20","author":"SA Yousefi","year":"2012","unstructured":"Yousefi SA, Lesnic D, Barikbin Z (2012) Satisfier function in ritz-galerkin method for the identification of a time-dependent diffusivity. J Inverse Ill-Posed Prbl 20(5\u20136):701\u2013722","journal-title":"J Inverse Ill-Posed Prbl"},{"key":"3246_CR41","doi-asserted-by":"publisher","DOI":"10.1016\/j.ijheatmasstransfer.2019.07.080","volume":"142","author":"L Zhong-Rong","year":"2019","unstructured":"Zhong-Rong L, Pan T, Wang L (2019) A sparse regularization approach to inverse heat source identification. Int J Heat Mass Transf 142:118430","journal-title":"Int J Heat Mass Transf"},{"issue":"10","key":"3246_CR42","doi-asserted-by":"publisher","first-page":"1741","DOI":"10.1080\/00036811.2018.1434512","volume":"98","author":"J Zhou","year":"2019","unstructured":"Zhou J, Li H, Yongzhi X (2019) Ritz-galerkin method for solving an inverse problem of parabolic equation with moving boundaries and integral condition. Appl Anal 98(10):1741\u20131755","journal-title":"Appl Anal"}],"container-title":["Computational and Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s40314-025-03246-3.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s40314-025-03246-3\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s40314-025-03246-3.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,9,6]],"date-time":"2025-09-06T14:52:49Z","timestamp":1757170369000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s40314-025-03246-3"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,5,14]]},"references-count":42,"journal-issue":{"issue":"6","published-print":{"date-parts":[[2025,9]]}},"alternative-id":["3246"],"URL":"https:\/\/doi.org\/10.1007\/s40314-025-03246-3","relation":{},"ISSN":["2238-3603","1807-0302"],"issn-type":[{"value":"2238-3603","type":"print"},{"value":"1807-0302","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,5,14]]},"assertion":[{"value":"13 January 2025","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"10 April 2025","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"28 April 2025","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"14 May 2025","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors declare that they have no Conflict of interest.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}],"article-number":"282"}}