{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,8]],"date-time":"2026-01-08T07:11:28Z","timestamp":1767856288159,"version":"3.49.0"},"reference-count":26,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2023,1,6]],"date-time":"2023-01-06T00:00:00Z","timestamp":1672963200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Algorithms"],"abstract":"<jats:p>The boundary value problem, BVP, for the PDE heat equation is studied and explained in this article. The problem declaration comprises two intervals; the (0, T) is the first interval and labels the heating of the inside burning chamber, and the second (T, \u221e) interval defines the normal cooling of the chamber wall when the chamber temperature concurs with the ambient temperature. It is necessary to prove the boundary function of this problem has its place in the space H10,\u221e in order to successfully apply the Fourier transform method. The applicability of the Fourier transform for time to this problem is verified. The method of projection regularization is used to solve the inverse boundary value problem for the heat equation and to obtain an evaluation for the error between the approximate and the real solution. These results are new and of practical interest as shown in the numerical case study.<\/jats:p>","DOI":"10.3390\/a16010033","type":"journal-article","created":{"date-parts":[[2023,1,9]],"date-time":"2023-01-09T04:47:08Z","timestamp":1673239628000},"page":"33","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":21,"title":["Solving of the Inverse Boundary Value Problem for the Heat Conduction Equation in Two Intervals of Time"],"prefix":"10.3390","volume":"16","author":[{"given":"Bashar Talib","family":"Al-Nuaimi","sequence":"first","affiliation":[{"name":"Computer Science Department, University of Diyala, Diyala 32001, Iraq"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9845-6599","authenticated-orcid":false,"given":"H.K.","family":"Al-Mahdawi","sequence":"additional","affiliation":[{"name":"Electronic Computer Centre, University of Diyala, Diyala 32001, Iraq"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Zainalabideen","family":"Albadran","sequence":"additional","affiliation":[{"name":"Computer Technical Engineering Department, University of Alkafeel, Najaf 60061, Iraq"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0281-082X","authenticated-orcid":false,"given":"Hussein","family":"Alkattan","sequence":"additional","affiliation":[{"name":"Department of System Programming, South Ural State University, 454080 Chelyabinsk, Russia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3442-6865","authenticated-orcid":false,"given":"Mostafa","family":"Abotaleb","sequence":"additional","affiliation":[{"name":"Department of System Programming, South Ural State University, 454080 Chelyabinsk, Russia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9221-7658","authenticated-orcid":false,"given":"El-Sayed M.","family":"El-kenawy","sequence":"additional","affiliation":[{"name":"Department of Communications and Electronics, Delta Higher Institute of Engineering and Technology, Mansoura 35111, Egypt"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2023,1,6]]},"reference":[{"key":"ref_1","first-page":"49","article-title":"On the Regularization of Ill-Posed Problems","volume":"153","author":"Tikhonov","year":"1963","journal-title":"Proc. USSR Acad. Sci."},{"key":"ref_2","unstructured":"Lavrent\u2019ev, M.M. (1962). On Certain Ill-Posed Problems of Mathematical Physics, Sobolev Division of the USSR Academy Science."},{"key":"ref_3","unstructured":"Ivanov, V.K., Vasin, V.V., and Tanana, V.P. (1978). Theory of Linear Ill-Posed Problem and Application, Nauok."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"129","DOI":"10.1007\/BF02416528","article-title":"Numerical solution of the equation of heat conduction for preceding times","volume":"40","author":"John","year":"1955","journal-title":"Ann. di Mat. pura ed Appl."},{"key":"ref_5","first-page":"12","article-title":"Numerical method for the heat equation with Dirichlet and Neumann conditions","volume":"1","author":"Cheniguel","year":"2014","journal-title":"Proc. Int. MultiConference Eng. Comput. Sci."},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Alifanov, O.M., Artioukhine, E.A., and Rumyantsev, S.V. (1995). Extreme Methods for Solving Ill-Posed Problems with Applications to Inverse Heat Transfer Problems, Begell House.","DOI":"10.1615\/978-1-56700-038-2.0"},{"key":"ref_7","unstructured":"Bergman, T.L., Incropera, F.P., DeWitt, D.P., and Lavine, A.S. (2011). Fundamentals of Heat and Mass Transfer, John Wiley & Sons."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"141","DOI":"10.1007\/s10598-017-9352-7","article-title":"Numerical method for the inverse boundary-value problem of the heat equation","volume":"28","author":"Dmitriev","year":"2017","journal-title":"Comput. Math. Model."},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Sidikova, A.I. (2018, January 9\u201316). Approximate Solution of Inverse Boundary Value Problem for Heat Conduction Equation. Proceedings of the 2018 International Russian Automation Conference (RusAutoCon), Sochi, Russia.","DOI":"10.1109\/RUSAUTOCON.2018.8501766"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"70","DOI":"10.1134\/S1995423919010063","article-title":"A Study of an Inverse Boundary Value Problem for the Heat Conduction Equation","volume":"12","author":"Sidikova","year":"2019","journal-title":"Numer. Anal. Appl."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"12049","DOI":"10.1088\/1742-6596\/1715\/1\/012049","article-title":"The control problem for the heat equation in the case of a composite material","volume":"1715","author":"Tanana","year":"2021","journal-title":"J. Phys. Conf. Ser."},{"key":"ref_12","unstructured":"Makhtoumi, M. (2018). Numerical solutions of heat diffusion equation over one dimensional rod region. arXiv."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"60050","DOI":"10.1063\/5.0094343","article-title":"The solution of inverse boundary problem for the heat exchange for the hollow cylinder","volume":"2398","author":"Sidikova","year":"2022","journal-title":"AIP Conf. Proc."},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Al-Mahdawi, H.K.I., Abotaleb, M., Alkattan, H., Tareq, A.-M.Z., Badr, A., and Kadi, A. (2022). Multigrid Method for Solving Inverse Problems for Heat Equation. Mathematics, 10.","DOI":"10.3390\/math10152802"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"60080","DOI":"10.1063\/5.0093361","article-title":"Iterated Lavrent\u2019ev regularization with the finite-dimensional approximation for inverse problem","volume":"2398","author":"Sidikova","year":"2022","journal-title":"AIP Conf. Proc."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"101887","DOI":"10.1016\/j.mex.2022.101887","article-title":"Parallel Multigrid Method for Solving Inverse Problems","volume":"9","author":"Sidikova","year":"2022","journal-title":"MethodsX"},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Al-Mahdawi, H.K.I., Alkattan, H., Abotaleb, M., Kadi, A., and El-kenawy, E.-S.M. (2022). Updating the Landweber Iteration Method for Solving Inverse Problems. Mathematics, 10.","DOI":"10.3390\/math10152798"},{"key":"ref_18","first-page":"768","article-title":"An inverse problem of heat conductivity","volume":"19","author":"Glasko","year":"1979","journal-title":"Zhurnal Vychislitel\u2019noi Mat. I Mat. Fiz."},{"key":"ref_19","first-page":"22","article-title":"Continuation problem for the parabolic equation with the data on the part of the boundary","volume":"11","author":"Belonosov","year":"2014","journal-title":"Siber. Electron. Math. Rep."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"34","DOI":"10.1134\/S1995423908010047","article-title":"A gradient descent method for solving an inverse coefficient heat conduction problem","volume":"1","author":"Kabanikhin","year":"2008","journal-title":"Numer. Anal. Appl."},{"key":"ref_21","unstructured":"Yagola, A.G., Stepanova, I.E., Van, Y., and Titarenko, V.N. (2014). Obratnye zadachi i metody ikh resheniya. Prilozheniya k geofizike. Inverse Problems and Methods for their Solution: Applications to Geophysics, BKL Publishers."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"34","DOI":"10.1134\/S1995423913010047","article-title":"A numerical method for solving an inverse thermoacoustic problem","volume":"6","author":"Kabanikhin","year":"2013","journal-title":"Numer. Anal. Appl."},{"key":"ref_23","first-page":"117","article-title":"On the order-optimality of the projection regularization method in solving inverse problems","volume":"7","author":"Tanana","year":"2004","journal-title":"Sib. Zhurnal Ind. Mat."},{"key":"ref_24","unstructured":"Farlow, S.J. (1993). Partial Differential Equations for Scientists and Engineers, Courier Corporation."},{"key":"ref_25","first-page":"281","article-title":"On an error estimate for an approximate solution for an inverse problem in the class of piecewise smooth functions","volume":"18","author":"Tanana","year":"2012","journal-title":"Tr. Inst. Mat. I Mekhaniki UrO RAN"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"935","DOI":"10.1515\/jiip.2011.012","article-title":"The optimum of the M. M. Lavrent\u2019ev method","volume":"18","author":"Tanana","year":"2011","journal-title":"J. Inverse Ill-Posed Probl."}],"container-title":["Algorithms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1999-4893\/16\/1\/33\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T18:02:00Z","timestamp":1760119320000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1999-4893\/16\/1\/33"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,1,6]]},"references-count":26,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2023,1]]}},"alternative-id":["a16010033"],"URL":"https:\/\/doi.org\/10.3390\/a16010033","relation":{},"ISSN":["1999-4893"],"issn-type":[{"value":"1999-4893","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,1,6]]}}}