{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,5,12]],"date-time":"2025-05-12T15:10:05Z","timestamp":1747062605533,"version":"3.40.5"},"reference-count":28,"publisher":"Springer Science and Business Media LLC","issue":"6","license":[{"start":{"date-parts":[[2025,3,1]],"date-time":"2025-03-01T00:00:00Z","timestamp":1740787200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2025,3,1]],"date-time":"2025-03-01T00:00:00Z","timestamp":1740787200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"funder":[{"name":"Natural Science Foundation of Shaanxi Province of China","award":["2022JQ-042"],"award-info":[{"award-number":["2022JQ-042"]}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Soft Comput"],"published-print":{"date-parts":[[2025,3]]},"DOI":"10.1007\/s00500-025-10560-9","type":"journal-article","created":{"date-parts":[[2025,4,29]],"date-time":"2025-04-29T08:31:49Z","timestamp":1745915509000},"page":"2653-2668","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Applications of integrals and extreme value of solution of uncertain age-dependent population equation with migration source"],"prefix":"10.1007","volume":"29","author":[{"given":"Lu","family":"Yang","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2025,4,29]]},"reference":[{"issue":"1","key":"10560_CR1","doi-asserted-by":"publisher","first-page":"69","DOI":"10.1007\/s10700-010-9073-2","volume":"9","author":"X Chen","year":"2010","unstructured":"Chen X, Liu B (2010) Existence and uniqueness theorem for uncertain differential equations. Fuzzy Optim Decis Making 9(1):69\u201381","journal-title":"Fuzzy Optim Decis Making"},{"issue":"2","key":"10560_CR2","doi-asserted-by":"publisher","first-page":"189","DOI":"10.1007\/s10700-020-09342-9","volume":"20","author":"X Chen","year":"2021","unstructured":"Chen X, Li J, Xiao C, Yang P (2021) Numerical solution and parameter estimation for uncertain SIR model with application to COVID-19. Fuzzy Optim Decis Making 20(2):189\u2013208","journal-title":"Fuzzy Optim Decis Making"},{"issue":"7","key":"10560_CR3","doi-asserted-by":"publisher","first-page":"1323","DOI":"10.1109\/TFUZZ.2018.2877628","volume":"27","author":"R Gao","year":"2019","unstructured":"Gao R, Ralescu D (2019) Uncertain wave equation for vibrating string. IEEE Trans Fuzzy Syst 27(7):1323\u20131331","journal-title":"IEEE Trans Fuzzy Syst"},{"key":"10560_CR4","doi-asserted-by":"publisher","DOI":"10.1007\/s10700-023-09408-4","author":"L He","year":"2023","unstructured":"He L, Zhu Y, Gu Y (2023) Nonparametric estimation for uncertain differential equations. Fuzzy Optim Decis Making. https:\/\/doi.org\/10.1007\/s10700-023-09408-4","journal-title":"Fuzzy Optim Decis Making"},{"issue":"2","key":"10560_CR5","doi-asserted-by":"publisher","first-page":"243","DOI":"10.1007\/s10700-020-09341-w","volume":"20","author":"L Jia","year":"2021","unstructured":"Jia L, Chen W (2021) Uncertain SEIAR model for COVID-19 cases in China. Fuzzy Optim Decis Making 20(2):243\u2013259","journal-title":"Fuzzy Optim Decis Making"},{"issue":"2","key":"10560_CR6","doi-asserted-by":"publisher","first-page":"177","DOI":"10.1007\/s10700-020-09337-6","volume":"20","author":"W Lio","year":"2021","unstructured":"Lio W, Liu B (2021) Initial value estimation of uncertain differential equations and zero-day of COVID-19 spread in China. Fuzzy Optim Decis Making 20(2):177\u2013188","journal-title":"Fuzzy Optim Decis Making"},{"key":"10560_CR7","volume-title":"Uncertainty theory","author":"B Liu","year":"2007","unstructured":"Liu B (2007) Uncertainty theory, 2nd edn. Springer, Berlin","edition":"2"},{"key":"10560_CR8","unstructured":"Liu B (2008) Fuzzy process, hybrid process and uncertain process. J Uncertain Syst 2(1):3\u201316"},{"issue":"1","key":"10560_CR9","first-page":"3","volume":"3","author":"B Liu","year":"2009","unstructured":"Liu B (2009) Some research problems in uncertainty theory. J Uncertain Syst 3(1):3\u201310","journal-title":"J Uncertain Syst"},{"issue":"4","key":"10560_CR10","doi-asserted-by":"publisher","first-page":"2243001","DOI":"10.1142\/S1752890922430012","volume":"15","author":"Y Liu","year":"2022","unstructured":"Liu Y (2022) Analysis of China\u2019s population with uncertain statistics. J Uncertain Syst 15(4):2243001","journal-title":"J Uncertain Syst"},{"issue":"6","key":"10560_CR11","doi-asserted-by":"publisher","first-page":"2773","DOI":"10.1007\/s00500-022-06766-w","volume":"26","author":"Y Liu","year":"2022","unstructured":"Liu Y, Liu B (2022) Estimating unknown parameters in uncertain differential equation by maximum likelihood estimation. Soft Comput 26(6):2773\u20132780","journal-title":"Soft Comput"},{"key":"10560_CR12","doi-asserted-by":"crossref","unstructured":"Liu Y, Liu B (2022) Residual analysis and parameter estimation of uncertain differential equation. Fuzzy Optim Decis Making 21(4):513\u2013530","DOI":"10.1007\/s10700-021-09379-4"},{"key":"10560_CR13","doi-asserted-by":"publisher","DOI":"10.1080\/03610926.2023.2269451","author":"Y Liu","year":"2024","unstructured":"Liu Y, Liu B (2024) Estimation of uncertainty distribution function by the principle of least squares. Commun Stat Theory Methods. https:\/\/doi.org\/10.1080\/03610926.2023.2269451","journal-title":"Commun Stat Theory Methods"},{"key":"10560_CR14","doi-asserted-by":"publisher","unstructured":"Liu Y, Liu B (2024) A modified uncertain maximum likelihood estimation with applications in uncertain statistics. Commun Stat Theory Methods. https:\/\/doi.org\/10.1080\/03610926.2023.2248534","DOI":"10.1080\/03610926.2023.2248534"},{"issue":"2","key":"10560_CR15","doi-asserted-by":"publisher","first-page":"1881","DOI":"10.1016\/j.apm.2020.08.061","volume":"89","author":"Z Liu","year":"2021","unstructured":"Liu Z, Yang X (2021) A linear uncertain pharmacokinetic model driven by Liu process. Appl Math Model 89(2):1881\u20131899","journal-title":"Appl Math Model"},{"issue":"1","key":"10560_CR16","doi-asserted-by":"publisher","first-page":"123","DOI":"10.1007\/s10700-022-09386-z","volume":"22","author":"F Mehrdoust","year":"2023","unstructured":"Mehrdoust F, Noorani I, Xu W (2023) Uncertain energy model for electricity and gas futures with application in spark-spread option price. Fuzzy Optim Decis Making 22(1):123\u2013148","journal-title":"Fuzzy Optim Decis Making"},{"issue":"2","key":"10560_CR17","first-page":"1","volume":"33","author":"Y Sheng","year":"2017","unstructured":"Sheng Y, Gao R, Zhang Z (2017) Uncertain population model with age-structure. J Intell Fuzzy Syst 33(2):1\u20136","journal-title":"J Intell Fuzzy Syst"},{"issue":"6","key":"10560_CR18","doi-asserted-by":"publisher","first-page":"7411","DOI":"10.1007\/s12652-022-04448-x","volume":"14","author":"L Yang","year":"2023","unstructured":"Yang L (2023) Uncertain age-dependent population equation with migration source. J Ambient Intell Humaniz Comput 14(6):7411\u20137425","journal-title":"J Ambient Intell Humaniz Comput"},{"key":"10560_CR19","doi-asserted-by":"publisher","DOI":"10.1007\/s10700-023-09415-5","author":"L Yang","year":"2024","unstructured":"Yang L, Liu Y (2024) Solution method and parameter estimation of uncertain partial differential equation with application to China\u2019s population. Fuzzy Optim Decis Making. https:\/\/doi.org\/10.1007\/s10700-023-09415-5","journal-title":"Fuzzy Optim Decis Making"},{"issue":"3","key":"10560_CR20","doi-asserted-by":"publisher","first-page":"383","DOI":"10.1007\/s10700-021-09370-z","volume":"21","author":"L Yang","year":"2022","unstructured":"Yang L, Ye T, Yang H (2022) Uncertain seepage equation in fissured porous media. Fuzzy Optim Decis Making 21(3):383\u2013403","journal-title":"Fuzzy Optim Decis Making"},{"issue":"3","key":"10560_CR21","doi-asserted-by":"publisher","first-page":"447","DOI":"10.1007\/s10700-022-09399-8","volume":"22","author":"X Yang","year":"2023","unstructured":"Yang X, Ke H (2023) Uncertain interest rate model for Shanghai interbank offered rate and pricing of American swaption. Fuzzy Optim Decis Making 22(3):447\u2013462","journal-title":"Fuzzy Optim Decis Making"},{"issue":"5","key":"10560_CR22","doi-asserted-by":"publisher","first-page":"717","DOI":"10.1007\/s12652-017-0479-3","volume":"8","author":"X Yang","year":"2017","unstructured":"Yang X, Ni Y (2017) Existence and uniqueness theorem for uncertain heat equation. J Ambient Intell Humaniz Comput 8(5):717\u2013725","journal-title":"J Ambient Intell Humaniz Comput"},{"issue":"3","key":"10560_CR23","doi-asserted-by":"publisher","first-page":"379","DOI":"10.1007\/s10700-016-9253-9","volume":"16","author":"X Yang","year":"2017","unstructured":"Yang X, Yao K (2017) Uncertain partial differential equation with application to heat conduction. Fuzzy Optim Decis Making 16(3):379\u2013403","journal-title":"Fuzzy Optim Decis Making"},{"issue":"3","key":"10560_CR24","first-page":"825","volume":"25","author":"K Yao","year":"2013","unstructured":"Yao K, Chen X (2013) A numerical method for solving uncertain differential equations. J Intell Fuzzy Syst 25(3):825\u2013832","journal-title":"J Intell Fuzzy Syst"},{"issue":"2","key":"10560_CR25","doi-asserted-by":"publisher","first-page":"195","DOI":"10.1007\/s10700-022-09389-w","volume":"22","author":"T Ye","year":"2023","unstructured":"Ye T, Liu B (2023) Uncertain hypothesis test for uncertain differential equations. Fuzzy Optim Decis Making 22(2):195\u2013211","journal-title":"Fuzzy Optim Decis Making"},{"issue":"9","key":"10560_CR26","doi-asserted-by":"publisher","first-page":"5277","DOI":"10.1007\/s00500-023-07892-9","volume":"27","author":"T Ye","year":"2023","unstructured":"Ye T (2023) Applications of three-dimensional uncertain heat equations. Soft Comput 27(9):5277\u20135292","journal-title":"Soft Comput"},{"key":"10560_CR27","doi-asserted-by":"publisher","DOI":"10.1007\/s10700-023-09417-3","author":"T Ye","year":"2024","unstructured":"Ye T (2024) Partial derivatives of uncertain fields and uncertain partial differential equations. Fuzzy Optim Decis Making. https:\/\/doi.org\/10.1007\/s10700-023-09417-3","journal-title":"Fuzzy Optim Decis Making"},{"key":"10560_CR28","doi-asserted-by":"publisher","DOI":"10.1007\/s10700-023-09418-2","author":"Y Zhu","year":"2024","unstructured":"Zhu Y (2024) On uncertain partial differential equations. Fuzzy Optim Decis Making. https:\/\/doi.org\/10.1007\/s10700-023-09418-2","journal-title":"Fuzzy Optim Decis Making"}],"container-title":["Soft Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00500-025-10560-9.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s00500-025-10560-9\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00500-025-10560-9.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,5,12]],"date-time":"2025-05-12T14:28:32Z","timestamp":1747060112000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s00500-025-10560-9"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,3]]},"references-count":28,"journal-issue":{"issue":"6","published-print":{"date-parts":[[2025,3]]}},"alternative-id":["10560"],"URL":"https:\/\/doi.org\/10.1007\/s00500-025-10560-9","relation":{},"ISSN":["1432-7643","1433-7479"],"issn-type":[{"type":"print","value":"1432-7643"},{"type":"electronic","value":"1433-7479"}],"subject":[],"published":{"date-parts":[[2025,3]]},"assertion":[{"value":"26 January 2025","order":1,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"29 April 2025","order":2,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The author declares that she has no Conflict of interest.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}},{"value":"This article does not contain any studies with human participants or animals performed by the author.","order":3,"name":"Ethics","group":{"name":"EthicsHeading","label":"Ethical approval"}},{"value":"Informed consent was obtained from all individual participants included in the study.","order":4,"name":"Ethics","group":{"name":"EthicsHeading","label":"Informed consent"}}]}}