{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,23]],"date-time":"2026-07-23T02:15:23Z","timestamp":1784772923145,"version":"3.55.0"},"reference-count":31,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2026,2,12]],"date-time":"2026-02-12T00:00:00Z","timestamp":1770854400000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2026,2,12]],"date-time":"2026-02-12T00:00:00Z","timestamp":1770854400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["72101080"],"award-info":[{"award-number":["72101080"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Fuzzy Optim Decis Making"],"published-print":{"date-parts":[[2026,6]]},"DOI":"10.1007\/s10700-025-09472-y","type":"journal-article","created":{"date-parts":[[2026,2,12]],"date-time":"2026-02-12T12:22:30Z","timestamp":1770898950000},"page":"165-195","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Research on the reliability of uncertain wave equation models"],"prefix":"10.1007","volume":"25","author":[{"given":"Rong","family":"Gao","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Shengxue","family":"Wang","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Yan","family":"Li","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2026,2,12]]},"reference":[{"key":"9472_CR1","doi-asserted-by":"publisher","DOI":"10.1016\/j.chaos.2023.113604","volume":"173","author":"X Chen","year":"2023","unstructured":"Chen, X., Li, F. Z., Yuan, D. M., Wang, J., & Shao, Y. (2023). Optimal control problems subject to uncertain random discrete-time noncausal systems. Chaos, Solitons & Fractals, 173, Article 113604.","journal-title":"Chaos, Solitons & Fractals"},{"key":"9472_CR2","doi-asserted-by":"publisher","first-page":"599","DOI":"10.1007\/s10700-024-09435-9","volume":"23","author":"L Chen","year":"2024","unstructured":"Chen, L., Wang, Y. L., Peng, J., & Xiao, Q. Z. (2024). Supply chain management based on uncertainty theory: A bibliometric analysis and future prospects. Fuzzy Optimization and Decision Making, 23, 599\u2013636.","journal-title":"Fuzzy Optimization and Decision Making"},{"key":"9472_CR3","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1016\/j.matcom.2025.01.015","volume":"233","author":"X Chen","year":"2025","unstructured":"Chen, X., Zhang, Z. Y., & Huang, P. Q. (2025). Uncertain stochastic linear quadratic control subject to forward and backward multi-stage systems. Mathematics and Computers in Simulation, 233, 1\u201320.","journal-title":"Mathematics and Computers in Simulation"},{"key":"9472_CR4","doi-asserted-by":"publisher","first-page":"28","DOI":"10.1016\/j.amc.2016.12.003","volume":"304","author":"R Gao","year":"2017","unstructured":"Gao, R. (2017). Uncertain wave equation with infinite half-boundary. Applied Mathematics and Computation,304, 28\u201340.","journal-title":"Applied Mathematics and Computation"},{"key":"9472_CR5","doi-asserted-by":"publisher","DOI":"10.1016\/j.chaos.2023.113976","volume":"175","author":"R Gao","year":"2023","unstructured":"Gao, R., & Hua, K. X. (2023). A numerical method for solving uncertain wave equation. Chaos, Solitons & Fractals, 175, Article 113976.","journal-title":"Chaos, Solitons & Fractals"},{"key":"9472_CR6","doi-asserted-by":"publisher","DOI":"10.1016\/j.seps.2025.102203","volume":"99","author":"R Gao","year":"2025","unstructured":"Gao, R., Hua, K. X., Wang, X. S., & Wei, J. (2025). Analysis of green e-commerce supply chain considering delivery time under reward-penalty mechanism based on confidence level. Socio-Economic Planning Sciences, 99, Article 102203.","journal-title":"Socio-Economic Planning Sciences"},{"key":"9472_CR7","doi-asserted-by":"publisher","DOI":"10.1016\/j.cie.2024.110583","volume":"197","author":"R Gao","year":"2024","unstructured":"Gao, R., & Li, X. Y. (2024). Reliability modeling and analysis of uncertain competing failure systems. Computers & Industrial Engineering, 197, Article 110583.","journal-title":"Computers & Industrial Engineering"},{"key":"9472_CR8","first-page":"469","volume":"356","author":"R Gao","year":"2019","unstructured":"Gao, R., Ma, N. N., & Sun, G. J. (2019). Stability of solution for uncertain wave equation. Applied Mathematics and Computing,356, 469\u201378.","journal-title":"Applied Mathematics and Computing"},{"issue":"7","key":"9472_CR9","doi-asserted-by":"publisher","first-page":"1323","DOI":"10.1109\/TFUZZ.2018.2877628","volume":"27","author":"R Gao","year":"2019","unstructured":"Gao, R., & Ralescu, A. D. (2019). Uncertain wave equation for vibrating string. IEEE Transactions on Fuzzy Systems,27(7), 1323\u201331.","journal-title":"IEEE Transactions on Fuzzy Systems"},{"key":"9472_CR10","doi-asserted-by":"crossref","unstructured":"Gao, R., & Zhang, S. J. (2024). Computing the reliability of mixed uncertain random $$k$$-out-of-$$n$$ systems with multiple possible states. Journal of Computational and Applied Mathematics,438, Article 115567.","DOI":"10.1016\/j.cam.2023.115567"},{"key":"9472_CR11","doi-asserted-by":"crossref","unstructured":"Gao, R., & Zhang, S. J. (2024). Reliability importance analysis of uncertain random $$k$$-out-of-$$n$$ systems with multiple states. Reliability Engineering & System Safety,243, Article 109901.","DOI":"10.1016\/j.ress.2023.109901"},{"key":"9472_CR12","doi-asserted-by":"publisher","first-page":"403","DOI":"10.1016\/j.apm.2021.01.029","volume":"94","author":"XY Li","year":"2021","unstructured":"Li, X. Y., Chen, W. B., Li, F. R., & Kang, R. (2021). Reliability evaluation with limited and censored time-to-failure data based on uncertainty distributions. Applied Mathematical Modelling,94, 403\u201320.","journal-title":"Applied Mathematical Modelling"},{"key":"9472_CR13","doi-asserted-by":"publisher","first-page":"249","DOI":"10.1016\/j.ins.2022.05.022","volume":"604","author":"YY Li","year":"2022","unstructured":"Li, Y. Y., Chen, Y., Zhang, Q. Y., & Kang, R. (2022). Belief reliability analysis of multi-state deteriorating systems under epistemic uncertainty. Information Sciences,604, 249\u201366.","journal-title":"Information Sciences"},{"key":"9472_CR14","doi-asserted-by":"publisher","DOI":"10.1016\/j.chaos.2025.116330","volume":"195","author":"B Li","year":"2025","unstructured":"Li, B., Tao, Z. Y., & Shu, Y. D. (2025). Uncertain Bass diffusion model and modeling the purchase volume of private cargo vehicles in China. Chaos Solitons Fractals,195, Article 116330.","journal-title":"Chaos Solitons Fractals"},{"key":"9472_CR15","doi-asserted-by":"publisher","DOI":"10.1016\/j.engfailanal.2020.104931","volume":"119","author":"XY Li","year":"2021","unstructured":"Li, X. Y., Tao, Z., Wu, J. P., & Zhang, W. (2021). Uncertainty theory based reliability modeling for fatigue. Engineering Failure Analysis,119, Article 104931.","journal-title":"Engineering Failure Analysis"},{"issue":"1","key":"9472_CR16","first-page":"3","volume":"2","author":"BD Liu","year":"2008","unstructured":"Liu, B. D. (2008). Fuzzy process, hybrid process and uncertain process. Journal of Uncertain Systems,2(1), 3\u201316.","journal-title":"Journal of Uncertain Systems"},{"issue":"1","key":"9472_CR17","first-page":"3","volume":"3","author":"BD Liu","year":"2009","unstructured":"Liu, B. D. (2009). Some research problems in uncertainty theory. Journal of Uncertain Systems, 3(1), 3\u201310.","journal-title":"Journal of Uncertain Systems"},{"key":"9472_CR18","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-662-44354-5","volume-title":"Uncertainty Theory","author":"BD Liu","year":"2015","unstructured":"Liu, B. D. (2015). Uncertainty Theory (4th ed). Springer-Verlag.","edition":"4th ed"},{"issue":"2","key":"9472_CR19","doi-asserted-by":"publisher","first-page":"735","DOI":"10.1109\/TR.2022.3158336","volume":"71","author":"Z Liu","year":"2022","unstructured":"Liu, Z., & Kang, R. (2022). Imperfect debugging software belief reliability growth model based on uncertain differential equation. IEEE Transactions on Reliability,71(2), 735\u201346.","journal-title":"IEEE Transactions on Reliability"},{"key":"9472_CR20","doi-asserted-by":"publisher","DOI":"10.1016\/j.ress.2022.108641","volume":"225","author":"Z Liu","year":"2022","unstructured":"Liu, Z., Li, X. Y., & Kang, R. (2022). Uncertain differential equation based accelerated degradation modeling. Reliability Engineering & System Safety, 225, Article 108641.","journal-title":"Reliability Engineering & System Safety"},{"key":"9472_CR21","doi-asserted-by":"publisher","DOI":"10.1016\/j.chaos.2023.114293","volume":"178","author":"Y Liu","year":"2024","unstructured":"Liu, Y., & Lio, W. (2024). Power option pricing problem of uncertain exponential Ornstein-Uhlenbeck model. Chaos, Solitons & Fractals, 178, Article 114293.","journal-title":"Chaos, Solitons & Fractals"},{"key":"9472_CR22","doi-asserted-by":"publisher","DOI":"10.1016\/j.chaos.2023.114178","volume":"176","author":"Z Liu","year":"2023","unstructured":"Liu, Z., Wang, S. H., Liu, B., & Kang, R. (2023). Change point software belief reliability growth model considering epistemic uncertainties. Chaos, Solitons & Fractals, 176, Article 114178.","journal-title":"Chaos, Solitons & Fractals"},{"issue":"2","key":"9472_CR23","doi-asserted-by":"publisher","first-page":"775","DOI":"10.1109\/TR.2022.3154770","volume":"71","author":"Z Liu","year":"2022","unstructured":"Liu, Z., Yang, S. K., Yang, M. H., & Kang, R. (2022). Software belief reliability growth model based on uncertain differential equation. IEEE Transactions on Reliability, 71(2), 775\u201387.","journal-title":"IEEE Transactions on Reliability"},{"key":"9472_CR24","doi-asserted-by":"publisher","DOI":"10.1016\/j.amc.2025.129340","volume":"495","author":"Z Liu","year":"2025","unstructured":"Liu, Z., & Zhang, Q. Y. (2025). Remaining useful life estimation considering threshold epistemic uncertainty with uncertain differential equation. Applied Mathematics and Computation, 495, Article 129340.","journal-title":"Applied Mathematics and Computation"},{"key":"9472_CR25","doi-asserted-by":"publisher","first-page":"398","DOI":"10.1016\/j.apm.2019.05.036","volume":"75","author":"BL Liu","year":"2019","unstructured":"Liu, B. L., Zhang, Z. Q., & Wen, Y. Q. (2019). Reliability analysis for devices subject to competing failure processes based on chance theory. Applied Mathematical Modelling, 75, 398\u2013413.","journal-title":"Applied Mathematical Modelling"},{"issue":"8","key":"9472_CR26","doi-asserted-by":"publisher","first-page":"1532","DOI":"10.1109\/TFUZZ.2018.2883016","volume":"27","author":"XY Li","year":"2019","unstructured":"Li, X. Y., Wu, J. P., Liu, L., Wen, M. L., & Kang, R. (2019). Modeling accelerated degradation data based on the uncertain process. IEEE Transactions on Fuzzy Systems, 27(8), 1532\u201342.","journal-title":"IEEE Transactions on Fuzzy Systems"},{"key":"9472_CR27","doi-asserted-by":"publisher","first-page":"131","DOI":"10.1016\/j.camwa.2022.05.033","volume":"119","author":"CL Tian","year":"2022","unstructured":"Tian, C. L., Jin, T., Yang, X. F., & Liu, Q. Y. (2022). Reliability analysis of the uncertain heat conduction model. Computers & Mathematics with Applications, 119, 131\u201340.","journal-title":"Computers & Mathematics with Applications"},{"key":"9472_CR28","doi-asserted-by":"publisher","DOI":"10.1016\/j.ress.2020.106967","volume":"201","author":"JP Wu","year":"2020","unstructured":"Wu, J. P., Kang, R., & Li, X. Y. (2020). Uncertain accelerated degradation modeling and analysis considering epistemic uncertainties in time and unit dimension. Reliability Engineering & System Safety, 201, Article 106967.","journal-title":"Reliability Engineering & System Safety"},{"key":"9472_CR29","volume":"450","author":"QQ Xu","year":"2023","unstructured":"Xu, Q. Q., & Zhu, Y. G. (2023). Reliability analysis of uncertain random systems based on uncertain differential equation. Applied Mathematical Modelling, 450, Article 128008.","journal-title":"Applied Mathematical Modelling"},{"key":"9472_CR30","doi-asserted-by":"publisher","first-page":"26","DOI":"10.1016\/j.ins.2017.10.050","volume":"429","author":"ZG Zeng","year":"2018","unstructured":"Zeng, Z. G., Kang, R., Wen, M. L., & Zio, E. (2018). Uncertainty theory as a basis for belief reliability. Information Sciences, 429, 26\u201336.","journal-title":"Information Sciences"},{"issue":"1","key":"9472_CR31","doi-asserted-by":"publisher","first-page":"15","DOI":"10.1007\/s10700-012-9138-5","volume":"12","author":"ZG Zeng","year":"2013","unstructured":"Zeng, Z. G., Wen, M. L., & Kang, R. (2013). Belief reliability: A new metrics for products\u2019 reliability. Fuzzy Optimization and Decision Making, 12(1), 15\u201327.","journal-title":"Fuzzy Optimization and Decision Making"}],"container-title":["Fuzzy Optimization and Decision Making"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10700-025-09472-y.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10700-025-09472-y","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10700-025-09472-y.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,6,9]],"date-time":"2026-06-09T03:50:37Z","timestamp":1780977037000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10700-025-09472-y"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,2,12]]},"references-count":31,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2026,6]]}},"alternative-id":["9472"],"URL":"https:\/\/doi.org\/10.1007\/s10700-025-09472-y","relation":{},"ISSN":["1568-4539","1573-2908"],"issn-type":[{"value":"1568-4539","type":"print"},{"value":"1573-2908","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,2,12]]},"assertion":[{"value":"29 September 2025","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"23 December 2025","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"12 February 2026","order":3,"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 known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of Interest"}},{"value":"This work did not involve any active collection of human data.","order":3,"name":"Ethics","group":{"name":"EthicsHeading","label":"Ethics statement"}},{"value":"The authors declare no competing interests.","order":4,"name":"Ethics","group":{"name":"EthicsHeading","label":"Competing interests"}}]}}