Abstract
This paper initializes higher-order uncertain calculus that deals with higher-order differentiation and multiple integration of uncertain process based on uncertainty theory. Fubini theorem and fundamental theorem of higher-order uncertain calculus are derived. Finally, this paper rigorously defines higher-order uncertain differential equations and introduces some analytic methods for solving these equations.
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References
Chen, X. W., & Ralescu, D. (2013). Liu process and uncertain calculus. Journal of Uncertainty Analysis and Applications, 1(1), 1–12.
Gao, R., & Ralescu, D. A. (2019). Uncertain wave equation for vibrating string. IEEE Transactions on Fuzzy Systems, 27(7), 1323–1331.
Jia, L. F., & Dai, W. (2019). Uncertain spring vibration equation. Journal of Industrial and Management Optimization, 18(4), 2401–2414.
Lio, W., & Liu, B. (2021). Initial value estimation of uncertain differential equations and zero-day of COVID-19 spread in China. Fuzzy Optimization and Decision Making, 20(2), 177–188.
Liu, B. (2007). Uncertainty theory (2nd ed.). Springer.
Liu, B. (2008). Fuzzy process, hybrid process and uncertain process. Journal of Uncertain Systems, 2(1), 3–16.
Liu, B. (2009). Some research problems in uncertainty theory. Journal of Uncertain Systems, 3, 3–10.
Liu, B. (2015). Uncertainty theory (4th ed.). Berlin: Springer.
Liu, Y., & Liu, B. (2022). Residual analysis and parameter estimation of uncertain differential equation. Fuzzy Optimization and Decision Making, 21(4), 513–530.
Liu, Y., & Liu, B. (2022). Estimating unknown parameters in uncertain differential equation by maximum likelihood estimation. Soft Computing, 26(6), 2773–2780.
Liu, Y., & Liu, B. (2023). Estimation of uncertainty distribution function by the principle of least squares. Communications in Statistics-Theory and Methods. https://doi.org/10.1080/03610926.2023.2269451
Liu, Y., & Liu, B. (2024). A modified uncertain maximum likelihood estimation with applications in uncertain statistics. Communications in Statistics-Theory and Methods. https://doi.org/10.1080/03610926.2023.2248534
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), 775–787.
Mehrdoust, F., Noorani, I., & Xu, W. (2023). Uncertain energy model for electricity and gas futures with application in spark-spread option price. Fuzzy Optimization and Decision Making, 22(1), 123–148.
Yang, X. F., & Ke, H. (2022). Uncertain interest rate model for Shanghai interbank offered rate and pricing of American swaption. Fuzzy Optimization and Decision Making, 22, 447–462.
Yang, L., & Liu, Y. (2023). Solution method and parameter estimation of uncertain partial differential equation with application to China’s population. Fuzzy Optimization and Decision Making. https://doi.org/10.1007/s10700-023-09415-5
Yang, X. F., & Yao, K. (2017). Uncertain partial differential equation with application to heat conduction. Fuzzy Optimization and Decision Making, 16(3), 379–403.
Yang, L., Ye, T. Q., & Yang, H. (2022). Uncertain seepage equation in fissured porous media. Fuzzy Optimization and Decision Making, 21(3), 383–403.
Yao, K. (2016). Uncertain differential equations. Springer.
Yao, K., & Chen, X. W. (2013). A numerical method for solving uncertain differential equations. Journal of Intelligent and Fuzzy Systems, 25(3), 825–832.
Yao, K., & Liu, B. (2020). Parameter estimation in uncertain differential equations. Fuzzy Optimization and Decision Making, 19(1), 1–12.
Ye, T. Q. (2021). A rigorous proof of fundamental theorem of uncertain calculus. Journal of Uncertain Systems, 14(2), 2150009.
Ye, T. Q. (2023). Partial derivatives of uncertain fields and uncertain partial differential equations. Fuzzy Optimization and Decision Making. https://doi.org/10.1007/s10700-023-09417-3
Ye, T. Q., & Liu, B. (2023). Uncertain hypothesis test for uncertain differential equations. Fuzzy Optimization and Decision Making, 22(2), 195–211.
Ye, T. Q., & Zheng, H. R. (2023). Analysis of birth rates in China with uncertain statistics. Journal of Intelligent and Fuzzy Systems, 44(6), 10621–10632.
Zhu, Y. G. (2023). On uncertain partial differential equations. Fuzzy Optimization and Decision Making. https://doi.org/10.1007/s10700-023-09418-2
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Zhang, K., Liu, B. Higher-order derivative of uncertain process and higher-order uncertain differential equation. Fuzzy Optim Decis Making 23, 295–318 (2024). https://doi.org/10.1007/s10700-024-09422-0
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DOI: https://doi.org/10.1007/s10700-024-09422-0