The inverse uncertainty distribution of the solutions to a class of higher-order uncertain differential equations

Zeman Wang; Zhong Liu; Zikun Han; Xiuying Guo; Qiubao Wang; Zeman Wang; Zhong Liu; Zikun Han; Xiuying Guo; Qiubao Wang

doi:10.3934/math.20241579

AIMS Mathematics

2024, Volume 9, Issue 11: 33023-33061. doi: 10.3934/math.20241579

Previous Article Next Article

Research article

The inverse uncertainty distribution of the solutions to a class of higher-order uncertain differential equations

1.
Department of Mathematics and Physics, Shijiazhuang Tiedao University, Shijiazhuang 050043, China
2.
Department of Engineering Mechanics, Shijiazhuang Tiedao University, Shijiazhuang 050043, China

Received: 19 September 2024 Revised: 26 October 2024 Accepted: 06 November 2024 Published: 21 November 2024
MSC : 47E05, 34A12

In this paper, we study the higher-order uncertain differential equations (UDEs) as defined by Kaixi Zhang ^[11], mainly focus on the second-order case. We propose a pivotal condition (monotonicity in some sense, see more details in Section 3), introduce the concept of $ \alpha $-paths of UDEs, and demonstrate its properties. Based on this, we derive the inverse uncertainty distribution of the solution. Finally, we present numerical examples to substantiate the rationality of the condition.
- higher-order uncertain differential equations,
- inverse uncertainty distribution,
- $ \alpha $-paths
Citation: Zeman Wang, Zhong Liu, Zikun Han, Xiuying Guo, Qiubao Wang. The inverse uncertainty distribution of the solutions to a class of higher-order uncertain differential equations[J]. AIMS Mathematics, 2024, 9(11): 33023-33061. doi: 10.3934/math.20241579

Related Papers:

Abstract

In this paper, we study the higher-order uncertain differential equations (UDEs) as defined by Kaixi Zhang ^[11], mainly focus on the second-order case. We propose a pivotal condition (monotonicity in some sense, see more details in Section 3), introduce the concept of $ \alpha $-paths of UDEs, and demonstrate its properties. Based on this, we derive the inverse uncertainty distribution of the solution. Finally, we present numerical examples to substantiate the rationality of the condition.

References

[1]	B. Liu, Uncertainty theory, 2 Eds., Berlin: Springer Berlin Heidelberg, 2007.
[2]	K. Yao, X. Chen, A numerical method for solving uncertain differential equations, J. Intell. Fuzzy Syst., 25 (2013), 825–832. http://doi.org/10.3233/IFS-120688 doi: 10.3233/IFS-120688
[3]	B. Liu, Toward uncertain finance theory, J. Uncertain. Anal. Appl., 1 (2013), 1. https://doi.org/10.1186/2195-5468-1-1 doi: 10.1186/2195-5468-1-1
[4]	L. Sheng, Y. Zhu, Optimistic value model of uncertain optimal control, Int. J. Uncert. Fuzz. Knowledge-Based Syst., 21 (2013), 75–87. https://doi.org/10.1142/S0218488513400060 doi: 10.1142/S0218488513400060
[5]	Z. Zhang, X. Yang, Uncertain population model, Soft Comput., 24 (2020), 2417–2423. https://doi.org/10.1007/s00500-018-03678-6 doi: 10.1007/s00500-018-03678-6
[6]	Z. Liu, X. Yang, A linear uncertain pharmacokinetic model driven by Liu process, Appl. Math. Model., 89 (2021), 1881–1899. https://doi.org/10.1016/j.apm.2020.08.061 doi: 10.1016/j.apm.2020.08.061
[7]	Z. Liu, R. Kang, Pharmacokinetics with intravenous infusion of two-compartment model based on Liu process, Commun. Statistics-Theory Meth., 53 (2024), 4975–4990. https://doi.org/10.1080/03610926.2023.2198626 doi: 10.1080/03610926.2023.2198626
[8]	Z. Li, Y. Sheng, Z. Teng, H. Miao, An uncertain differential equation for SIS epidemic model, J. Intell. Fuzzy Syst., 33 (2017), 2317–2327. https://doi.org/10.3233/JIFS-17354 doi: 10.3233/JIFS-17354
[9]	C. Tian, T. Jin, X. Yang, Q. Liu, Reliability analysis of the uncertain heat conduction model, Comput. Math. Appl., 119 (2022), 131–140. https://doi.org/10.1016/j.camwa.2022.05.033 doi: 10.1016/j.camwa.2022.05.033
[10]	X. Yang, K. Yao, Uncertain partial differential equation with application to heat conduction, Fuzzy Optim. Decis. Making, 16 (2017), 379–403. https://doi.org/10.1007/s10700-016-9253-9 doi: 10.1007/s10700-016-9253-9
[11]	K. Zhang, B. Liu, Higher-order derivative of uncertain process and higher-order uncertain differential equation, Fuzzy Optim. Decis. Making, 23 (2024), 295–318. https://doi.org/10.1007/s10700-024-09422-0 doi: 10.1007/s10700-024-09422-0
[12]	B. Liu, Uncertainty theory, 5 Eds., Berlin: Springer, 2024.
[13]	B. Liu, Uncertainty theory, 3 Eds., Berlin: Springer, 2010.
[14]	B. Liu, Some research problems in uncertainty theory, J. Uncertain Syst., 3 (2009), 3–10.
[15]	X. Chen, D. A. Ralescu, Liu process and uncertain calculus, J. Uncertain. Anal. Appl., 1 (2013), 3. https://doi.org/10.1186/2195-5468-1-3 doi: 10.1186/2195-5468-1-3
[16]	T. Ye, A rigorous proof of fundamental theorem of uncertain calculus, J. Uncertain Syst., 14 (2021), 2150009. https://doi.org/10.1142/S1752890921500094 doi: 10.1142/S1752890921500094
[17]	B. Liu, Uncertainty theory, 4 Eds., Berlin: Springer Berlin Heidelberg, 2015.
[18]	L. Jia, W. Dai, Uncertain spring vibration equation, J. Indust. Manage. Optim., 18 (2022), 2401. https://doi.org/10.3934/jimo.2021073 doi: 10.3934/jimo.2021073
[19]	A. Shekhovtsov, Higher order maximum persistency and comparison theorems, Comput. Vision Image Underst., 143 (2016), 54–79. https://doi.org/10.1016/j.cviu.2015.05.002 doi: 10.1016/j.cviu.2015.05.002
[20]	W. Walter, Ordinary differential equations, Berlin: Springer Science & Business Media, 2013.

Reader Comments

Your name:*

Email:*
© 2024 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)