October  2019, 15(4): 1995-2008. doi: 10.3934/jimo.2018133

Extreme values problem of uncertain heat equation

School of Information Technology & Management, University of International, Business & Economics, Beijing 100029, China

* Corresponding author: Yaodong Ni

Received  March 2018 Revised  April 2018 Published  October 2019 Early access  August 2018

Fund Project: The second author is supported by National Natural Science Foundation of China (Grant No. 71471038).

Uncertain heat equation is a class of uncertain partial differential equations involving Liu processes. This paper first gives the uncertainty distributions and the inverse uncertainty distributions of extreme values of solutions for uncertain heat equations. Numerical methods are designed to gain the inverse uncertainty distributions of extreme values of solutions.

Citation: Xiangfeng Yang, Yaodong Ni. Extreme values problem of uncertain heat equation. Journal of Industrial and Management Optimization, 2019, 15 (4) : 1995-2008. doi: 10.3934/jimo.2018133
References:
[1]

X. Chen and B. Liu, Existence and uniqueness theorem for uncertain differential equations, Fuzzy Optimization and Decision Making, 9 (2010), 69-81.  doi: 10.1007/s10700-010-9073-2.

[2]

X. Chen and J. Gao, Uncertain term structure model of interest rate, Soft Computing, 17 (2013), 597-604.  doi: 10.1007/s00500-012-0927-0.

[3]

B. Liu, Uncertainty Theory, 2nd edition, Springer-Verlag, Berlin, 2007.

[4]

B. Liu, Fuzzy process, hybrid process and uncertain process, Journal of Uncertain Systems, 2 (2008), 3-16. 

[5]

B. Liu, Some research problems in uncertainty theory, Journal of Uncertain Systems, 3 (2009), 3-10.  doi: 10.1007/978-3-662-44354-5.

[6]

B. Liu, Uncertainty Theory: A Branch of Mathematics for Modeling Human Uncertainty, Springer-Verlag, Berlin, 2010.

[7]

B. Liu, Toward uncertain finance theory, Journal of Uncertainty Analysis and Applications, 1 (2013), Article 1. doi: 10.1186/2195-5468-1-1.

[8]

B. Liu, Uncertainty distribution and independence of uncertain processes, Fuzzy Optimization and Decision Making, 13 (2014), 259-271.  doi: 10.1007/s10700-014-9181-5.

[9]

Y. LiuX. Chen and D. A. Ralescu, Uncertain currency model and currency option pricing, International Journal of Intelligent Systems, 30 (2015), 40-51.  doi: 10.1002/int.21680.

[10]

X. Yang and J. Gao, Uncertain differential games with application to capitalism, Journal of Uncertainty Analysis and Applications, 1 (2013), Article 17. doi: 10.1186/2195-5468-1-17.

[11]

X. Yang and J. Gao, Linear-quadratic uncertain differential games with application to resource extraction problem, IEEE Transactions on Fuzzy Systems, 24 (2016), 819-826.  doi: 10.1109/TFUZZ.2015.2486809.

[12]

X. Yang and K. Yao, Uncertain partial differential equation with application to heat conduction, Fuzzy Optimization and Decision Making, 16 (2017), 379-403.  doi: 10.1007/s10700-016-9253-9.

[13]

X. Yang and Y. Ni, Existence and uniqueness theorem for uncertain heat equation, Journal of Ambient Intelligence and Humanized Computing, 8 (2017), 717-725.  doi: 10.1007/s12652-017-0479-3.

[14]

X. Yang, A numerical method for solving uncertain heat equation, Applied Mathematics and Computation, 329 (2018), 92-104.  doi: 10.1016/j.amc.2018.01.055.

[15]

K. Yao and X. Chen, A numerical method for solving uncertain differential equations, Journal of Intelligent & Fuzzy Systems, 25 (2013), 825-832. 

[16]

Y. Zhu, Uncertain optimal control with application to a portfolio selection model, Cybernetics and Systems, 41 (2010), 535-547. 

show all references

References:
[1]

X. Chen and B. Liu, Existence and uniqueness theorem for uncertain differential equations, Fuzzy Optimization and Decision Making, 9 (2010), 69-81.  doi: 10.1007/s10700-010-9073-2.

[2]

X. Chen and J. Gao, Uncertain term structure model of interest rate, Soft Computing, 17 (2013), 597-604.  doi: 10.1007/s00500-012-0927-0.

[3]

B. Liu, Uncertainty Theory, 2nd edition, Springer-Verlag, Berlin, 2007.

[4]

B. Liu, Fuzzy process, hybrid process and uncertain process, Journal of Uncertain Systems, 2 (2008), 3-16. 

[5]

B. Liu, Some research problems in uncertainty theory, Journal of Uncertain Systems, 3 (2009), 3-10.  doi: 10.1007/978-3-662-44354-5.

[6]

B. Liu, Uncertainty Theory: A Branch of Mathematics for Modeling Human Uncertainty, Springer-Verlag, Berlin, 2010.

[7]

B. Liu, Toward uncertain finance theory, Journal of Uncertainty Analysis and Applications, 1 (2013), Article 1. doi: 10.1186/2195-5468-1-1.

[8]

B. Liu, Uncertainty distribution and independence of uncertain processes, Fuzzy Optimization and Decision Making, 13 (2014), 259-271.  doi: 10.1007/s10700-014-9181-5.

[9]

Y. LiuX. Chen and D. A. Ralescu, Uncertain currency model and currency option pricing, International Journal of Intelligent Systems, 30 (2015), 40-51.  doi: 10.1002/int.21680.

[10]

X. Yang and J. Gao, Uncertain differential games with application to capitalism, Journal of Uncertainty Analysis and Applications, 1 (2013), Article 17. doi: 10.1186/2195-5468-1-17.

[11]

X. Yang and J. Gao, Linear-quadratic uncertain differential games with application to resource extraction problem, IEEE Transactions on Fuzzy Systems, 24 (2016), 819-826.  doi: 10.1109/TFUZZ.2015.2486809.

[12]

X. Yang and K. Yao, Uncertain partial differential equation with application to heat conduction, Fuzzy Optimization and Decision Making, 16 (2017), 379-403.  doi: 10.1007/s10700-016-9253-9.

[13]

X. Yang and Y. Ni, Existence and uniqueness theorem for uncertain heat equation, Journal of Ambient Intelligence and Humanized Computing, 8 (2017), 717-725.  doi: 10.1007/s12652-017-0479-3.

[14]

X. Yang, A numerical method for solving uncertain heat equation, Applied Mathematics and Computation, 329 (2018), 92-104.  doi: 10.1016/j.amc.2018.01.055.

[15]

K. Yao and X. Chen, A numerical method for solving uncertain differential equations, Journal of Intelligent & Fuzzy Systems, 25 (2013), 825-832. 

[16]

Y. Zhu, Uncertain optimal control with application to a portfolio selection model, Cybernetics and Systems, 41 (2010), 535-547. 

Figure 1.  Inverse Uncertainty Distributions of Extreme Values in Example 4.1
Figure 2.  Inverse Uncertainty Distributions of Extreme Values in Example 4.2
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