# American Institute of Mathematical Sciences

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  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 & 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.  Google Scholar [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.  Google Scholar [3] B. Liu, Uncertainty Theory, 2nd edition, Springer-Verlag, Berlin, 2007. Google Scholar [4] B. Liu, Fuzzy process, hybrid process and uncertain process, Journal of Uncertain Systems, 2 (2008), 3-16.   Google Scholar [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.  Google Scholar [6] B. Liu, Uncertainty Theory: A Branch of Mathematics for Modeling Human Uncertainty, Springer-Verlag, Berlin, 2010. Google Scholar [7] B. Liu, Toward uncertain finance theory, Journal of Uncertainty Analysis and Applications, 1 (2013), Article 1. doi: 10.1186/2195-5468-1-1.  Google Scholar [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.  Google Scholar [9] Y. Liu, X. 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [15] K. Yao and X. Chen, A numerical method for solving uncertain differential equations, Journal of Intelligent & Fuzzy Systems, 25 (2013), 825-832.   Google Scholar [16] Y. Zhu, Uncertain optimal control with application to a portfolio selection model, Cybernetics and Systems, 41 (2010), 535-547.   Google Scholar

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.  Google Scholar [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.  Google Scholar [3] B. Liu, Uncertainty Theory, 2nd edition, Springer-Verlag, Berlin, 2007. Google Scholar [4] B. Liu, Fuzzy process, hybrid process and uncertain process, Journal of Uncertain Systems, 2 (2008), 3-16.   Google Scholar [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.  Google Scholar [6] B. Liu, Uncertainty Theory: A Branch of Mathematics for Modeling Human Uncertainty, Springer-Verlag, Berlin, 2010. Google Scholar [7] B. Liu, Toward uncertain finance theory, Journal of Uncertainty Analysis and Applications, 1 (2013), Article 1. doi: 10.1186/2195-5468-1-1.  Google Scholar [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.  Google Scholar [9] Y. Liu, X. 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [15] K. Yao and X. Chen, A numerical method for solving uncertain differential equations, Journal of Intelligent & Fuzzy Systems, 25 (2013), 825-832.   Google Scholar [16] Y. Zhu, Uncertain optimal control with application to a portfolio selection model, Cybernetics and Systems, 41 (2010), 535-547.   Google Scholar
Inverse Uncertainty Distributions of Extreme Values in Example 4.1
Inverse Uncertainty Distributions of Extreme Values in Example 4.2
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