# American Institute of Mathematical Sciences

## Stability in measure for uncertain heat equations

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

Received  December 2018 Revised  January 2019 Published  July 2019

Uncertain heat equation is a type of uncertain partial differential equations driven by Liu processes. As an important part in uncertain heat equation, stability analysis has not been researched as yet. This paper first introduces a concept of stability in measure for uncertain heat equation, and proves a stability theorem under strong Lipschitz condition that provides a sufficient for an uncertain heat equation being stable in measure. Moreover, some examples are given.

Citation: Xiangfeng Yang. Stability in measure for uncertain heat equations. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2019152
##### 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] R. Gao, Milne method for solving uncertain differential equations, Applied Mathematics and Computation, 274 (2016), 774-785. doi: 10.1016/j.amc.2015.11.043. Google Scholar [3] B. Liu, Uncertainty Theory, 2nd edn, Springer-Verlag, Berlin, 2004. doi: 10.1007/978-3-540-39987-2. 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. Google Scholar [6] B. Liu, Uncertainty Theory: A Branch of Mathematics for Modeling Human Uncertainty, Springer-Verlag, Berlin, 2010.Google Scholar [7] 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 [8] H. J. Liu, H. Ke and W. Y. Fei, Almost sure stability for uncertain differential equation, Fuzzy Optimization and Decision Making, 13 (2014), 463-473. doi: 10.1007/s10700-014-9188-y. Google Scholar [9] Y. Liu, An analytic method for solving uncertain differential equations, Journal of Uncertain Systems, 6 (2012), 244-249. Google Scholar [10] Y. H. Sheng and C. G. Wang, Stability in the $p$-th moment for uncertain differential equation, Journal of Intelligent & Fuzzy Systems, 26 (2014), 1263-1271. Google Scholar [11] Y. H. Sheng and J. Gao, Exponential stability of uncertain differential equation, Soft Computing, 20 (2016), 3673-3678. Google Scholar [12] N. Tao and Y. Zhu, Attractivity and stability analysis of uncertain differential systems, International Journal of Bifurcation and Chaos, 25 (2015), 1550022 (10 pages). doi: 10.1142/S0218127415500224. Google Scholar [13] X. Wang, Y. F. Ning, A. Tauqir Moughal and X. M. Chen, Adams-Simpson method for solving uncertain differential equation, Applied Mathematics and Computation, 271 (2015), 209-219. doi: 10.1016/j.amc.2015.09.009. Google Scholar [14] X. Yang and Y. Y. Shen, Runge-Kutta method for solving uncertain differential equations, Journal of Uncertainty Analysis and Applications, 3 (2015), Article 17.Google Scholar [15] X. Yang and D. A. Ralescu, Adams method for solving uncertain differential equations, Applied Mathematics and Computation, 270 (2015), 993-1003. doi: 10.1016/j.amc.2015.08.109. Google Scholar [16] 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 [17] X. Yang, Y. Ni and Y. Zhang, Stability in inverse distribution for uncertain differential equations, Journal of Intelligent & Fuzzy Systems, 32 (2017), 2051-2059. Google Scholar [18] X. Yang and Y. Ni, Existence and uniqueness theorem for uncertain heat equation, Journal of Ambient Intelligence and Humanized Computing, 8 (2017), 717-725. Google Scholar [19] X. Yang, Solving uncertain heat equation via numerical method, Applied Mathematics and Computation, 329 (2018), 92-104. doi: 10.1016/j.amc.2018.01.055. Google Scholar [20] X. Yang and Y. Ni, Extreme values problem of uncertain heat equation, Journal of Industrial & Management Optimization, 2018. doi: 10.3934/jimo.2018133. Google Scholar [21] K. Yao and X. Chen, A numerical method for solving uncertain differential equations, Journal of Intelligent & Fuzzy Systems, 25 (2013), 825-832. Google Scholar [22] K. Yao, Extreme values and integral of solution of uncertain differential equation, Journal of Uncertainty Analysis and Applications, 1 (2013), Article 2.Google Scholar [23] K. Yao, J. Gao and Y. Gao, Some stability theorems of uncertain differential equation, Fuzzy Optimization and Decision Making, 12 (2013), 3-13. doi: 10.1007/s10700-012-9139-4. Google Scholar [24] K. Yao, A type of nonlinear uncertain differential equations with analytic solution, Journal of Uncertainty Analysis and Applications, 1 (2013), Article 8.Google Scholar [25] K. Yao, H. Ke and Y. H. Sheng, Stability in mean for uncertain differential equation, Fuzzy Optimization and Decision Making, 14 (2015), 365-379. doi: 10.1007/s10700-014-9204-2. Google Scholar [26] K. Yao, Uncertain Differential Equations, Springer-Verlag, Berlin, 2016. doi: 10.1007/978-3-662-52729-0. Google Scholar

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##### 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] R. Gao, Milne method for solving uncertain differential equations, Applied Mathematics and Computation, 274 (2016), 774-785. doi: 10.1016/j.amc.2015.11.043. Google Scholar [3] B. Liu, Uncertainty Theory, 2nd edn, Springer-Verlag, Berlin, 2004. doi: 10.1007/978-3-540-39987-2. 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. Google Scholar [6] B. Liu, Uncertainty Theory: A Branch of Mathematics for Modeling Human Uncertainty, Springer-Verlag, Berlin, 2010.Google Scholar [7] 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 [8] H. J. Liu, H. Ke and W. Y. Fei, Almost sure stability for uncertain differential equation, Fuzzy Optimization and Decision Making, 13 (2014), 463-473. doi: 10.1007/s10700-014-9188-y. Google Scholar [9] Y. Liu, An analytic method for solving uncertain differential equations, Journal of Uncertain Systems, 6 (2012), 244-249. Google Scholar [10] Y. H. Sheng and C. G. Wang, Stability in the $p$-th moment for uncertain differential equation, Journal of Intelligent & Fuzzy Systems, 26 (2014), 1263-1271. Google Scholar [11] Y. H. Sheng and J. Gao, Exponential stability of uncertain differential equation, Soft Computing, 20 (2016), 3673-3678. Google Scholar [12] N. Tao and Y. Zhu, Attractivity and stability analysis of uncertain differential systems, International Journal of Bifurcation and Chaos, 25 (2015), 1550022 (10 pages). doi: 10.1142/S0218127415500224. Google Scholar [13] X. Wang, Y. F. Ning, A. Tauqir Moughal and X. M. Chen, Adams-Simpson method for solving uncertain differential equation, Applied Mathematics and Computation, 271 (2015), 209-219. doi: 10.1016/j.amc.2015.09.009. Google Scholar [14] X. Yang and Y. Y. Shen, Runge-Kutta method for solving uncertain differential equations, Journal of Uncertainty Analysis and Applications, 3 (2015), Article 17.Google Scholar [15] X. Yang and D. A. Ralescu, Adams method for solving uncertain differential equations, Applied Mathematics and Computation, 270 (2015), 993-1003. doi: 10.1016/j.amc.2015.08.109. Google Scholar [16] 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 [17] X. Yang, Y. Ni and Y. Zhang, Stability in inverse distribution for uncertain differential equations, Journal of Intelligent & Fuzzy Systems, 32 (2017), 2051-2059. Google Scholar [18] X. Yang and Y. Ni, Existence and uniqueness theorem for uncertain heat equation, Journal of Ambient Intelligence and Humanized Computing, 8 (2017), 717-725. Google Scholar [19] X. Yang, Solving uncertain heat equation via numerical method, Applied Mathematics and Computation, 329 (2018), 92-104. doi: 10.1016/j.amc.2018.01.055. Google Scholar [20] X. Yang and Y. Ni, Extreme values problem of uncertain heat equation, Journal of Industrial & Management Optimization, 2018. doi: 10.3934/jimo.2018133. Google Scholar [21] K. Yao and X. Chen, A numerical method for solving uncertain differential equations, Journal of Intelligent & Fuzzy Systems, 25 (2013), 825-832. Google Scholar [22] K. Yao, Extreme values and integral of solution of uncertain differential equation, Journal of Uncertainty Analysis and Applications, 1 (2013), Article 2.Google Scholar [23] K. Yao, J. Gao and Y. Gao, Some stability theorems of uncertain differential equation, Fuzzy Optimization and Decision Making, 12 (2013), 3-13. doi: 10.1007/s10700-012-9139-4. Google Scholar [24] K. Yao, A type of nonlinear uncertain differential equations with analytic solution, Journal of Uncertainty Analysis and Applications, 1 (2013), Article 8.Google Scholar [25] K. Yao, H. Ke and Y. H. Sheng, Stability in mean for uncertain differential equation, Fuzzy Optimization and Decision Making, 14 (2015), 365-379. doi: 10.1007/s10700-014-9204-2. Google Scholar [26] K. Yao, Uncertain Differential Equations, Springer-Verlag, Berlin, 2016. doi: 10.1007/978-3-662-52729-0. Google Scholar
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