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December  2019, 24(12): 6533-6540. doi: 10.3934/dcdsb.2019152

Stability in measure for uncertain heat equations

Xiangfeng Yang

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

Received  December 2018 Revised  January 2019 Published  December 2019 Early access  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.

Keywords: Uncertainty theory, Liu process, heat equation, stability in measure.
Mathematics Subject Classification: Primary: 35B35, 35K05; Secondary: 93E15.
Citation: Xiangfeng Yang. Stability in measure for uncertain heat equations. Discrete & Continuous Dynamical Systems - B, 2019, 24 (12) : 6533-6540. 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

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]

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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