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Stability in measure for uncertain heat equations
School of Information Technology & Management, University of International, Business & Economics, Beijing 100029, China |
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.
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] |
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. |
[3] |
B. Liu, Uncertainty Theory, 2nd edn, Springer-Verlag, Berlin, 2004.
doi: 10.1007/978-3-540-39987-2. |
[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.
|
[6] |
B. Liu, Uncertainty Theory: A Branch of Mathematics for Modeling Human Uncertainty, Springer-Verlag, Berlin, 2010. |
[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. |
[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. |
[9] |
Y. Liu,
An analytic method for solving uncertain differential equations, Journal of Uncertain Systems, 6 (2012), 244-249.
|
[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.
|
[11] |
Y. H. Sheng and J. Gao,
Exponential stability of uncertain differential equation, Soft Computing, 20 (2016), 3673-3678.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[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.
|
[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.
|
[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. |
[20] |
X. Yang and Y. Ni, Extreme values problem of uncertain heat equation, Journal of Industrial & Management Optimization, 2018.
doi: 10.3934/jimo.2018133. |
[21] |
K. Yao and X. Chen,
A numerical method for solving uncertain differential equations, Journal of Intelligent & Fuzzy Systems, 25 (2013), 825-832.
|
[22] |
K. Yao, Extreme values and integral of solution of uncertain differential equation, Journal of Uncertainty Analysis and Applications, 1 (2013), Article 2. |
[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. |
[24] |
K. Yao, A type of nonlinear uncertain differential equations with analytic solution, Journal of Uncertainty Analysis and Applications, 1 (2013), Article 8. |
[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. |
[26] |
K. Yao, Uncertain Differential Equations, Springer-Verlag, Berlin, 2016.
doi: 10.1007/978-3-662-52729-0. |
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] |
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. |
[3] |
B. Liu, Uncertainty Theory, 2nd edn, Springer-Verlag, Berlin, 2004.
doi: 10.1007/978-3-540-39987-2. |
[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.
|
[6] |
B. Liu, Uncertainty Theory: A Branch of Mathematics for Modeling Human Uncertainty, Springer-Verlag, Berlin, 2010. |
[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. |
[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. |
[9] |
Y. Liu,
An analytic method for solving uncertain differential equations, Journal of Uncertain Systems, 6 (2012), 244-249.
|
[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.
|
[11] |
Y. H. Sheng and J. Gao,
Exponential stability of uncertain differential equation, Soft Computing, 20 (2016), 3673-3678.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[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.
|
[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.
|
[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. |
[20] |
X. Yang and Y. Ni, Extreme values problem of uncertain heat equation, Journal of Industrial & Management Optimization, 2018.
doi: 10.3934/jimo.2018133. |
[21] |
K. Yao and X. Chen,
A numerical method for solving uncertain differential equations, Journal of Intelligent & Fuzzy Systems, 25 (2013), 825-832.
|
[22] |
K. Yao, Extreme values and integral of solution of uncertain differential equation, Journal of Uncertainty Analysis and Applications, 1 (2013), Article 2. |
[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. |
[24] |
K. Yao, A type of nonlinear uncertain differential equations with analytic solution, Journal of Uncertainty Analysis and Applications, 1 (2013), Article 8. |
[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. |
[26] |
K. Yao, Uncertain Differential Equations, Springer-Verlag, Berlin, 2016.
doi: 10.1007/978-3-662-52729-0. |
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