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doi: 10.3934/jimo.2021073
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## Uncertain spring vibration equation

 1 School of Management and Engineering, Capital University of Economics and Business, Beijing, China 2 Department of Financial Engineering, Central University of Finance and Economics, Beijing 100081, China

* Corresponding author: Wei Dai

Received  July 2020 Revised  January 2021 Early access April 2021

Fund Project: The first author is supported by the Beijing Municipal Education Commission Foundation of China (No. KM202110038001), the Young Academic Innovation Team of Capital University of Economics and Business (No. QNTD202002), and the special fund of basic scientific research business fees of Beijing Municipal University of Capital University of Economics and Business (No. XRZ2020016)

The spring vibration equation is to model the behavior of a spring which has a time varying force acting on it. The stochastic spring vibration equation was proposed for modeling spring vibration phenomena with noise described by Wiener process. However, there exists a paradox in some cases. Thus, as a counterpart, this paper proposes uncertain spring vibration equation driven by Liu process to describe the noise. Moreover, the analytic solution of uncertain spring vibration equation is derived and the inverse uncertainty distribution of solution is proved. At last, this paper presents a paradox of stochastic spring vibration equation.

Citation: Lifen Jia, Wei Dai. Uncertain spring vibration equation. Journal of Industrial and Management Optimization, doi: 10.3934/jimo.2021073
##### References:
 [1] R. Cao, W. Hou and Y. Gao, An entropy-based three-stage approach for multi-objective system reliability optimization considering uncertainty, Engineering Optimization, 50 (2018), 1453-1469.  doi: 10.1080/0305215X.2017.1402014. [2] T. K. Caughey, Derivation and application of the fokker-planck equation to discrete nonlinear dynamic systems subjected to white random excitation, Journal of the Acoustical Society of America, 35 (1963), 1683-1692.  doi: 10.1121/1.1918788. [3] 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. [4] S. Crandall, Random Vibration, Technology Press of MIT and John Wiley and Sons, New York, 1958. [5] A. Einstein, On the movement of small particles suspended in stationary liquids required by the molecular-kinetic theory of heat, Annalen der Physik, 17 (1905), 549-560. [6] X. Gao and L. Jia, Degree-constrained minimum spanning tree problem with uncertain edge weights, Applied Soft Computing, 56 (2017), 580-588.  doi: 10.1016/j.asoc.2016.07.054. [7] R. Gao, Uncertain wave equation with infinite half-boundary, Applied Mathematics and Computation, 304 (2017), 28-40.  doi: 10.1016/j.amc.2016.12.003. [8] T. Gard, Introduction to Stoachastic Differential Equations, Marcel Dekker, 1988. [9] B. Li and Y. Zhu, Parametric optimal control of uncertain systems under an optimistic value criterion, Engineering Optimization, 50 (2018), 55-69.  doi: 10.1080/0305215X.2017.1303054. [10] B. Liu, Uncertainty Theory. An introduction to its axiomatic foundations, Studies in Fuzziness and Soft Computing, 154. Springer-Verlag, Berlin, 2004. doi: 10.1007/978-3-540-39987-2. [11] B. Liu, Fuzzy process, hybrid process and uncertain process, Journal of Uncertain Systems, 2 (2008), 3-16. [12] B. Liu, Some research problems in uncertainty theory, Journal of Uncertain Systems, 3 (2009), 3-10. [13] B. Liu, Uncertainty Theory: A Branch of Mathematics for Modeling Human Uncertainty, Springer-Verlag, Berlin, 2010. [14] B. Liu, Toward ucertain finance theory, Journal of Uncertainty Analysis and Applications, 1 (2013), Ariticle 1. doi: 10.1186/2195-5468-1-1. [15] W. T. Thomson and M. V. Barton, The response of mechanical systems to random excitations, Journal of Applied Mechanics, 24 (1957), 248-251. [16] G. E. Uhlenbeck and L. S. Ornstein, On the theory of the Brownian motion, Physical Review, 36 (1930), 823-841.  doi: 10.1103/PhysRev.36.823. [17] 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. [18] 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. [19] 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. [20] 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. [21] 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. [22] X. Yang, Stability in measure for uncertain heat equations, Discrete and Continuous Dynamical Systems Series B, 24 (2019), 6533-6540.  doi: 10.3934/dcdsb.2019152. [23] X. Yang and Y. Ni, Extreme values problem of uncertain heat equation, Journal of Industrial and Management Optimization, 15 (2019), 1995-2008.  doi: 10.3934/jimo.2018133. [24] K. Yao and X. Chen, A numerical method for solving uncertain differential equations, Journal of Intelligent and Fuzzy Systems, 25 (2013), 825-832.  doi: 10.3233/IFS-120688. [25] K. Yao, Uncertainty Differential Equation, Springer-Verlag, Berlin, 2016. doi: 10.1007/978-3-662-52729-0. [26] K. Yao and B. Liu, Parameter estimation in uncertain differential equations, Fuzzy Optimization and Decision Making, 19 (2020), 1-12.  doi: 10.1007/s10700-019-09310-y. [27] Y. Zhu, Uncertain optimal control with application to a portfolio selection model, Cybernetics and Systems, 41 (2010), 535-547.  doi: 10.1080/01969722.2010.511552.

show all references

##### References:
 [1] R. Cao, W. Hou and Y. Gao, An entropy-based three-stage approach for multi-objective system reliability optimization considering uncertainty, Engineering Optimization, 50 (2018), 1453-1469.  doi: 10.1080/0305215X.2017.1402014. [2] T. K. Caughey, Derivation and application of the fokker-planck equation to discrete nonlinear dynamic systems subjected to white random excitation, Journal of the Acoustical Society of America, 35 (1963), 1683-1692.  doi: 10.1121/1.1918788. [3] 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. [4] S. Crandall, Random Vibration, Technology Press of MIT and John Wiley and Sons, New York, 1958. [5] A. Einstein, On the movement of small particles suspended in stationary liquids required by the molecular-kinetic theory of heat, Annalen der Physik, 17 (1905), 549-560. [6] X. Gao and L. Jia, Degree-constrained minimum spanning tree problem with uncertain edge weights, Applied Soft Computing, 56 (2017), 580-588.  doi: 10.1016/j.asoc.2016.07.054. [7] R. Gao, Uncertain wave equation with infinite half-boundary, Applied Mathematics and Computation, 304 (2017), 28-40.  doi: 10.1016/j.amc.2016.12.003. [8] T. Gard, Introduction to Stoachastic Differential Equations, Marcel Dekker, 1988. [9] B. Li and Y. Zhu, Parametric optimal control of uncertain systems under an optimistic value criterion, Engineering Optimization, 50 (2018), 55-69.  doi: 10.1080/0305215X.2017.1303054. [10] B. Liu, Uncertainty Theory. An introduction to its axiomatic foundations, Studies in Fuzziness and Soft Computing, 154. Springer-Verlag, Berlin, 2004. doi: 10.1007/978-3-540-39987-2. [11] B. Liu, Fuzzy process, hybrid process and uncertain process, Journal of Uncertain Systems, 2 (2008), 3-16. [12] B. Liu, Some research problems in uncertainty theory, Journal of Uncertain Systems, 3 (2009), 3-10. [13] B. Liu, Uncertainty Theory: A Branch of Mathematics for Modeling Human Uncertainty, Springer-Verlag, Berlin, 2010. [14] B. Liu, Toward ucertain finance theory, Journal of Uncertainty Analysis and Applications, 1 (2013), Ariticle 1. doi: 10.1186/2195-5468-1-1. [15] W. T. Thomson and M. V. Barton, The response of mechanical systems to random excitations, Journal of Applied Mechanics, 24 (1957), 248-251. [16] G. E. Uhlenbeck and L. S. Ornstein, On the theory of the Brownian motion, Physical Review, 36 (1930), 823-841.  doi: 10.1103/PhysRev.36.823. [17] 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. [18] 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. [19] 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. [20] 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. [21] 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. [22] X. Yang, Stability in measure for uncertain heat equations, Discrete and Continuous Dynamical Systems Series B, 24 (2019), 6533-6540.  doi: 10.3934/dcdsb.2019152. [23] X. Yang and Y. Ni, Extreme values problem of uncertain heat equation, Journal of Industrial and Management Optimization, 15 (2019), 1995-2008.  doi: 10.3934/jimo.2018133. [24] K. Yao and X. Chen, A numerical method for solving uncertain differential equations, Journal of Intelligent and Fuzzy Systems, 25 (2013), 825-832.  doi: 10.3233/IFS-120688. [25] K. Yao, Uncertainty Differential Equation, Springer-Verlag, Berlin, 2016. doi: 10.1007/978-3-662-52729-0. [26] K. Yao and B. Liu, Parameter estimation in uncertain differential equations, Fuzzy Optimization and Decision Making, 19 (2020), 1-12.  doi: 10.1007/s10700-019-09310-y. [27] Y. Zhu, Uncertain optimal control with application to a portfolio selection model, Cybernetics and Systems, 41 (2010), 535-547.  doi: 10.1080/01969722.2010.511552.
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