-
Previous Article
Optimizing micro-algae production in a raceway pond with variable depth
- JIMO Home
- This Issue
-
Next Article
On the optimal control problems with characteristic time control constraints
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 |
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.
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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[12] |
B. Liu, Some research problems in uncertainty theory, Journal of Uncertain Systems, 3 (2009), 3-10. Google Scholar |
[13] |
B. Liu, Uncertainty Theory: A Branch of Mathematics for Modeling Human Uncertainty, Springer-Verlag, Berlin, 2010. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[12] |
B. Liu, Some research problems in uncertainty theory, Journal of Uncertain Systems, 3 (2009), 3-10. Google Scholar |
[13] |
B. Liu, Uncertainty Theory: A Branch of Mathematics for Modeling Human Uncertainty, Springer-Verlag, Berlin, 2010. Google Scholar |
[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. |
[1] |
Andrea Tosin, Mattia Zanella. Uncertainty damping in kinetic traffic models by driver-assist controls. Mathematical Control & Related Fields, 2021 doi: 10.3934/mcrf.2021018 |
[2] |
Qi Lü, Xu Zhang. A concise introduction to control theory for stochastic partial differential equations. Mathematical Control & Related Fields, 2021 doi: 10.3934/mcrf.2021020 |
[3] |
Nhu N. Nguyen, George Yin. Stochastic partial differential equation models for spatially dependent predator-prey equations. Discrete & Continuous Dynamical Systems - B, 2020, 25 (1) : 117-139. doi: 10.3934/dcdsb.2019175 |
[4] |
Abdulrazzaq T. Abed, Azzam S. Y. Aladool. Applying particle swarm optimization based on Padé approximant to solve ordinary differential equation. Numerical Algebra, Control & Optimization, 2021 doi: 10.3934/naco.2021008 |
[5] |
Seddigheh Banihashemi, Hossein Jafaria, Afshin Babaei. A novel collocation approach to solve a nonlinear stochastic differential equation of fractional order involving a constant delay. Discrete & Continuous Dynamical Systems - S, 2021 doi: 10.3934/dcdss.2021025 |
[6] |
Changpin Li, Zhiqiang Li. Asymptotic behaviors of solution to partial differential equation with Caputo–Hadamard derivative and fractional Laplacian: Hyperbolic case. Discrete & Continuous Dynamical Systems - S, 2021 doi: 10.3934/dcdss.2021023 |
[7] |
Zhiming Guo, Zhi-Chun Yang, Xingfu Zou. Existence and uniqueness of positive solution to a non-local differential equation with homogeneous Dirichlet boundary condition---A non-monotone case. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1825-1838. doi: 10.3934/cpaa.2012.11.1825 |
[8] |
Enkhbat Rentsen, N. Tungalag, J. Enkhbayar, O. Battogtokh, L. Enkhtuvshin. Application of survival theory in Mining industry. Numerical Algebra, Control & Optimization, 2021, 11 (3) : 443-448. doi: 10.3934/naco.2020036 |
[9] |
Alexandr Mikhaylov, Victor Mikhaylov. Dynamic inverse problem for Jacobi matrices. Inverse Problems & Imaging, 2019, 13 (3) : 431-447. doi: 10.3934/ipi.2019021 |
[10] |
Armin Lechleiter, Tobias Rienmüller. Factorization method for the inverse Stokes problem. Inverse Problems & Imaging, 2013, 7 (4) : 1271-1293. doi: 10.3934/ipi.2013.7.1271 |
[11] |
Andrey Kovtanyuk, Alexander Chebotarev, Nikolai Botkin, Varvara Turova, Irina Sidorenko, Renée Lampe. Modeling the pressure distribution in a spatially averaged cerebral capillary network. Mathematical Control & Related Fields, 2021 doi: 10.3934/mcrf.2021016 |
[12] |
Ricardo A. Podestá, Denis E. Videla. The weight distribution of irreducible cyclic codes associated with decomposable generalized Paley graphs. Advances in Mathematics of Communications, 2021 doi: 10.3934/amc.2021002 |
[13] |
Guillaume Bal, Wenjia Jing. Homogenization and corrector theory for linear transport in random media. Discrete & Continuous Dynamical Systems, 2010, 28 (4) : 1311-1343. doi: 10.3934/dcds.2010.28.1311 |
[14] |
Felix Finster, Jürg Fröhlich, Marco Oppio, Claudio F. Paganini. Causal fermion systems and the ETH approach to quantum theory. Discrete & Continuous Dynamical Systems - S, 2021, 14 (5) : 1717-1746. doi: 10.3934/dcdss.2020451 |
[15] |
Fioralba Cakoni, Shixu Meng, Jingni Xiao. A note on transmission eigenvalues in electromagnetic scattering theory. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2021025 |
[16] |
Rabiaa Ouahabi, Nasr-Eddine Hamri. Design of new scheme adaptive generalized hybrid projective synchronization for two different chaotic systems with uncertain parameters. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2361-2370. doi: 10.3934/dcdsb.2020182 |
[17] |
Sergei Avdonin, Julian Edward. An inverse problem for quantum trees with observations at interior vertices. Networks & Heterogeneous Media, 2021, 16 (2) : 317-339. doi: 10.3934/nhm.2021008 |
[18] |
Hirofumi Notsu, Masato Kimura. Symmetry and positive definiteness of the tensor-valued spring constant derived from P1-FEM for the equations of linear elasticity. Networks & Heterogeneous Media, 2014, 9 (4) : 617-634. doi: 10.3934/nhm.2014.9.617 |
[19] |
Ritu Agarwal, Kritika, Sunil Dutt Purohit, Devendra Kumar. Mathematical modelling of cytosolic calcium concentration distribution using non-local fractional operator. Discrete & Continuous Dynamical Systems - S, 2021 doi: 10.3934/dcdss.2021017 |
[20] |
W. Cary Huffman. On the theory of $\mathbb{F}_q$-linear $\mathbb{F}_{q^t}$-codes. Advances in Mathematics of Communications, 2013, 7 (3) : 349-378. doi: 10.3934/amc.2013.7.349 |
2019 Impact Factor: 1.366
Tools
Metrics
Other articles
by authors
[Back to Top]