×

zbMATH — the first resource for mathematics

Stability of solution for uncertain wave equation. (English) Zbl 1428.35153
Summary: The uncertain wave equation is an important type of uncertain partial differential equation driven by Liu process which is a special type of uncertain process with independent and stationary increments. For an uncertain wave equation, it is head for us to get its solution. What is more, even if we get it, we still should know whether the obtained solution is stable or not. So, this paper puts forward the concept of stability of uncertain wave equations in the sense of convergence in uncertain measure. Then, we discuss the condition for an uncertain wave equation being stable and prove the stability theorem. In addition, some examples are given to show what is the concept of stability exactly and how to judge an uncertain wave equation being stable.

MSC:
35K20 Initial-boundary value problems for second-order parabolic equations
35B35 Stability in context of PDEs
35R60 PDEs with randomness, stochastic partial differential equations
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
PDF BibTeX Cite
Full Text: DOI
References:
[1] Cabaña, E. M., The vibrating string forced by white noise, Z. Wahrscheinlichkeitsthrorie Verwandte Geb., 15, 111-130 (1970) · Zbl 0193.45101
[2] Chen, X. W.; Liu, B., Existence and uniqueness theorem for uncertain differential equations, Fuzzy Optim. Decis. Mak., 9, 1, 69-81 (2010) · Zbl 1196.34005
[3] Czapor, S. R.; Mclenaghan, R. G.; Sasse, F. D., Complete solution of Hadamard’s problem for the scalar wave equation on Petrov type III space times, Ann. l’Inst. Henri Poincaré, 71, 6, 595-620 (1999) · Zbl 0951.35131
[4] D’Alembert, J. R., Researches on the curve that a tense cord forms set into vibration, Hist. R. Acad. Sci. Belles-Lett. Berl., 3, 214-219 (1747)
[5] Gao, R., Milne method for solving uncertain differential equations, Appl. Math. Comput., 274, 774-785 (2016) · Zbl 1410.60053
[6] Gao, R., Uncertain wave equation with infinite half-boundary, Appl. Math. Comput., 304, 28-40 (2017) · Zbl 1411.35190
[8] Gao, R.; Chen, X. W., Some concepts and properties of uncertain fields, J. Intell. Fuzzy Syst., 32, 4367-4378 (2017) · Zbl 1386.60014
[10] Gao, R.; Ralescu, A. D., Uncertain wave equation for string vibration, IEEE Trans. Fuzzy Syst. (2018)
[11] Gao, R.; Yao, K., Importance index of components in uncertain random systems, Knowl.-Based Syst., 109, 208-217 (2016)
[12] Liu, B., Uncertainty Theory (2007), Springer-Verlag: Springer-Verlag Berlin
[13] Liu, B., Fuzzy process, hybrid process and uncertain process, J. Uncertain Syst., 2, 1, 3-16 (2008)
[14] Liu, B., Some research problems in uncertainty theory, J. Uncertain Syst., 3, 1, 3-10 (2009)
[15] Liu, B., Theory and Practice of Uncertain Programming (2009), Springer-Verlag: Springer-Verlag Berlin · Zbl 1158.90010
[16] Liu, B., Uncertainty Theory: A Branch of Mathematics for Modeling Human Uncertainty (2010), Springer-Verlag: Springer-Verlag Berlin
[17] Liu, B., Toward uncertain finance theory, J. Uncertain. Anal. Appl., 1 (2013)
[18] Liu, H. J.; Ke, H.; Fei, W. Y., Almost sure stability for uncertain differential equation, Fuzzy Optim. Decis. Mak., 13, 4, 463-473 (2014) · Zbl 1449.60118
[19] Marcus, M.; Mizel, V. J., Stochastic hyperbolic systems and the wave equation, Stoch. Stoch. Rep., 36, 225-244 (1991) · Zbl 0739.60059
[20] Orsingher, E., Randomly forced vibrations of a string, Ann. l’Inst. Henri Poincaré-Sect. B, 18, 4, 367-394 (1982) · Zbl 0493.60067
[21] Wiener, N., Differential space, J. Math. Phys., 2, 131-174 (1923)
[22] Yao, K.; Chen, X. W., A numerical method for solving uncertain differential equations, J. Intell. Fuzzy Syst., 25, 3, 825-832 (2013) · Zbl 1291.65025
[23] Yao, K.; Gao, J. W.; Gao, Y., Some stability theorems of uncertain differential equation, Fuzzy Optim. Decis. Mak., 12, 1, 3-13 (2013) · Zbl 1412.60104
[24] Zhang, Q. Y.; Kang, R.; Wen, M. L., Belief reliability for uncertain random systems, IEEE Trans. Fuzzy Syst., 26, 6, 3605-3614 (2018)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.