Abstract
High-order uncertain differential equations are used to model differentiable uncertain systems with high-order differentials. Since most of high-order differential equations have no analytic solutions, how to design numerical methods for solving the high-order uncertain differential equation has always been a core issue in practice. This paper designs a numerical method for solving high-order uncertain differential equations via Adams–Simpson method. A procedure is designed, and some numerical experiments are given to illustrate the efficiency and effectiveness of the Adams–Simpson method. Furthermore, this paper gives how to calculate the expected value, the inverse uncertainty distributions of the extreme value, and the integral of the solution of the high-order uncertain differential equation with the aid of Adams–Simpson method.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Chen XW, Liu YH, Ralescu DA (2013) Uncertain stock model with periodic dividends. Fuzzy Optim Decis Mak 12(1):111–123
Chen XW, Liu B (2010) Existence and uniqueness theorem for uncertain differential equations. Fuzzy Optim Decis Mak 9(1):69–81
Gao R (2016) Milne method for solving uncertain differential equations. Appl Math Comput 274:774–785
Ito K (1944) Stochastic integral. Proc Imp Acad 20(8):519–524
Ito K (1951) On stochastic differential equations. Mem Am Math Soc 4
Ji XY, Zhou J (2018) Solving high-order uncertain differential equations via Runge–Kutta method. IEEE Trans Fuzzy Syst 26(3):1379–1386
Kahneman D, Tversky A (1979) Prospect theory: an analysis of decision under risk. Econometrica 47(2):263–291
Liu B (2007) Uncertainty theory, 2nd edn. Springer, Berlin
Liu B (2009) Some research problems in uncertainty theory. J Uncertain Syst 3(1):3–10
Liu B (2008) Fuzzy process, hybrid process and uncertain process. J Uncertain Syst 2(1):3–16
Liu B (2010) Uncertain theory: a branch of mathematics for modeling human uncertainty. Springer, Berlin
Liu YH, Chen XW, Ralescu DA (2015) Uncertain currency model and currency option pricing. Int J Intell Syst 1:40–51
Liu YH (2012) An analytic method for solving uncertain differential equations with analytic solutions. J Uncertain Syst 6(4):244–249
Peng J, Yao K (2011) A new option pricing model for stocks in uncertainty markets. Int J Oper Res 8(2):18–26
Shen YY, Yao K (2016) A mean-reverting currency model in an uncertain environment. Soft Comput 20(10):4131–4138
Sheng YH (2017) Stability of high-order uncertain differential equations. J Intell Fuzzy Syst 33:1363–1373
Wiener N (1923) Differential-space. J Math Phys 2(1–4):131–174
Wang X, Ning YF, Moughal TA, Chen XM (2015) Adams–Simpson method for solving uncertain differential equations. Appl Math Comput 271:209–219
Yao K (2015) A no-arbitrage theorem for uncertain stock model. Fuzzy Optim Decis Mak 14(2):227–242
Yao K (2013) A type nonlinear of uncertain differential equations with analytic solution. J Uncertain Anal Appl 1 Article 8
Yao K, Chen XW (2013) A numerical method for solving uncertain differential equations. J Intell Fuzzy Syst 25(3):825–832
Yao K (2013) Extreme values and integral of solution of uncertain differential equations. J Uncertain Anal Appl 1:2
Yao K (2016) Uncertain differential equations, 2nd edn. Springer, Berlin
Yang XF, Shen YY (2015) Runge–Kutta method for solving uncertain differential equations. J Uncertain Anal Appl 3 Article 17
Yang XF, Ralescu DA (2015) Adams method for solving uncertain differential equations. Appl Math Comput 270:993–1003
Zhu YG (2010) Uncertain optimal control with application to a portfolio selection model. Cybern Syst 41(7):535–547
Zhang Y, Gao JW, Huang ZY (2017) Hamming method for solving uncertain differential equations. Appl Math Comput 313:331–341
Acknowledgements
This research is funded by the Natural Science Foundation of Xinjiang (Grant No. 2020D01C017), National Natural Science Foundation of China (Grants Nos. 12061072, 11861064), and National Natural Science Foundation of China—Joint Key Program of Xinjiang (Grant No. U1703262).
Author information
Authors and Affiliations
Corresponding authors
Additional information
Communicated by Valeria Neves Domingos Cavalcanti.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Supplementary Information
Below is the link to the electronic supplementary material.
Rights and permissions
About this article
Cite this article
Hou, Y., Wu, L. & Sheng, Y. Solving high-order uncertain differential equations via Adams–Simpson method. Comp. Appl. Math. 40, 252 (2021). https://doi.org/10.1007/s40314-020-01408-z
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s40314-020-01408-z