Skip to main content
SpringerLink
Log in

On uncertain partial differential equations

  • Published:
Fuzzy Optimization and Decision Making Aims and scope Submit manuscript

Abstract

Uncertain partial differential equation (UPDE) was introduced in literature. But the solution of a UPDE was not defined well. In this article, we will rigorously give a suitable concept of a UPDE and define its solution by an integral equation. Then, some examples are given to show the rationality of the definition. Uncertain heat conduction equation is presented as an application of UPDE. For those UPDEs having no analytic solutions, \(\alpha\)-path method is introduced to obtain the inverse uncertainty distributions of solutions to UPDEs.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1

Similar content being viewed by others

References

  • Chen, X. (2011). American option pricing formula for uncertain financial market. International Journal of Operations Research, 8(2), 32–37.

    MathSciNet  Google Scholar 

  • Gu, Y., & Zhu, Y. (2022). Optimal control for parabolic uncertain system based on wavelet transformation. Axioms, 11(9), 453–470.

    Article  Google Scholar 

  • Liu, B. (2007). Uncertainty Theory (2nd ed.). Berlin: Springer-Verlag.

    Google Scholar 

  • Liu, B. (2008). Fuzzy process, hybrid process and uncertain process. Journal of Uncertain Systems, 2(1), 3–16.

    Google Scholar 

  • Liu, B. (2009). Some research problems in uncertainty theory. Journal of Uncertain Systems, 3(1), 3–10.

    Google Scholar 

  • Liu, B. (2015). Uncertainty Theory (4th ed.). Berlin: Springer-Verlag.

    Book  Google Scholar 

  • Liu, H., Ke, H., & Fei, W. (2014). Almost sure stability for uncertain differential equation. Fuzzy Optimization and Decision Making, 13(4), 463–473.

    Article  MathSciNet  Google Scholar 

  • Liu, Y., & Liu, B. (2022). Residual analysis and parameter estimation of uncertain differential equation. Fuzzy Optimization and Decision Making, 21(4), 513–530.

    Article  MathSciNet  Google Scholar 

  • Sheng, Y., & Gao, J. (2016). Exponential stability of uncertain differential equation. Soft Computing, 20(9), 3673–3678.

    Article  Google Scholar 

  • Sheng, W., & Wang, C. (2014). Stability in the p-th moment for uncertain differential equation. Journal of Intelligent & Fuzzy Systems, 26(3), 1263–1271.

    Article  MathSciNet  Google Scholar 

  • Sun, J., & Chen, X. (2015). Asian option pricing formula for uncertain financial market. Journal of Uncertainty Analysis and Applications, 3, 11.

    Article  Google Scholar 

  • Yang, L., & Liu, Y. (2023). Solution method and parameter estimation of uncertain partial differential equation with application to China’s population. Fuzzy Optimization and Decision Making. https://doi.org/10.1007/s10700-023-09415-5

    Article  Google Scholar 

  • Yang, X., & Ke, H. (2023). Uncertain interest rate model for Shanghai interbank offered rate and pricing of American swaption. Fuzzy Optimization and Decision Making, 22(3), 447–462.

    Article  MathSciNet  Google Scholar 

  • Yang, X., Ni, Y., & Zhang, Y. (2017). Stability in inverse distribution for uncertain differential equations. Journal of Intelligent & Fuzzy Systems, 32(3), 2051–2059.

    Article  Google Scholar 

  • Yang, X., & Yao, K. (2017). Uncertain partial differential equation with application to heat conduction. Fuzzy Optimization and Decision Making, 16(3), 379–403.

    Article  MathSciNet  Google Scholar 

  • Yao, K., & Chen, X. (2013). A numerical method for solving uncertain differential equations. Journal of Intelligent and Fuzzy Systems, 25, 825–832.

    Article  MathSciNet  Google Scholar 

  • Yao, K., Ke, H., & Sheng, Y. (2015). Stability in mean for uncertain differential equation. Fuzzy Optimization and Decision Making, 14(3), 365–379.

    Article  MathSciNet  Google Scholar 

  • Ye, T., & Liu, B. (2023). Uncertain hypothesis test for uncertain differential equations. Fuzzy Optimization and Decision Making, 22(2), 195–211.

    Article  MathSciNet  Google Scholar 

  • Zhu, Y. (2010). Uncertain optimal control with application to a portfolio selection model. Cybernetics and Systems, 41(7), 535–547.

    Article  Google Scholar 

  • Zhu, Y. (2019). Uncertain Optimal Control. Singapore: Springer Nature.

    Book  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, Nanjing University of Science and Technology, Nanjing, 210094, China

    Yuanguo Zhu

Authors
  1. Yuanguo Zhu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yuanguo Zhu.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zhu, Y. On uncertain partial differential equations. Fuzzy Optim Decis Making 23, 219–237 (2024). https://doi.org/10.1007/s10700-023-09418-2

Download citation

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10700-023-09418-2

Keywords

Search

Navigation