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Numerical solution of uncertain partial differential equations and its applications

Author

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  • Lu Yang

    (Xi’an University of Finance and Economics)

  • Yang Liu

    (Beihang University)

Abstract

Uncertain partial differential equations are widely used in practice, such as demography, traffic flows and so on. This paper proves an existence and uniqueness theorem for a class of uncertain partial differential equations. Then the properties of $$\alpha$$ α -path are given based on linear growth, Lipschitz and regular conditions. Since uncertain partial differential equations are difficult to get analytical solutions, this paper presents a formula which combines an uncertain partial differential equation with a class of classical partial differential equations. Based on the formula, an algorithm for calculating the inverse uncertainty distribution of solution of an uncertain partial differential equation is also deduced. Finally, expected value, extreme value, first hitting time, time and spatial integrals of the solution of uncertain partial differential equation are also discussed.

Suggested Citation

  • Lu Yang & Yang Liu, 2025. "Numerical solution of uncertain partial differential equations and its applications," Fuzzy Optimization and Decision Making, Springer, vol. 24(1), pages 69-98, March.
    • Handle: RePEc:spr:fuzodm:v:24:y:2025:i:1:d:10.1007_s10700-025-09439-z
    DOI: 10.1007/s10700-025-09439-z
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    References listed on IDEAS

    as
    1. Tingqing Ye & Baoding Liu, 2023. "Uncertain hypothesis test for uncertain differential equations," Fuzzy Optimization and Decision Making, Springer, vol. 22(2), pages 195-211, June.
    2. Tingqing Ye, 2024. "Partial derivatives of uncertain fields and uncertain partial differential equations," Fuzzy Optimization and Decision Making, Springer, vol. 23(2), pages 199-217, June.
    3. Xiangfeng Yang & Hua Ke, 2023. "Uncertain interest rate model for Shanghai interbank offered rate and pricing of American swaption," Fuzzy Optimization and Decision Making, Springer, vol. 22(3), pages 447-462, September.
    4. Chen, Dan & Liu, Yang, 2023. "Uncertain Gordon-Schaefer model driven by Liu process," Applied Mathematics and Computation, Elsevier, vol. 450(C).
    5. Yuanguo Zhu, 2024. "On uncertain partial differential equations," Fuzzy Optimization and Decision Making, Springer, vol. 23(2), pages 219-237, June.
    6. Tingqing Ye & Xiangfeng Yang, 2022. "Three-dimensional uncertain heat equation," International Journal of Modern Physics C (IJMPC), World Scientific Publishing Co. Pte. Ltd., vol. 33(01), pages 1-19, January.
    7. Xiangfeng Yang & Kai Yao, 2017. "Uncertain partial differential equation with application to heat conduction," Fuzzy Optimization and Decision Making, Springer, vol. 16(3), pages 379-403, September.
    8. Lu Yang & Yang Liu, 2024. "Solution method and parameter estimation of uncertain partial differential equation with application to China’s population," Fuzzy Optimization and Decision Making, Springer, vol. 23(1), pages 155-177, March.
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