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Solution method and parameter estimation of uncertain partial differential equation with application to China’s population

Author

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  • Lu Yang

    (Xi’an University of Finance and Economics)

  • Yang Liu

    (Beihang University)

Abstract

Since the concept of uncertain partial differential equations (UPDEs) was proposed, it has been developed significantly and led us to study parameter estimation for UPDEs. This paper proposes a concept of residual of a class of UPDEs, which follows a linear uncertainty distribution. Afterwards, an $$\alpha$$ α -path of a class of UPDEs is introduced and the important result that the inverse uncertainty distribution of solution of a class of UPDEs is just the $$\alpha$$ α -path of the corresponding UPDEs is reached. And a numerical method is designed to obtain the inverse uncertainty distribution of solution of UPDEs. In addition, based on the $$\alpha$$ α -path and the inverse uncertainty distribution, an algorithm is designed for calculating the residuals of UPDEs corresponding to the observed data. Then a method of moments to estimate unknown parameters in UPDEs is provided. Furthermore, uncertain hypothesis test is recast to evaluate whether an uncertain partial differential equation fits the observed data. Finally, the method of moments is applied to modeling China’s population and the fitness of the estimated parameters is verified by using uncertain hypothesis test.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:fuzodm:v:23:y:2024:i:1:d:10.1007_s10700-023-09415-5
    DOI: 10.1007/s10700-023-09415-5
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    References listed on IDEAS

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    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. Xiaowei Chen & Jing Li & Chen Xiao & Peilin Yang, 2021. "Numerical solution and parameter estimation for uncertain SIR model with application to COVID-19," Fuzzy Optimization and Decision Making, Springer, vol. 20(2), pages 189-208, June.
    3. 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.
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    Cited by:

    1. Binhao He & Baoding Liu, 2025. "Higher-order partial derivatives of uncertain field with application to higher-order uncertain partial differential equation," Fuzzy Optimization and Decision Making, Springer, vol. 24(2), pages 317-342, June.
    2. Liu, Zhe & Li, Yanbin, 2025. "Carbon option pricing using uncertain differential equation," Energy, Elsevier, vol. 320(C).
    3. 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.
    4. Haoxuan Li & Jianlin He & Qi Zhang & Chuanmei Zhang & Wusheng Wang & Xianshuai Cao & Tongjie Zhang & Xiangfeng Yang, 2025. "Modeling product sales with uncertain differential equation: an application to Meituan," Fuzzy Optimization and Decision Making, Springer, vol. 24(2), pages 271-292, June.

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