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Solving uncertain heat equation via numerical method

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  • Yang, Xiangfeng

Abstract

Uncertain heat equation is a type of uncertain partial differential equations driven by Liu processes. This paper proposes a concept of α-path for uncertain heat equation, and shows that the solution of an uncertain heat equation can be represented by a family of solutions of ordinary heat equations. And, a formula is derived to calculate expected value of solution of uncertain heat equation. Moreover, a numerical method is designed to solve uncertain heat equation. Several examples are given to illustrate the efficiency of the numerical method.

Suggested Citation

  • Yang, Xiangfeng, 2018. "Solving uncertain heat equation via numerical method," Applied Mathematics and Computation, Elsevier, vol. 329(C), pages 92-104.
    • Handle: RePEc:eee:apmaco:v:329:y:2018:i:c:p:92-104
    DOI: 10.1016/j.amc.2018.01.055
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    References listed on IDEAS

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    1. Yang, Xiangfeng & Ralescu, Dan A., 2015. "Adams method for solving uncertain differential equations," Applied Mathematics and Computation, Elsevier, vol. 270(C), pages 993-1003.
    2. Wang, Xiao & Ning, Yufu & Moughal, Tauqir A. & Chen, Xiumei, 2015. "Adams–Simpson method for solving uncertain differential equation," Applied Mathematics and Computation, Elsevier, vol. 271(C), pages 209-219.
    3. Gao, Rong, 2016. "Milne method for solving uncertain differential equations," Applied Mathematics and Computation, Elsevier, vol. 274(C), pages 774-785.
    4. 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. Yang, Xiangfeng & Ralescu, Dan A., 2021. "A Dufort–Frankel scheme for one-dimensional uncertain heat equation," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 181(C), pages 98-112.
    2. Jia, Lifen & Lio, Waichon & Yang, Xiangfeng, 2018. "Numerical method for solving uncertain spring vibration equation," Applied Mathematics and Computation, Elsevier, vol. 337(C), pages 428-441.
    3. Yang, Meng & Ni, Yaodong & Song, Qinyu, 2022. "Optimizing driver consistency in the vehicle routing problem under uncertain environment," Transportation Research Part E: Logistics and Transportation Review, Elsevier, vol. 164(C).
    4. Jian Zhou & Yujiao Jiang & Athanasios A. Pantelous & Weiwen Dai, 2023. "A systematic review of uncertainty theory with the use of scientometrical method," Fuzzy Optimization and Decision Making, Springer, vol. 22(3), pages 463-518, September.
    5. Lu Yang & Tingqing Ye & Haizhong Yang, 2022. "Uncertain seepage equation in fissured porous media," Fuzzy Optimization and Decision Making, Springer, vol. 21(3), pages 383-403, September.
    6. Jia, Lifen & Sheng, Yuhong, 2019. "Stability in distribution for uncertain delay differential equation," Applied Mathematics and Computation, Elsevier, vol. 343(C), pages 49-56.
    7. Abdulsalam, Athraa & Senu, Norazak & Majid, Zanariah Abdul & Long, Nik Mohd Asri Nik, 2024. "Development of high-order adaptive multi-step Runge–Kutta–Nyström method for solving special second-order ODEs," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 216(C), pages 104-125.
    8. Chen, Xin & Zhu, Yuanguo & Sheng, Linxue, 2021. "Optimal control for uncertain stochastic dynamic systems with jump and application to an advertising model," Applied Mathematics and Computation, Elsevier, vol. 407(C).

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