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A Dufort–Frankel scheme for one-dimensional uncertain heat equation

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  • Yang, Xiangfeng
  • Ralescu, Dan A.

Abstract

Uncertain heat equation (for short, UHE) is a type of second-order uncertain PDEs driven by Liu processes. In most cases, since it is tough to get the analytic solution for a UHE, we must find a way to obtain its numerical solution. A forward difference scheme has been designed to solve UHE, but our paper will show that this method may exhibit instability in some situations. We explore another approach, unconditionally stable and namely Dufort–Frankel method. Moreover, this paper will use Dufort–Frankel method to calculate the expected value and extreme value of the solution for a UHE.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:matcom:v:181:y:2021:i:c:p:98-112
    DOI: 10.1016/j.matcom.2020.09.022
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    References listed on IDEAS

    as
    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. Yang, Xiangfeng, 2018. "Solving uncertain heat equation via numerical method," Applied Mathematics and Computation, Elsevier, vol. 329(C), pages 92-104.
    3. Gao, Rong, 2016. "Milne method for solving uncertain differential equations," Applied Mathematics and Computation, Elsevier, vol. 274(C), pages 774-785.
    4. Zhang, Yi & Gao, Jinwu & Huang, Zhiyong, 2017. "Hamming method for solving uncertain differential equations," Applied Mathematics and Computation, Elsevier, vol. 313(C), pages 331-341.
    5. Gao, Rong, 2017. "Uncertain wave equation with infinite half-boundary," Applied Mathematics and Computation, Elsevier, vol. 304(C), pages 28-40.
    6. 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.
    7. 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.
    8. 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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