Three-dimensional uncertain heat equation

Tingqing Ye and Xiangfeng Yang ()
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Tingqing Ye: Department of Mathematical Sciences, Tsinghua University, Beijing 100084, P. R. China
Xiangfeng Yang: School of Information Technology and Management, University of International Business and Economics, Beijing 100029, P. R. China

International Journal of Modern Physics C (IJMPC), 2022, vol. 33, issue 01, 1-19

Abstract: Heat equation is a partial differential equation describing the temperature change of an object with time. In the traditional heat equation, the strength of heat source is assumed to be certain. However, in practical application, the heat source is usually influenced by noise. To describe the noise, some researchers tried to employ a tool called Winner process. Unfortunately, it is unreasonable to apply Winner process in probability theory to modeling noise in heat equation because the change rate of temperature will tend to infinity. Thus, we employ Liu process in uncertainty theory to characterize the noise. By modeling the noise via Liu process, the one-dimensional uncertain heat equation was constructed. Since the real world is a three-dimensional space, the paper extends the one-dimensional uncertain heat equation to a three-dimensional uncertain heat equation. Later, the solution of the three-dimensional uncertain heat equation and the inverse uncertainty distribution of the solution are given. At last, a paradox of stochastic heat equation is introduced.

Keywords: Uncertainty theory; uncertain process; uncertain partial differential equation; funcertain heat equation; three-dimensional space (search for similar items in EconPapers)
Date: 2022
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DOI: 10.1142/S0129183122500127

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