非Fourier温度场分布的奇摄动解

包立平; 李文彦; 吴立群

doi:10.21656/1000-0887.390112

非Fourier温度场分布的奇摄动解

doi: 10.21656/1000-0887.390112

¹杭州电子科技大学理学院, 杭州 310018;²杭州电子科技大学机械工程学院, 杭州 310018

基金项目: 国家自然科学基金（51175134）；浙江省重点自然科学基金（LZ15E050004）

详细信息

作者简介:
包立平(1962—)，男，副教授，博士(通讯作者. E-mail: baolp@hdu.edu.cn)；李文彦(1993—)，女，硕士生.

中图分类号: TK12
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- 被引次数: 0
出版历程
- 收稿日期: 2018-04-11
- 修回日期: 2018-11-12
- 刊出日期: 2019-05-01

Singularly Perturbed Solutions of NonFourier Temperature Field Distribution in Single-Layer Materials

¹School of Sciences, Hangzhou Dianzi University, Hangzhou 310018, P.R.China;²School of Mechanical Engineering, Hangzhou Dianzi University, Hangzhou 310018, P.R.China

Funds: The National Natural Science Foundation of China（51175134）

摘要

摘要: 应用非Fourier热传导定律构建了单层材料中温度场模型，即一类在无界域上带小参数的奇摄动双曲方程，通过奇摄动展开方法，得到了该问题的渐近解.首先应用奇摄动方法得到了该问题的外解和边界层矫正项，通过对内解和外解的最大模估计和关于时间导数的最大模估计以及线性抛物方程理论，得到了内外解的存在唯一性，从而得到了解的形式渐近展开式.通过余项估计，给出了渐近解的L²估计，得到了渐近解的一致有效性，从而得到了无界域上温度场的分布.通过奇摄动分析，给出了非Fourier 温度场与Fourier 温度场的关系，描述了非Fourier温度场的具体形态.
- 奇摄动 /
- 热传导方程 /
- 无界域 /
- 一致有效估计
Abstract: A temperature field model for single-layer materials was constructed with the non-Fourier heat conduction law, i.e. a type of singularly perturbed hyperbolic equations with small parameters in an unbounded domain. The asymptotic solution to the problem was obtained with the singularly perturbed expansion method. Firstly, the singular perturbation method was used to obtain the external solution and boundary layer correction terms of the problem. Through estimation of the maximum norms of the internal solution and the external solution, and the maximum norms of the time derivative, and under the theory of linear parabolic equations, the existence and uniqueness of the internal and external solutions were obtained, and the formal asymptotic expansion of the solution was obtained. The L² estimator of the asymptotic solution was given with the remainder estimator. The uniform validity of the asymptotic solution and the distribution of the temperature field in the unbounded domain were got. Through singular perturbation analysis, the relationship between the non-Fourier temperature field and the Fourier temperature field was given, and the specific behaviors of the non-Fourier temperature field were described.
- singular perturbation /
- heat conduction equation /
- unbounded domain /
- uniformly valid estimate

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参考文献(16)

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