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应用非Fourier热传导定律构建了温度场模型, 即一类在无界域上的三维奇摄动双曲抛物方程的初边值问题. 随着温度急剧变化, 热传导系数发生跳跃, 相应可以用非线性的具有间断系数的奇摄动双参数双曲方程表示. 通过奇摄动双参数展开方法, 得到了该问题的渐近解. 首先应用奇摄动方法得到该问题的展开式, 通过对解做出估计以及古典解的存在唯一性定理给出了内解和外解的存在性、唯一性. 其次, 由奇摄动理论, 得到该类奇摄动双曲方程进行了初始层矫正, 得到了解关于时间的导数的估计. 并且通过用Fourier 变换确定了热传导系数跳跃的位置表达式, 从而得到了解的形式渐近展开式. 最后通过余项估计, 得到了渐近解的一致有效性, 从而得到了热传导系数间断的温度场的分布.Thermoelastic coupling model excited by laser is of great significance in engineering. To study the thermoelastic coupling model, the distribution of temperature field must be determined firstly. Because the laser excitation time is short (usually femtosecond), the traditional Fourier heat conduction law is no longer suitable. Therefore, it is necessary to establish the distribution of temperature field by using the non-Fourier heat conduction law. Previous studies on the temperature field model mostly use numerical analysis and computer simulation to discuss its numerical solution, but few can directly solve the analytical solution of the model. Up to now, there are few reports about using singularly perturbed analysis method to solve the asymptotic solution of temperature field model and determine the jumping position of heat conductivity coefficient. In this paper, a temperature field model is constructed by using the non-Fourier heat conduction law, i.e. a class of singularly perturbed hyperbolic equations with small parameters in an unbounded domain. The nonlinear singularly perturbed two-parameter hyperbolic equations with discontinuous coefficients are obtained when the heat conduction coefficients jump due to sharp temperature changes. By using the singularly perturbed biparametric expansion method, the asymptotic solution of the problem is obtained. First, the expansion of the problem is obtained by using singularly perturbed method. The existence and uniqueness of the internal and external solutions are obtained by estimating the maximum modulus of the internal and external solutions and the maximum modulus estimates of the time derivatives, and the formal asymptotic expansion of the solutions is obtained. Secondly, the singularly perturbed hyperbolic equation is corrected by the singular perturbation theory, and the derivative of the solution is estimated. The position expression of the jump of the thermal conductivity coefficient is determined by the Fourier transform, and the seam method is used to connect the seams of the two sides of the jump position of the thermal conductivity coefficient, thus the form asymptotic expansion of the solution is obtained. Finally, the uniform validity of the asymptotic solution is obtained by estimating the residual term, and the distribution of the temperature field with discontinuous heat conduction coefficient is obtained. In this paper, we have synthetically applied the knowledge of ordinary differential equations, partial differential equations, mathematical and physical equations, nonlinear acoustics, mathematical analysis, singular perturbation theory and so on, which enriched the study of non-Fourier temperature field model.
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Keywords:
- heat conduction equation /
- discontinuous coefficient /
- uniformly valid estimate /
- two parameters
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[1] 关建飞, 沈中华, 许伯强 等 2005 光电子·激光 16 231
Google Scholar
Guan J F, Shen Z H, Xu B Q, et al. 2005 Photoelectronics and Laser 16 231
Google Scholar
[2] 沈中华, 许伯强, 倪晓武 等 2004 中国激光 31 1275
Google Scholar
Shen Z H, Xu B Q, Ni X W, et al. 2004 China Laser 31 1275
Google Scholar
[3] Tzou D Y 1995 Int. J. Heat. Mass. Transf. 38 3231
Google Scholar
[4] Tzou D Y 1995 ASME J. Heat. Mass. Transf. 117 8
Google Scholar
[5] 李金娥, 王保林, 常冬梅 2011 固体力学学报 s1 248
Li J E, Wang B L, Chang D M 2011 J. Solid Mechanics s1 248
[6] 张浙, 刘登瀛 2000 力学进展 30 446
Google Scholar
Zhang Z, Liu D Y 2000 Progress in Mechanics 30 446
Google Scholar
[7] Liu Y, Li H, He S 2010 Numer. Math. A: J. Chin. Univ. 171 1
[8] Amirov R 2014 Inter. Conference on Non. Differential and Difference Equations May 2 7
[9] Farrell P A, O’Riordan E, Shishkin G I 2005 Math. Computation 74 1759
Google Scholar
[10] Teixeira M A, Silva P R D 2012 Physica D: Non. Phenomena 241 1948
Google Scholar
[11] Teixeira M A 2012 Perturbation Theory for Non-smooth Systems (New York: Springer) pp32−49
[12] 林娟 2011 武汉大学学报(理学版) 57 109
Lin J 2011 J. Wuhan University (Science Edition)
57 109 [13] 兴梅 2005 数学物理学报 25 685
Google Scholar
Xing M 2005 J. Math. Phys. 25 685
Google Scholar
[14] Tan Q J, Leng Z J 2008 Math. Research. Rev. (English Edition)
4 41 [15] Llibre J, Silva P R D, Teixeira M A 2006 J. Dyn. Differ. Equ. 19 309
[16] 康连城 1989 数学年刊 A辑 (中文版) 13 529
Kang L C 1989 Math. Annual Series A (Chinese Edition)
13 529 [17] 康连城 1992 应用数学和力学 13 135
Kang L C 1992 Appl. Math. Mechanics 13 135
[18] 包立平 , 李文彦, 吴立群 2019 应用数学和力学 40 536
Bao L P, Li W Y, Wu L Q 2019 Appl. Math. Mech. 40 536
[19] Zhang J Q, Nie L R, Chen C Y, et al. 2016 AIP Advances 6 075212
Google Scholar
[20] Chen R, Nie L, Chen C 2018 Chaos 28 053115
Google Scholar
[21] Chen R, Nie L, Chen C, et al. 2017 J. Stat. Mech Theory. E 14 013201
[22] Nie L, Yu L, Zheng Z, et al. 2013 Phys. Rev. 062142
[23] 伍卓群, 尹景学, 王春明 2003 椭圆与抛物型方程引论 (北京: 科学出版社) 第152−184页
Wu Z Q, Yin J X, Wang C M 2003 Introduction to Elliptic and Parabolic Equations Fristplace (Beijing: Science Press) (in Chinese) pp152−184 (in Chinese)
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