TRXB 土壤学报 Acta Pedologica Sinica 0564-3929 土壤学报编辑部 中国南京 trxb-60-3-599 10.11766/trxb202210220581 S153 A 综述与评论 Reviews and Comments 土壤温度预报方程研究进展 Progress of Soil Temperature Prediction Equation 建彬 ZHANG Jianbin

20211103015@nuist.edu.cn 1

Soil temperature (especially surface temperature) is a key physical quantity in the interaction between land and atmosphere, and plays a very important role in the earth system. Soil temperature prediction technology has always been the core scientific problem in land surface model, numerical weather prediction and climate prediction. This paper systematically reviews the research progress of soil temperature prediction equation, from the classical heat conduction equation to the heat conduction convection equation that takes into account the physical process of vertical movement of soil moisture, from the single sine wave approximation to the Fourier series approximation of the daily change of surface temperature, from the assumption that the diurnal change of convection parameters is constant to the consideration of its diurnal change, and emphatically summarizes the creation, improvement and solution of the soil heat conduction convection equation. Finally, this paper reviews the application of heat conduction convection equation in the study of surface energy balance, vertical movement of soil moisture, water flux, earthquake and frozen soil heat transfer. At the same time, it is pointed out that the influences of soil water phases and plant roots on the heat conduction-convection equation is warranted for the future research of soil temperature prediction equation.

\begin{document} ${Q_{\text{g}}} = - \lambda \partial T/\partial z,$ \end{document}

\begin{document} ${C_{\text{g}}}\partial T/\partial t = - \partial {Q_{\text{g}}}/\partial z,$ \end{document}

\begin{document} $\partial T/\partial t = k{\partial ^2}T/\partial {z^2}$ \end{document}

\begin{document} ${Q_{\text{v}}} = {C_{\text{w}}}W\theta \Delta T,$ \end{document}

\begin{document} ${C_{\text{g}}}\partial T/\partial t = - \partial {Q_{\text{v}}}/\partial z,$ \end{document}

\begin{document} $\partial T/\partial t = W\partial T/\partial z,$ \end{document}

\begin{document} $\frac{{\partial T}}{{\partial t}} = k\frac{{{\partial ^2}T}}{{\partial {z^2}}} + W\frac{{\partial T}}{{\partial z}},$ \end{document}

1）谐波法

\begin{document} $\left\{ {\begin{array}{*{20}{l}} {\frac{{\partial T}}{{\partial t}} = k\frac{{{\partial ^2}T}}{{\partial {z^2}}} + W\frac{{\partial T}}{{\partial z}}(t > 0, z > 0)} \\ {{{\left. T \right|}_{z = 0}} = {T_0} + A\sin \omega t\left( {t \geqslant 0} \right)} \end{array}} \right.,$ \end{document}

\begin{document} $\begin{gathered} T\left( {z, t} \right) = {T_0} + A\exp \left[ {\left( { - \frac{W}{{2k}} - \frac{{\sqrt 2 }}{{4k}}\sqrt {{W^2} + \sqrt {{W^4} + 16{k^2}{\omega ^2}} } } \right)z} \right] \hfill \\ \;\;\;\;\;\;\;\;\;\;\;\;\sin \left[ {\omega t - z\frac{{\sqrt 2 \omega }}{{\sqrt {{W^2} + \sqrt {{W^4} + 16{k^2}{\omega ^2}} } }}} \right], \hfill \\ \end{gathered}$ \end{document}

\begin{document} $\left\{ {\begin{array}{*{20}{l}} {k = \frac{{{{\left( {{z_1} - {z_2}} \right)}^2}\omega \ln \left( {\frac{{{A_2}}}{{{A_1}}}} \right)}}{{\left( {{\Phi _2} - {\Phi _1}} \right)\left[ {{{\left( {{\Phi _2} - {\Phi _1}} \right)}^2} + {{\left( {\ln \left( {\frac{{{A_2}}}{{{A_1}}}} \right)} \right)}^2}} \right]}}} \\ {W = \frac{{\omega \left( {{z_2} - {z_1}} \right)}}{{{\Phi _2} - {\Phi _1}}}\left[ {\frac{{2{{\ln }^2}\left( {\frac{{{A_2}}}{{{A_1}}}} \right)}}{{{{\left( {{\Phi _2} - {\Phi _1}} \right)}^2} + {{\left( {\ln \left( {\frac{{{A_2}}}{{{A_1}}}} \right)} \right)}^2}}} - 1} \right]} \end{array}} \right.$ \end{document}

2）拉普拉斯变换法

\begin{document} $\left\{ {\begin{array}{*{20}{l}} {\frac{{\partial T}}{{\partial t}} = k\frac{{{\partial ^2}T}}{{\partial {z^2}}} + W\frac{{\partial T}}{{\partial z}}(t > 0, z > 0)} \\ {{{\left. T \right|}_{t = 0}} = {T_0} - \gamma z, \left( {z \geqslant 0} \right)} \\ {{{\left. T \right|}_{z = 0}} = {T_0} + A\sin \omega t\left( {t \geqslant 0} \right)} \end{array}} \right.,$ \end{document}

\begin{document} $\left\{ {\begin{array}{*{20}{l}} {\frac{{\partial {T^{**}}}}{{\partial t}} = k\frac{{{\partial ^2}{T^{**}}}}{{\partial {z^2}}}} \\ {t = 0, {T^{**}} = 0} \\ {z = 0, {T^{**}} = \left( {A\sin \omega t + \gamma Wt} \right){e^{\frac{{{W^2}}}{{4k}}t}}} \end{array}} \right.,$ \end{document}

\begin{document} $T = {T_0} - \gamma z - \gamma Wt + {T^*},$ \end{document}

\begin{document} $\begin{gathered} T\left( {z, t} \right) = {T_0} - \gamma z - \gamma Wt + \frac{{z{e^{\frac{W}{{2k}}z\frac{{{W^2}}}{{4k}}t}}}}{{2\sqrt {k\pi } }} \hfill \\ \;\;\;\;\;\;\;\;\;\;\;\mathop \smallint \limits_0 \frac{{Asin(\omega \tau + \Phi + \gamma W\tau ){e^{\frac{{{W^2}}}{{4k}}\tau }}}}{{{{\left( {t - \tau } \right)}^{3/2}}}}{e^{\frac{{{z^2}}}{{4k\left( {t - \tau } \right)}}}}d\tau . \hfill \\ \end{gathered}$ \end{document}

\begin{document} $\left\{ {\begin{array}{*{20}{l}} {\frac{{\partial T}}{{\partial t}} = k\frac{{{\partial ^2}T}}{{\partial {z^2}}} + \left( {a + b\sin (\omega t)} \right)\frac{{\partial T}}{{\partial z}}} \\ {T\left( {z, 0} \right) = f\left( z \right)} \\ {T\left( {\infty , t} \right) = {T_1}} \\ {T\left( {0, t} \right) = {T_0} + \mathop \sum \limits_i^n {A_i}\sin \left( {i\omega t + {\phi _i}} \right), \left( {t \geqslant 0} \right)} \end{array}} \right.,$ \end{document}

\begin{document}${T^*} = T\left( {z, t} \right) - {T_1}$\end{document}可得到同性的边界条件，经此变换上述方程、边界条件和初始条件变为，

\begin{document} $\left\{ {\begin{array}{*{20}{l}} {\frac{{\partial {T^*}}}{{\partial t}} = k\frac{{{\partial ^2}{T^*}}}{{\partial {z^2}}} + \left( {a + b\sin \omega t} \right)\frac{{\partial {T^*}}}{{\partial z}}} \\ {{T^*}\left( {0, t} \right) = \left( {{T_0} - {T_1}} \right) + \mathop \sum \limits_{i = 1}^n {A_i}\sin (i\omega t + {\Phi _i}), i = 1, 2, ..., n} \\ {{T^*}\left( {\infty , t} \right) = 0} \\ {{T^*}\left( {z, 0} \right) = f\left( z \right) - {T_1}} \end{array}} \right.,$ \end{document}

\begin{document} $U\left( {z, t} \right) = \frac{2}{\pi }\mathop \smallint \limits_0^\infty V\left( {p, t} \right)\sin (p\left( {x - {p_1}\left( t \right)} \right)dp,$ \end{document}

Soil temperature prediction equation and its boundary conditions, initial conditions and analytical solutions

 序号 No. 方程Equation 边界和初始条件Boundary and initial conditions 解析解Analytical solution 参考文献Reference 热传导方程 1 \begin{document}$\frac{{\partial T}}{{\partial t}} = k\frac{{{\partial ^2}T}}{{\partial {z^2}}}$\end{document} \begin{document}$T\left( {0, t} \right) = {T_0} + A\sin (\omega t + \phi ), \left( {t \geqslant 0} \right)$\end{document} \begin{document}$T\left( {z, t} \right) = {T_0} + A\exp \left( { - z\sqrt {\omega /2k} } \right)\sin \left( {\omega t - z\sqrt {\omega /2k} } \right)$\end{document}  2 \begin{document}$\frac{{\partial T}}{{\partial t}} = k\frac{{{\partial ^2}T}}{{\partial {z^2}}}$\end{document} \begin{document}$T\left( {z, t} \right) = {T_0} - \gamma z, z \geqslant 0$\end{document}\begin{document}$T\left( {0, t} \right) = {T_0} + A\sin (\omega t + \phi ), \left( {t \geqslant 0} \right)$\end{document} \begin{document}$T\left( {t, z} \right) = {T_0} - \gamma z + \frac{z}{{2\sqrt {k\pi } }}\mathop \smallint \limits_0^t \frac{{A\sin (\omega t + \phi )}}{{{{(t - \tau )}^{3/2}}}} \times {e^{ - \frac{{{z^2}}}{{4{k^2}\left( {t - \tau } \right)}}}}d\tau$\end{document}  3 \begin{document}$\frac{{\partial T}}{{\partial t}} = k\frac{{{\partial ^2}T}}{{\partial {z^2}}}$\end{document} \begin{document}$T\left( {0, t} \right) = {T_0} + \mathop \sum \limits_i^n {A_i}\sin \left( {i\omega t + {\phi _i}} \right)\left( {t \geqslant 0} \right)$\end{document} \begin{document}$T\left( {z, t} \right) = {T_0} + \mathop \sum \limits_i^n {A_i}\exp \left( { - z\sqrt {i\omega /2k} } \right) \times \sin \left( {i\omega t + {\phi _i} - z\sqrt {i\omega /2k} } \right)$\end{document} [4, 32] 4 \begin{document}$\frac{{\partial T}}{{\partial t}} = k\frac{{{\partial ^2}T}}{{\partial {z^2}}}$\end{document} \begin{document}$T\left( {z, t} \right) = {T_0} - \gamma z, z \geqslant 0$\end{document}\begin{document}$T\left( {0, t} \right) = {T_0} + \mathop \sum \limits_i^n {A_i}\sin \left( {i\omega t + {\phi _i}} \right), \left( {t \geqslant 0} \right)$\end{document} \begin{document}$T\left( {t, z} \right) = {T_0} - \gamma z + \frac{z}{{2\sqrt {k\pi } }}\mathop \smallint \limits_0^t \frac{{\mathop \sum \nolimits_i^n {A_i}\sin \left( {i\omega t + {\phi _i}} \right)}}{{{{(t - \tau )}^{3/2}}}} \times {e^{ - \frac{{{z^2}}}{{4{k^2}\left( {t - \tau } \right)}}}}d\tau$\end{document}  热传导-对流方程 5 \begin{document}$\frac{{\partial T}}{{\partial t}} = k\frac{{{\partial ^2}T}}{{\partial {z^2}}} + W\frac{{\partial T}}{{\partial z}}$\end{document} \begin{document}$T\left( {0, t} \right) = {T_0} + A\sin \left( {\omega t + \phi } \right), \left( {t \geqslant 0} \right)$\end{document} \begin{document}$\begin{gathered} T\left( {z, t} \right) = {T_0} + A\exp \left[ {\left( { - \frac{W}{{2k}} - \frac{{\sqrt 2 }}{{4k}}\sqrt {{W^2} + \sqrt {{W^4} + 16{k^2}{\omega ^2}} } } \right)z} \right] \times \hfill \\ \sin \left[ {\omega t + \phi - z\frac{{\sqrt 2 \omega }}{{\sqrt {{W^2} + \sqrt {{W^4} + 16{k^2}{\omega ^2}} } }}} \right] \hfill \\ \end{gathered}$\end{document} [3, 7] 6 \begin{document}$\frac{{\partial T}}{{\partial t}} = k\frac{{{\partial ^2}T}}{{\partial {z^2}}} + W\frac{{\partial T}}{{\partial z}}$\end{document} \begin{document}$T\left( {z, t} \right) = {T_0} - \gamma z, z \geqslant 0$\end{document}\begin{document}$T\left( {0, t} \right) = {T_0} + A\sin \omega t, \left( {t \geqslant 0} \right)$\end{document} \begin{document}$\begin{gathered} T\left( {z, t} \right) = {T_0} - \gamma z - \gamma Wt + \frac{{z{e^{\frac{W}{{2k}}z\frac{{{W^2}}}{{4k}}t}}}}{{2\sqrt {k\pi } }}\mathop \smallint \limits_0^t \frac{{A\sin (\omega \tau + \Phi + \gamma W\tau ){e^{\frac{{{W^2}}}{{4k}}\tau }}}}{{{{\left( {t - \tau } \right)}^{3/2}}}} \hfill \\ \;\;\;\;\;\;\;\;\;\;\;\;{e^{\frac{{{z^2}}}{{4k\left( {t - \tau } \right)}}}}d\tau \hfill \\ \end{gathered}$\end{document}  7 \begin{document}$\frac{{\partial T}}{{\partial t}} = k\frac{{{\partial ^2}T}}{{\partial {z^2}}} + W\frac{{\partial T}}{{\partial z}}$\end{document} \begin{document}$T\left( {0, t} \right) = {T_0} + \mathop \sum \limits_{i = 1}^n {A_i}\sin \left( {i\omega t + {\phi _i}} \right)\left( {t \geqslant 0} \right)$\end{document} \begin{document}$\begin{gathered} T\left( {z, t} \right) = {T_0} + \mathop \sum \limits_{i = 1}^n {A_i}\exp \left[ {\left( { - \frac{W}{{2k}} - \frac{{\sqrt 2 }}{{4k}}\sqrt {{W^2} + \sqrt {{W^4} + 16{i^2}{k^2}{\omega ^2}} } } \right)z} \right] \times \hfill \\ \sin \left[ {i\omega t + {\phi _i} - z\frac{{\sqrt 2 \omega }}{{\sqrt {{W^2} + \sqrt {{W^4} + 16{i^2}{k^2}{\omega ^2}} } }}} \right] \hfill \\ \end{gathered}$\end{document}  8 \begin{document}$\begin{gathered} \frac{{\partial T}}{{\partial t}} = k\frac{{{\partial ^2}T}}{{\partial {z^2}}} + \hfill \\ \;\;\;\;\;\;\;\;\left( {a + b\sin (\omega t)} \right)\frac{{\partial T}}{{\partial z}} \hfill \\ \end{gathered}$\end{document} \begin{document}$T\left( {z, 0} \right) = f\left( z \right)$\end{document}\begin{document}$T\left( {\infty , t} \right) = {T_1}$\end{document}\begin{document}$T\left( {0, t} \right) = {T_0} + A\sin (\omega t + \Phi ), \left( {t \geqslant 0} \right)$\end{document} \begin{document}$T\left( {z, t} \right) = {T_1} + \frac{2}{\pi }\mathop \smallint \limits_0^\infty V\left( {p, t} \right)\sin (p\left( {x - {p_1}\left( t \right)} \right)dp \times \exp \left( { - \frac{{{a_1}^2t}}{{4k}} - \frac{{{a_1}z}}{{2k}}} \right)$\end{document}  9 \begin{document}$\begin{gathered} \frac{{\partial T}}{{\partial t}} = k\frac{{{\partial ^2}T}}{{\partial {z^2}}} + \hfill \\ \;\;\;\;\;\;\;\;\left( {a + b\sin (\omega t)} \right)\frac{{\partial T}}{{\partial z}} \hfill \\ \end{gathered}$\end{document} \begin{document}$T\left( {z, 0} \right) = f\left( z \right)$\end{document}\begin{document}$T\left( {\infty , t} \right) = {T_1}$\end{document}\begin{document}$T\left( {0, t} \right) = {T_0} + \mathop \sum \limits_i^n {A_i}\sin \left( {i\omega t + {\phi _i}} \right)\left( {t \geqslant 0} \right)$\end{document} \begin{document}$T\left( {z, t} \right) = {T_1} + \frac{2}{\pi }\mathop \smallint \limits_0^\infty V\left( {p, t} \right)\sin (p\left( {x - {p_1}\left( t \right)} \right)dp \times \exp \left( { - \frac{{{a_1}^2t}}{{4k}} - \frac{{{a_1}z}}{{2k}}} \right)$\end{document}  强迫-恢复方法 10 \begin{document}$\frac{{\partial T}}{{\partial t}} = k\frac{{{\partial ^2}T}}{{\partial {z^2}}},$\end{document}\begin{document}$G = - \lambda \frac{{\partial T}}{{\partial z}}$\end{document} \begin{document}$T\left( {0, t} \right) = {T_0} + A\sin (\omega t + \Phi ), \left( {t \geqslant 0} \right)$\end{document} \begin{document}$\frac{{d{T_g}}}{{dt}} = \frac{{2G\left( {0, t} \right)}}{{{C_g}d}} - \omega \left( {{T_g} - \bar T} \right)$\end{document}\begin{document}$\overline T$\end{document}通过以下方程求出\begin{document}$\frac{{d\bar T}}{{dt}} = G\left( {0, t} \right)/\left[ {{C_g}{{\left( {365\pi } \right)}^{1/2}}d} \right]$\end{document} [20-22] 11 \begin{document}$\frac{{\partial T}}{{\partial t}} = k\frac{{{\partial ^2}T}}{{\partial {z^2}}} + W\frac{{\partial T}}{{\partial z}}$\end{document}\begin{document}$G=-\lambda \frac{\partial T}{\partial z}$\end{document} \begin{document}$T\left( {0, t} \right) = {T_0} + A\sin (\omega t + \Phi ), \left( {t \geqslant 0} \right)$\end{document} \begin{document}$\frac{{d{T_g}}}{{dt}} = \frac{{2G\left( {0, t} \right)}}{{{C_g}\left( {\frac{{2k}}{{N\omega }}} \right)}} - \omega \left( {\frac{N}{M}} \right)\left( {{T_g} - \bar T} \right)$\end{document}\begin{document}$\overline T$\end{document}通过以下方程求出\begin{document}$\frac{{d\bar T}}{{dt}} = G\left( {0, t} \right)/\left[ {{C_g}{{\left( {365\pi } \right)}^{1/2}}d} \right]$\end{document} 

Application of soil heat conduction-convection equation

 序号No. 应用对象Application object 土壤深度Soil depth/m 主要发现Major findings 参考文献Reference 1 地表能量平衡 0~0.05 改进地表土壤热通量的计算，提升了地表能量平衡闭合度（从0.7提升到0.9） [5-6] 2 土壤水运动趋势 0~0.20 首次发现热传导-对流方法可以捕捉沙漠土壤水月尺度的运动趋势  3 土壤水通量/地震 3.2~25.4 发现相位后移（\begin{document}${\varnothing _1} - {\varnothing _2}$\end{document}）和振幅衰减（（\begin{document}${A_1}/{A_2}$\end{document}）的空间变化特征与该地区的温泉分布相一致，从而为青藏高原东缘断裂带的热液活动提供了新的证据  4 土壤水通量/地震 0~20 估计了附近两次大地震后地下水通量变化，揭示了两次地震的不同水文响应 

Gao\begin{document}$\frac{{\partial T}}{{\partial t}} = k\frac{{{\partial ^2}T}}{{\partial {z^2}}} + W\frac{{\partial T}}{{\partial z}}$\end{document}的基础上提出了考虑热传导-对流过程的地表土壤热通量计算方法：

\begin{document} ${G_0} = {G_z} + {C_g}k\frac{{\partial T}}{{\partial z}} + {C_w}W\nabla T$ \end{document}

\begin{document} $\frac{{\partial {T^*}}}{{\partial t}} = \left( {\frac{{\partial U}}{{\partial t}} + Um} \right)\exp \left( {mt + nz} \right),$ \end{document}

\begin{document} $\frac{{\partial {T^*}}}{{\partial z}} = \left( {\frac{{\partial U}}{{\partial z}} + Un} \right)\exp \left( {mt + nz} \right),$ \end{document}

\begin{document} $\frac{{{\partial ^2}{T^*}}}{{\partial {z^2}}} = \left[ {\frac{{{\partial ^2}U}}{{\partial {z^2}}} + n\frac{{\partial U}}{{\partial z}} + n\left( {\frac{{\partial U}}{{\partial t}} + Un} \right)} \right]\exp \left( {mt + nz} \right),$ \end{document}

\begin{document} $\begin{gathered} \frac{{\partial U}}{{\partial t}} + Um = k\left( {\frac{{{\partial ^2}U}}{{\partial {z^2}}} + 2n\frac{{\partial U}}{{\partial z}} + U{n^2}} \right) + \left( {a + b\sin \omega t} \right) \hfill \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\left( {\frac{{\partial U}}{{\partial z}} + Un} \right), \hfill \\ \end{gathered}$ \end{document}

\begin{document} $\begin{gathered} \frac{{\partial U}}{{\partial t}} = k\frac{{{\partial ^2}U}}{{\partial {z^2}}} + b\sin \omega t\frac{{\partial U}}{{\partial z}} + Unb\sin \omega t + \hfill \\ \;\;\;\;\;\;\;\left( {2nk + a} \right)\frac{{\partial U}}{{\partial z}} + U\left( {k{n^2} + an - m} \right), \hfill \\ \end{gathered}$ \end{document}

\begin{document}$\left\{ {\begin{array}{*{20}{l}} {2nk + a = 0} \\ {k{n^2} + an - m = 0} \end{array}} \right.$\end{document}

\begin{document} $\frac{{\partial {T^*}}}{{\partial t}} = \left( {\frac{{\partial U}}{{\partial t}} - U\frac{{{a^2}}}{{4k}}} \right)\exp \left( { - \frac{{{a^2}t}}{{4k}} - \frac{{az}}{{2k}}} \right),$ \end{document}

\begin{document} $\frac{{\partial {T^*}}}{{\partial z}} = \left( {\frac{{\partial U}}{{\partial z}} - U\frac{a}{{2k}}} \right)\exp \left( { - \frac{{{a^2}t}}{{4k}} - \frac{{az}}{{2k}}} \right),$ \end{document}

\begin{document} $\frac{{{\partial ^2}{T^*}}}{{\partial {z^2}}} = \left( {\frac{{{\partial ^2}U}}{{\partial {z^2}}} - \frac{a}{k}\frac{{\partial U}}{{\partial z}} + \frac{{{a^2}}}{{4{k^2}}}U} \right)\exp \left( { - \frac{{{a^2}t}}{{4k}} - \frac{{az}}{{2k}}} \right),$ \end{document}

\begin{document} $\begin{gathered} \frac{{\partial U}}{{\partial t}} - U\frac{{{a^2}}}{{4k}} = k\left( {\frac{{{\partial ^2}U}}{{\partial {z^2}}} - \frac{a}{k}\frac{{\partial U}}{{\partial z}} + \frac{{{a^2}}}{{4{k^2}}}U} \right) + \hfill \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\left( {a + b\sin \omega t} \right)\left( {\frac{{\partial U}}{{\partial z}} - U\frac{a}{{2k}}} \right), \hfill \\ \end{gathered}$ \end{document}

\begin{document} $\begin{array}{l}\frac{\partial U}{\partial t}=k\left(\frac{{\partial }^{2}U}{\partial {z}^{2}}-\frac{a}{k}\frac{\partial U}{\partial z}+\frac{{a}^{2}}{4{k}^{2}}U\right)+a\frac{\partial U}{\partial z}-U\frac{{a}^{2}}{2k}+\\ b \ \mathrm{sin} \ (\omega t)\frac{\partial U}{\partial z}-b \ \mathrm{sin} \ (\omega t)\frac{a}{2k}U+U\frac{{a}^{2}}{4k}\\ =k\frac{{\partial }^{2}U}{\partial {z}^{2}}-\left(\frac{ab}{2k} \ \mathrm{sin} \ \omega t\right)U+b \ \mathrm{sin} \ \omega t\frac{\partial U}{\partial z},\end{array}$ \end{document}

\begin{document} $\left\{ \begin{array}{l} \frac{\partial U}{\partial t}=k\frac{{\partial }^{2}U}{\partial {z}^{2}}-\left(\frac{ab}{2k} \ \mathrm{sin} \ \omega t\right)U\left(z, t\right)+b \ \mathrm{sin} \ \omega t\frac{\partial U}{\partial z}\\ U\left(0, t\right)=\mathrm{exp}\left(\frac{{a}^{2}t}{4k}\right)[\left({T}_{0}-{T}_{1}\right)+ \sum\limits_{i = 1}^n {A}_{i} \ \mathrm{sin} \ (i\omega t+{\Phi }_{i})], i=1, 2, \mathrm{...}, n\\ U\left(\infty , t\right)=0\\ U\left(z, 0\right)=\left[f\left(z\right)-{T}_{1}\right]\mathrm{exp}\left(\frac{az}{2k}\right)\end{array} \right. ,$ \end{document}

\begin{document} ${p_1}\left( t \right) = \mathop \smallint \limits_0^t b\sin \omega tdt = \frac{b}{\omega }\left( {1 - \cos \omega t} \right),$ \end{document}

\begin{document}$Z = z + {p_1}\left( t \right)$\end{document}，得到\begin{document}$U\left( {z, t} \right) = U\left[ {Z - {p_1}\left( t \right), t} \right] = V\left( {Z, t} \right)$\end{document}

UVt求偏导，可得\begin{document}$\frac{{\partial U}}{{\partial t}} = \frac{{\partial V}}{{\partial t}} + \frac{{\partial V}}{{\partial Z}}\frac{{\partial Z}}{{\partial t}}$\end{document}由此可以得到

\begin{document} $\frac{{\partial U}}{{\partial t}} = \frac{{\partial V}}{{\partial t}} + \frac{{\partial {p_1}\left( t \right)}}{{\partial t}}\frac{{\partial V}}{{\partial Z}} = \frac{{\partial V}}{{\partial t}} + b\sin \omega t\frac{{\partial V}}{{\partial Z}},$ \end{document}

\begin{document} $\left\{ \begin{array}{l} \frac{\partial U}{\partial z}=\frac{\partial V}{\partial Z}\\ \frac{{\partial }^{2}U}{\partial {z}^{2}}=\frac{{\partial }^{2}V}{\partial {Z}^{2}}\end{array} \right.,$ \end{document}

\begin{document} $\left\{ {\begin{array}{*{20}{l}} {\frac{{\partial V}}{{\partial t}} = k\frac{{{\partial ^2}V}}{{\partial {Z^2}}} - \left( {\frac{{ab}}{{2k}}\sin \omega t} \right)V\left( {Z, t} \right)} \\ {V\left( {{p_1}\left( t \right), t} \right) = \exp \left( {\frac{{{a^2}t}}{{4k}}} \right)[\left( {{T_0} - {T_1}} \right) + \mathop \sum \limits_{i = 1}^n {A_i}\sin (i\omega t + {\Phi _i})], i = 1, 2, ..., n} \\ {V\left( {\infty , t} \right) = 0} \\ {V\left( {Z, 0} \right) = \left[ {f\left( Z \right) - {T_1}} \right]\exp \left( {\frac{{aZ}}{{2k}}} \right)} \end{array}} \right. ,$ \end{document}

\begin{document} $V\left( {p, t} \right) = \mathop \smallint \limits_0^\infty V\left( {Z, t} \right)\sin (pZ)dZ,$ \end{document}

\begin{document} $\left\{ {\begin{array}{*{20}{l}} {\frac{{dV\left( {p, t} \right)}}{{dt}} = - \left[ {k{p^2} + \frac{{ab}}{{2k}}\sin (\omega t} \right)]V\left( {p, t} \right) + kp\exp (\frac{{{a^2}t}}{{4k}})\left[ {\left( {{T_0} - {T_1}} \right) + \mathop \sum \limits_{i = 1}^n {A_i}\sin (i\omega t + {\Phi _i})} \right]} \\ {V\left( {p, 0} \right) = \mathop \smallint \limits_0^\infty [f\left( Z \right) - {T_1}]\exp \left( {\frac{{aZ}}{{2k}}} \right)\sin \left( {pZ} \right)dZ} \end{array}} \right.$ \end{document}

\begin{document} $\begin{array}{l}V=\mathrm{exp}(-k{p}^{2}t+{b}_{3} \ \mathrm{cos} \ \omega t)\left(M+J+c\right)=\\ \mathrm{exp}(-k{p}^{2}t+{b}_{3} \ \mathrm{cos} \ \omega t)\left({V}_{1}+{V}_{2}+{V}_{3}+{V}_{4}\right),\end{array}$ \end{document}

\begin{document} ${V}_{1}={b}_{1}\mathrm{exp}({b}_{2}t)\sum\limits_{i = 1}^n \frac{{b}_{2}{A}_{i} \ \mathrm{sin} \ (i\omega t+{\Phi }_{i})-i\omega {A}_{i} \ \mathrm{cos} \ (i\omega t+{\Phi }_{i})}{{b}_{2}{}^{2}+{(i\omega )}^{2}},$ \end{document}

\begin{document} ${V}_{2}=-{b}_{1}{b}_{3}\sum\limits_{i = 1}^n \left[ \ \mathrm{cos} \ {\Phi }_{i}{V}_{2}{}'\left(i\right)+ \ \mathrm{sin} \ {\Phi }_{i}{V}_{2}{}''\left(i\right)\right],$ \end{document}

\begin{document} $\begin{array}{l}{V}_{2}{}'\left(i\right)=\frac{1}{2}\frac{\mathrm{exp}({b}_{2}t)}{{b}_{2}{}^{2}+{(i+1)}^{2}{\omega }^{2}}\left\{{b}_{2}{A}_{i} \ \mathrm{sin} \ [\left(i+1\right)\omega t\left]-\left(i+1\right)\omega {A}_{i} \ \mathrm{cos} \ [\left(i+1\right)\omega t\right]\right\}\\ +\frac{1}{2}\frac{\mathrm{exp}({b}_{2}t)}{{b}_{2}{}^{2}+{(i-1)}^{2}{\omega }^{2}}\left\{{b}_{2}{A}_{i} \ \mathrm{sin} \ [\left(i-1\right)\omega t\left]-\left(i-1\right)\omega {A}_{i} \ \mathrm{cos} \ [\left(i-1\right)\omega t\right]\right\},\end{array}$ \end{document}

\begin{document} $\begin{array}{l}{V}_{2}{}''\left(i\right)=\frac{1}{2}\frac{\mathrm{exp}({b}_{2}t)}{{b}_{2}{}^{2}+{(i+1)}^{2}{\omega }^{2}}\left\{{b}_{2}{A}_{i} \ \mathrm{cos} \ [\left(i+1\right)\omega t\left]+\left(i+1\right)\omega {A}_{i} \ \mathrm{sin} \ [\left(i+1\right)\omega t\right]\right\}\\ +\frac{1}{2}\frac{\mathrm{exp}({b}_{2}t)}{{b}_{2}{}^{2}+{(i-1)}^{2}{\omega }^{2}}\left\{{b}_{2}{A}_{i} \ \mathrm{cos} \ [\left(i-1\right)\omega t\left]+\left(i-1\right)\omega {A}_{i} \ \mathrm{sin} \ [\left(i-1\right)\omega t\right]\right\},\end{array}$ \end{document}

\begin{document} ${V}_{3}=\frac{{b}_{4}}{{b}_{2}}\mathrm{exp}({b}_{2}t),$ \end{document}

\begin{document} ${V}_{4}=-{b}_{4}{b}_{3}\mathrm{exp}({b}_{2}t)\frac{{b}_{2} \ \mathrm{cos} \ (\omega t)+\omega \ \mathrm{sin} \ (\omega t)}{{b}_{2}{}^{2}+{\omega }^{2}}。$ \end{document}

\begin{document}$t = 0$\end{document}时，

\begin{document} $V\left( {p, 0} \right) = \exp (\frac{{{a_1}b}}{{2k\omega }})\left[ {{V_1}\left( {p, 0} \right) + {V_2}\left( {p, 0} \right) + {V_3}\left( {p, 0} \right) + {V_4}\left( {p, 0} \right) + c} \right]$ \end{document}

\begin{document} $c = \exp ( - \frac{{ab}}{{2k\omega }})V\left( {p, 0} \right) - \left[ {{V_1}\left( {p, 0} \right) + {V_2}\left( {p, 0} \right) + {V_3}\left( {p, 0} \right) + {V_4}\left( {p, 0} \right)} \right],$ \end{document}

\begin{document} $V\left( {p, 0} \right) = \mathop \smallint \limits_0^\infty [f\left( Z \right) - {T_1}]\exp \left( {\frac{{aZ}}{{2k}}} \right)\sin \left( {pZ} \right)dZ。$ \end{document}

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