User:Glengarry

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$\frac{dS_n(t)}{dt}+\gamma_nS_n(t)=\zeta_n\mathrm{e}^{\beta t}$

$S_n(t)=\frac{2}{L}\int_{0}^{L}\phi(x,t)\sin\left(\frac{n\pi x}{L}\right)dx$

$\gamma_n=\left(\frac{n\pi}{L}\right)^2\frac{k_s}{cp}\,$

$\zeta_n=\frac{2J^2\rho}{cpn\pi}[1-\cos (n\pi)]\,$

$S_n(t)=\frac{\zeta_n}{\gamma_n+\beta}(\mathrm{e}^{\beta t}-\mathrm{e}^{-\gamma_n t})\,$

$\phi(x,t)=\sum_{n=1}^{\infty} \frac{\zeta_n}{\gamma_n+\beta}(\mathrm{e}^{\beta t}-\mathrm{e}^{-\gamma_n t})\sin\left(\frac{n\pi x}{L}\right)\,$

$T(x,t)=T_\infty+\sum_{n=1}^{\infty} \frac{\zeta_n}{\gamma_n+\beta}[1-\mathrm{e}^{-(\gamma_n+\beta)t}]\sin\left(\frac{n\pi x}{L}\right)\,$

$\tau_1=(\gamma_1+\beta)^{-1}=\left(\frac{\pi^2 k_s}{cpL^2}+\frac{Sk_a}{ghcp}\right)^{-1}$

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