Heat Equation¶

• 1-D heat equation

• 3-D heat equation

Principle¶

• 沒微分兩端固定 → $\sin$
• 兩端固定一次微分等於0 → $\cos$

Finite Medium¶

take

沒微分兩端固定 = 0¶

• 沒微分兩端固定 → $\sin$
• $U(0,t)=U(L,t)=0$

兩端固定一次微分等於0¶

• 兩端固定一次微分等於0 → $\cos$
• $U_x(0,t)=U_x(L,t)=0$

有 A¶

• $U(0,t)=0;U_x(L,t)=-AU(L,t)$

• 用作圖解 $\tan (\alpha L)=-\frac{\alpha }{A}$, 令解為 $A_1,A_2,\dots$

• 解 case 3 (case 1, case 2 are trivial)

• $$f(x)=\sum _{n=1}^{\mathrm{\infty }}{c}_n\sin (\sqrt{\lambda }x)$$
• $$c_n=\frac{(f(x),\sin (\sqrt{\lambda }x))}{||\sin (\sqrt{\lambda }x)||}$$

• 參考 Eigenfunction expansion

• $$\begin{array}{c}U(x,t)=\sum _{n=1}^{\mathrm{\infty }}{c}_n\sin (\sqrt{\lambda }x)e^{-k\lambda t}\end{array}$$

兩端固定不為 0¶

• $U(0,t)=T_1,U(L,t)=T_2$

• Set $U(x,t)=u(x,t)+\varphi (x)$

• Then $u_t=k(u_{xx}+{\varphi }^{″}(x))$

• Try let ${\varphi }^{″}(x)=0$

• solve $u(x,t)$

有外力¶

• $U_t=k{U}_{xx}+F(x,t)$

• 且兩端固定為 0

• $U(x,t)=$ 外力修正 + 沒外力解

• $$B_n(t)=\frac{2}{L}{\int }_0^{L}F(x,t)\sin (\sqrt{\lambda }x)dx$$
• $$\begin{array}{c}{b}_n={\int }_0^t{e}^{-k\lambda (t-\tau )}{B}_n(\tau )d\tau \end{array}$$
• $$\begin{array}{c}U(x,t)=\sum _{n=1}^{\mathrm{\infty }}{b}_n\sin (\sqrt{{\lambda }_n}x)+\sum _{n=1}^{\mathrm{\infty }}{c}_n\sin \sqrt{{\lambda }_n}{e}^{-k{\lambda }_{n}t}\end{array}$$

Infinite Media¶

• $$\begin{array}{c}{\lambda }_n={\omega }^2\end{array}$$
• $$\begin{array}{c}U(x,t)={\int }_o^{\mathrm{\infty }}[a_{\omega }\cos (\omega x)+b_{\omega }\sin (\omega x)]e^{-k{\omega }^{2}t}d\omega \\ \\ a_{\omega }=\frac{1}{\pi }{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}f(x)\cos (\omega x)dx\\ \\ b_{\omega }=\frac{1}{\pi }{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}f(x)\sin (\omega x)dx\end{array}$$

Half infinite Media¶

• $x\in (0,\mathrm{\infty })$