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# 分离变量法 I: 齐次方程，齐次边界条件

$\begin{cases} u_{t}-a^2u_{xx}=0,\quad & 0\le x\le L, t\geq 0 \\ u(0,t)=0, u(L,t)=0,&t\geq 0\\ u(x,0)=x,& 0\leq x\leq L \end{cases}$

$T'(t)X(x)-a^2T(t)X^{\prime\prime}(x)=0$

$\frac{T'(t)}{a^2T(t)}=\frac{X^{\prime\prime}(x)}{X(x)}$

$\frac{T'(t)}{a^2T(t)}=\frac{X^{\prime\prime}(x)}{X(x)}=-\lambda$

$\begin{array}{l} X^{\prime\prime}(x)+\lambda X(x)=0\\ T'(t)+a^2\lambda T(t)=0 \end{array}$

$\begin{cases} X^{\prime\prime}(x)+\lambda X(x)=0, 0\le x\le L\\ X(0)=0, X(L)=0 \end{cases}$

$r^2+\lambda=0$

$r_{1,2}=\pm\sqrt{-\lambda}$

$X(x)=C_1e^{\mu x}+C_2e^{-\mu x}$

$X(x)=C_1\sinh(\mu x)+C_2\cosh(\mu x)$

$X(x)=C_1x+C_2.$

$X(x)=C_1\cos(\mu x)+C_2\sin(\mu x)$

$\mu=\frac{n\pi}{L}, n=1,2,\cdots$

$X_n=\sin(\frac{n\pi x}{L}),$

$\begin{cases} X”(x)+\lambda X(x)=0, 0\le x\le L\\ X(0)=0, X(L)=0 \end{cases}$

$T'(t)+a^2\lambda T(t)=0$

$T'(t)+a^2(\frac{n\pi}{L})^2 T(t)=0$

$T_n(t)=A_ne^{-\frac{a^2n^2\pi^2}{L^2}t}$

$u_n=T_n(t)X_n(x)=A_ne^{-\frac{a^2n^2\pi^2}{L^2}t}\sin(\frac{n\pi x}{L})，n=1,2,\cdots$

$u(x,t)=\sum_{n=1}^{\infty}u_n=\sum_{n=1}^{\infty}A_ne^{-\frac{a^2n^2\pi^2}{L^2}t}\sin(\frac{n\pi x}{L})$

$f(x)=\sum_{n=1}^{\infty}A_n\sin(\frac{n\pi x}{L})$

$A_n=\frac{2}{L}\int_0^L f(x)\sin(\frac{n\pi x}{L})dx$

$u(x,t)=\sum_{n=1}^{\infty}u_n=\sum_{n=1}^{\infty}(\frac{2}{L}\int_0^L f(x)\sin(\frac{n\pi x}{L})dx)e^{-\frac{a^2n^2\pi^2}{L^2}t}\sin(\frac{n\pi x}{L})$