Chap2

\[ \left\{ \begin{array}{rcl} \frac{\partial^2 u}{\partial t^2}=a^2\frac{\partial^2 u}{\partial x^2},(-\infty<x<+\infty,t>0) \\ u|_{t=0}=\varphi(x),(-\infty<x<+\infty) \\ \frac{\partial u}{\partial t} |_{t=0}=\phi(x),(-\infty<x<+\infty) \end{array} \right. \]

\[ u(x,t)=\frac{1}{2} (\varphi (x-at) \varphi (x+at)) + \frac{1}{2a} \int_{x-at}^{x+at} \varphi(\alpha) d\alpha \]

\[ u(x,t) = \left\{ \begin{aligned} \frac{1}{2} \left[ \varphi(x + at) + \varphi(x - at) \right] + \frac{1}{2a} \int_{x - at}^{x + at} \phi(\alpha) \, d\alpha, & x-at \ge 0 \\ \frac{1}{2} \left[ \varphi(x+at)-\varphi(at-x)\right]+\frac{1}{2a}\int_{at-x}^{x+at} \varphi(\alpha)d\alpha ,x-at<0 \end{aligned} \right. \]

\[ \left\{ \begin{aligned} \frac{\partial^2 \omega}{\partial t^2}=a^2 \frac{\partial \omega}{\partial x^2} \quad (-\infty<x<+\infty,t>\tau) \\ \omega|_{t=\tau}=0 \quad(-\infty<x<+\infty) \\ \frac{\partial \omega}{\partial t}|_{t=\tau} =f(x,\tau) (-\infty<x<+\infty) \end{aligned} \right. \]

\[ u(x,t)=\int_{0}^{t} \omega(x,t,\tau)d\tau \]

\[ \left\{ \begin{aligned} \frac{\partial^2 u}{\partial t^2}=a^2\frac{\partial^2 u}{\partial x^2}+f(x,t) \quad (-\infty<x<+\infty,t>0) \\ u|_{t=0}=0 \quad (-\infty<x<+\infty) \\ \frac{\partial u}{\partial t}|_{t=0}=0 \quad (-\infty<x<+\infty) \end{aligned} \right. \]

\[ u(x,t)=\frac{1}{2}[\varphi(x+at) +\varphi(x-at)] +\frac{1}{2a}\int_{x-at}^{x+at}\varphi(\xi)d \xi + \frac{1}{2a}\iint_G f(\xi,\tau)d\xi d\tau \]