方企勤 第五章 多元函数微分学 第5.5题
📝 题目
5.5.3 设 $f\left( x\right) \in {C}^{2}\left( {-\infty ,\infty }\right) ,F\left( x\right) \in {C}^{1}\left( {-\infty ,\infty }\right)$ ,
$$ u = \frac{1}{2}\left\lbrack {f\left( {x + {at}}\right) + f\left( {x - {at}}\right) }\right\rbrack + \frac{1}{2n}{\int }_{x - {at}}^{x + {at}}F\left( y\right) \mathrm{d}y. $$
求证: 当 $\displaystyle{- \infty < x < \infty ,t \geq 0}$ 时, $u\left( {x,t}\right) ,\frac{{\partial }^{2}u}{\partial {t}^{2}},\frac{{\partial }^{2}u}{\partial {x}^{2}}$ 连续,且满足
$$ \frac{{\partial }^{2}u}{\partial {t}^{2}} = {a}^{2}\frac{{\partial }^{2}v}{\partial {x}^{2}},\;u\left( {x,0}\right) = f\left( x\right) ,\;\frac{\partial u\left( {x,0}\right) }{\partial t} = F\left( x\right) . $$
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