4.2边值条件的齐次化方法习题参考解答
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# 边值条件的齐次化方法习题参考解答
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1、 利用函数变换将下面的边界条件齐次化:
(1)、 $\displaystyle u(0,t)=\mu\_1(t),\ u\_x(l,t)=\mu\_2(t)$.
[纸质资料](https://mp.weixin.qq.com/s/ycnPCSqWFlThEnq9ZZ6gBQ)/[答疑](https://mp.weixin.qq.com/s/JGYZG5rsshf7Z2Amo2di8A)/(https://mp.weixin.qq.com/s/Pt6\_h5MqtomrUDYiPEwkxg)/(https://mp.weixin.qq.com/s/dWvpeJFKnFr0WYPoidXXMA) /
\begin\{aligned\} U(x)=u(x)-[\mu\_1(t)+x\mu\_2(t)]. \tiny\boxed\{\begin\{array\}\{c\}\mbox\{跟锦数学微信公众号\}\\\\\mbox\{zhangzujin.cn\}\end\{array\}\}\end\{aligned\}
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(2)、 $\displaystyle (u\_x-\alpha\_1u)|\_\{x=0\}=\mu\_1(t), (u\_x-\alpha\_2 u)|\_\{x=l\}=\mu\_2(t)$, $\displaystyle \alpha\_1,\alpha\_2$ 为常数.
[纸质资料](https://mp.weixin.qq.com/s/ycnPCSqWFlThEnq9ZZ6gBQ)/[答疑](https://mp.weixin.qq.com/s/JGYZG5rsshf7Z2Amo2di8A)/(https://mp.weixin.qq.com/s/Pt6\_h5MqtomrUDYiPEwkxg)/(https://mp.weixin.qq.com/s/dWvpeJFKnFr0WYPoidXXMA) / 考虑一个二次函数 $\displaystyle ax^2+bx+c$, 使得它满足所给边界条件, 从而定出 $\displaystyle a,b,c$, 很多解, 取一个即可.
(2-1)、 当 $\displaystyle \alpha\_2=\frac\{2\}\{l\}$ 时, 取 $\displaystyle U(x)=u(x)-x^2$ 即可;
(2-2)、 当 $\displaystyle \alpha\_2\neq \frac\{2\}\{l\}$ 且 $\displaystyle \alpha\_1=0$ 时, 取
\begin\{aligned\} U(x)=u(x)-\left\[\frac\{\mu\_2(t)-\mu\_1(t)+\alpha\_2l\mu\_1(t)\}\{l(2-\alpha\_2l)\}x^2+\mu\_1x\right\]; \tiny\boxed\{\begin\{array\}\{c\}\mbox\{跟锦数学微信公众号\}\\\\\mbox\{zhangzujin.cn\}\end\{array\}\}\end\{aligned\}
(2-3)、 当 $\displaystyle \alpha\_2\neq \frac\{2\}\{l\}$ 且 $\displaystyle \alpha\_1\neq 0$ 时, 取
\begin\{aligned\} U(x)=u(x)-\left\[\frac\{\alpha\_1 \mu\_2(t)-\alpha\_2\mu\_1(t)\}\{\alpha\_1 l(2-\alpha\_2l)\}x^2-\frac\{\mu\_1\}\{\alpha\_1\}\right\]. \tiny\boxed\{\begin\{array\}\{c\}\mbox\{跟锦数学微信公众号\}\\\\\mbox\{zhangzujin.cn\}\end\{array\}\}\end\{aligned\}
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(3)、 $\displaystyle u\_x(1,t)=\mu\_1(t),\ u(2,t)=\mu\_2(t)$.
[纸质资料](https://mp.weixin.qq.com/s/ycnPCSqWFlThEnq9ZZ6gBQ)/[答疑](https://mp.weixin.qq.com/s/JGYZG5rsshf7Z2Amo2di8A)/(https://mp.weixin.qq.com/s/Pt6\_h5MqtomrUDYiPEwkxg)/(https://mp.weixin.qq.com/s/dWvpeJFKnFr0WYPoidXXMA) /
\begin\{aligned\} U(x)=u(x)-[(x-2)\mu\_1(t)+\mu\_2(t)]. \tiny\boxed\{\begin\{array\}\{c\}\mbox\{跟锦数学微信公众号\}\\\\\mbox\{zhangzujin.cn\}\end\{array\}\}\end\{aligned\}
跟锦数学微信公众号. [在线资料](https://mp.weixin.qq.com/s/F-TU-uzeo3EjxI5LzjUvRw)/[公众号](https://mp.weixin.qq.com/s/pdC49P5WZXTEpRBa0JBfow)/
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