Homework Nine.

89

(a)

$$u(x,t)=\frac{60 x}{l} + \sum\limits_{n=1}^\infty \left(\frac... ...\right) e^{-\frac{n^2\pi^2\alpha^2}{l^2}t}\sin\frac{n\pi x}{l},$$

(b)

$u(5,30)\simeq 12.6$ degree Centigrade

$u(5,60)\simeq 13.7$ degree Centigrade

(c)

$u(5,3)\simeq 14.1$ degree Centigrade

12 percent between one term and two terms, third term is $-0.005$ degrees.

(d) $t=16-$ seconds

90

$$u(x,t)=\frac{2}{\pi} - \frac{4}{\pi}\sum\limits_{n=1}^\infty ... ...ac{4 m^2 \pi^2 \alpha^2 t}{l^2}}}{4m^2-1}\cos\frac{2m\pi x}{l},$$

$$u(x,t\to\infty) =\frac{2}{\pi}.$$

91

Steady state is $u(x,t)=T.$

92

Steady state is

$$u(x,t) = -\frac{5}{4}x+30.$$

93

Steady state is $u(x,t)=T.$

94

Steady state is $u(x,t)=e^x -1.$

95

$$u(x,t)=\sum\limits_{n=1}^\infty c_n e^{-\frac{ (2 n - 1 )^2 \pi^2\alpha^2 t}{4 l^2}} \sin\frac{(2n-1)\pi x}{2 l},$$

$$c_n = \frac{2}{l}\int\limits_0^l f(x) \sin\frac{(2 n - 1)\pi x}{2 l} d x.$$