<Text-field style="Heading 1" layout="Heading 1">Both ends of the rod are in an ice bath (u(0,t)=u(L,t)=0)</Text-field>

According to the analysis of Section 9.5 (separation of variables), the temperature at time t at the position x on the bar is given by the Fourier series (here we compute only the partial sum up to N, N=50 is good enough to get a good approximation)

u := (x,t,N) -> 4*u0/Pi * sum(1/(2*j-1) * exp(-(2*j-1)^2*Pi^2*k*t/L^2) * sin((2*j-1)*Pi*x/L) , j=1..N);

Zio2JUkieEc2IkkidEdGJUkiTkdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLCQqKEkjdTBHRiUiIiJJI1BpRyUqcHJvdGVjdGVkRyEiIi1JJHN1bUc2JEYwSShfc3lzbGliR0YlNiQqKCwmSSJqR0YlIiIjRjFGLkYxLUkkZXhwR0Y0NiMsJCosRjhGOkYvRjpJImtHRiVGLjklRi5JIkxHRiUhIiNGMUYuLUkkc2luR0Y0NiMqKkY4Ri5GL0YuOSRGLkZCRjFGLi9GOTtGLjkmRi4iIiVGJUYlRiU=

The spatial distribution of the temperature on the rod after 30s is

plot(u(x,30,50), x=0..L);

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

We can find the time at which the rod midpoint reaches 50 degrees:

plot({u(L/2,t,L),50}, t=1..5000);

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

We could estimate the time graphically or just use the buit-in numerical methods in Maple

T50 := fsolve(u(L/2,t,L) = 50 , t=0..5000);

JCIrMnphY0QhIic=

The time in minutes and seconds is then:

printf("%d:%.2f\134n",floor(T50/60), frac(T50/60)*60);

42:36.55

<Text-field style="Heading 1" layout="Heading 1">End x=L is insulated while x=0 is in ice bath</Text-field>

Here we use the anaylsis from problem 9.5.24 to find that the temperature on the rod is given by

c:= n-> (2/L)*int(100*sin(n*Pi*x/2/L),x=0..L);

Zio2I0kibkc2IkYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLCQqJkkiTEdGJSEiIi1JJGludEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJTYkLCQtSSRzaW5HRi82IywkKio5JCIiIkkjUGlHRjBGOkkieEdGJUY6RitGLCNGOiIiIyIkKyIvRjw7IiIhRitGOkY+RiVGJUYl

ubis := (x,t,N) -> sum(c(2*j+1)*exp(-((2*j+1)*Pi/2/L)^2*k*t)*sin((2*j+1)*Pi*x/2/L),j=0..N);

Zio2JUkieEc2IkkidEdGJUkiTkdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLUkkc3VtRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlNiQqKC1JImNHRiU2IywmSSJqR0YlIiIjIiIiRjhGOC1JJGV4cEdGLTYjLCQqLEY1RjdJI1BpR0YuRjdJIkxHRiUhIiNJImtHRiVGODklRjgjISIiIiIlRjgtSSRzaW5HRi02IywkKipGNUY4Rj5GODkkRjhGP0ZEI0Y4RjdGOC9GNjsiIiE5JkYlRiVGJQ==

Here is the typical temperature distribution after 3000s

plot(ubis(x,3000,50), x=0..L);

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

We can see that the maximum temperature of the rod is at the insulated end. We can find the time at which the rod midpoint reaches 50 degrees:

plot({ubis(L,t,L),50}, t=1..30000);

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

We could estimate the time graphically or just use the buit-in numerical methods in Maple

T50 := fsolve(ubis(L,t,L) = 50 , t=0..30000);

JCIraSI+RS0iISIm

The time in minutes and seconds is then:

printf("%d:%.2f\134n",floor(T50/60), frac(T50/60)*60);

170:26.19