Application 9.5 Heated Rod (p 628) Diffusivity constant k:=0.15;

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Length of the rod L := 50;

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Initial temperature u0 := 100;

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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 (Fig. 9.5.7) plot(u(x,30,50), x=0..50);

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

and after 30 min (Fig. 9.5.8). Because of symmetries the maximum temperature will be at the center of the interval (x=25) plot(u(x,30*60,50),x=0..50);

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

We can also see how the temperature at a particular point of the bar changes with time. Here is how the temperature at the midpoint evolves. The temperature over the rest of the bar is smaller (Fig. 9.5.9) plot({u(25,t,50),50}, t=0..7000);

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

Thus if we want to know how long it takes for the temperature on the rod to be less than 50C (so that we don't burn by holding it, for example), it is sufficient to look at the temperature on the midpoint because it is the hottest point of the rod. According to the previous plot it takes the system between 1000 and 2000 seconds for this to happen. Let us zoom in to find a better value: plot({u(25,t,50),50}, t=1000..2000);

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

From the plot the time would be between 1500 and 1600 seconds. We could continue zooming in (narrowing the range of t around the intersection between the temperature curve and u=50), or we could also use the numerical methods built in Maple to find a numerical solution. Here we specify that the solution is in the range t=0..2000 (if you omit this argument to fsolve, Maple gets confused and gives you a negative time, which is non-physical! Of course, the range you specify here depends on the problem) T50 := fsolve(u(25,t,50) = 50 , t=0..2000);

JCIrIypmNnk6ISIn

The time in minutes and seconds is then: printf("%d:%.2f\134n",floor(T50/60), frac(T50/60)*60);

26:18.12