# Elementary differential equations 7th edition - Boyce W.E

ISBN 0-471-31999-6

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n=1

(x, t) = cne~n2n22T2500 sin , (22)

where, from Eq. (21),

"50 . nnx

. sin dx

5 J0 50

J

J0

40 ,, , \80/nn, n odd;

= (1 cos nn) = ! (23)

nn (0, n even.

Finally, by substituting for cn in Eq. (22) we obtain

u(x, t) = 80 y 1 en2n22t/2500 sin . (24)

n fr-t n 50

n=1,3,5,...

The expression (24) for the temperature is moderately complicated, but the negative exponential factor in each term of the series causes the series to converge quite rapidly,

u

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Chapter 10. Partial Differential Equations and Fourier Series

except for small values of t or a2. Therefore accurate results can usually be obtained by using only a few terms of the series.

In order to display quantitative results let us measure t in seconds; then

a2

has the

units of cm2/sec. If we choose a2 = 1 for convenience, this corresponds to a rod of a material whose thermal properties are somewhere between copper and aluminum. The behavior of the solution can be seen from the graphs in Figures 10.5.3 through 10.5.5. In Figure 10.5.3 we show the temperature distribution in the bar at several different times. Observe that the temperature diminishes steadily as heat in the bar is lost through the endpoints. The way in which the temperature decays at a given point in the bar is indicated in Figure 10.5.4, where temperature is plotted against time for a few selected points in the bar. Finally, Figure 10.5.5 is a three-dimensional plot of u versus both x and t. Observe that the graphs in Figures 10.5.3 and 10.5.4 are obtained by intersecting the surface in Figure 10.5.5 by planes on which either t or x is constant. The slight waviness in Figure 10.5.5 at t = 0 results from using only a finite number of terms in the series for u(x, t) and from the slow convergence of the series for t = 0.

FIGURE 10.5.3 Temperature distributions at several times for the heat conduction problem of Example 1.

FIGURE 10.5.4 Dependence of temperature on time at several locations for the heat conduction problem of Example 1.

10.5 Separation of Variables; Heat Conduction in a Rod

579

FIGURE 10.5.5 Plot of temperature u versus x and t for the heat conduction problem of Example 1.

A question with possible practical implications is to determine the time t at which the entire bar has cooled to a specified temperature. For example, when is the temperature in the entire bar no greater than 1 °C? Because of the symmetry of the initial temperature distribution and the boundary conditions, the warmest point in the bar is always the center. Thus, t is found by solving u(25, t) = 1 for t. Using one term in the series expansion (24), we obtain

2500 ,

t = ln(80/n) = 820 sec. n

PROBLEMS In each of Problems 1 through 6 determine whether the method of separation of variables can be

I used to replace the given partial differential equation by a pair of ordinary differential equations.

If so, find the equations.

1. XUxx + Ut = 0 2. tUxx + XUt = 0 3. Uxx + Uxt + Ut = 0

4. [p(x)ux]x r (x)Utt = 0 5. uxx + (x + y)Uyy = 0 6. uxx + Uyy + xu = 0

7. Find the solution of the heat conduction problem

100uxx = ut, 0 < x < 1, t > 0;

u(0, t) = 0, u(1, t) = 0, t > 0;

u(x, 0) = sin2nx sin5nx, 0 < x < 1.

8. Find the solution of the heat conduction problem

Uxx = 4Ut, 0 < x < 2, t > 0;

u(0, t) = 0, u(2, t) = 0, t > 0;

u(x , 0) = 2sin(nx /2) sin nx + 4sin2nx, 0 < x < 2.

Consider the conduction of heat in a rod 40 cm in length whose ends are maintained at 0°C for all t > 0. In each of Problems 9 through 12 find an expression for the temperature u(x , t) if the initial temperature distribution in the rod is the given function. Suppose that a2 = 1.

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Chapter 10. Partial Differential Equations and Fourier Series

9. u(x, 0) = 50, 0 < x < 40 10. u(x, 0) = P Ž < x < 20

v J 140 x, 20 < x < 40

0, 0 < x < 10,

11. u(x, 0) = 50, 10 < x < 30, 12. u(x, 0) = x, 0 < x < 40

0, 30 < x < 40

? 13. Consider again the rod in Problem 9. For t = 5 and x = 20 determine how many terms are

needed to find the solution correct to three decimal places. A reasonable way to do this is to find n so that including one more term does not change the first three decimal places of u (20, 5). Repeat for t = 20 and t = 80. Form a conclusion about the speed of convergence of the series for u(x, t).

? 14. For the rod in Problem 9:

(a) Plot u versus x for t = 5, 10, 20, 40, 100, and 200. Put all the graphs on the same set of axes and thereby obtain a picture of the way in which the temperature distribution changes with time.

(b) Plot u versus t for x = 5, 10, 15, and 20.

(c) Draw a three-dimensional plot of u versus x and t.

(d) How long does it take for the entire rod to cool off to a temperature of no more than 1°C?

? 15. Followthe instructions in Problem 14 for the rod in Problem 10.

? 16. Follow the instructions in Problem 14 for the rod in Problem 11.

? 17. For the rod in Problem 12:

(a) Plot u versus x for t = 5, 10, 20, 40, 100, and 200.

(b) For each value of t used in part (a) estimate the value of x for which the temperature is greatest. Plot these values versus t to see how the location of the warmest point in the rod changes with time.

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