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Xn (x) = sin(nn x/L), n = 1, 2, 3,..., (14)
associated with the eigenvalues
kn = n2n2 / L2, n = 1, 2, 3,.... (15)
Turning now to Eq. (10) for T(t) and substituting n2n2/L2 for ê, we have
T' + (n2n 2a2/L 2)T = 0. (16)
Thus T(t) is proportional to exp(—n2n2a2t/L2). Hence, multiplying solutions of
Eqs. (9) and (10) together, and neglecting arbitrary constants of proportionality, we conclude that the functions
un (x, t) = e~n n a t/L sin(nn x/L), n = 1, 2, 3,... (17)
satisfy the partial differential equation (1) and the boundary conditions (4) for each
positive integer value of n. The functions un are sometimes called fundamental solutions of the heat conduction problem (1), (3), and (4).
It remains only to satisfy the initial condition (3),
u(x, 0) = f (x),
0 x L .
10.5 Separation of Variables; Heat Conduction in a Rod
Recall that we have often solved initial value problems by forming linear combinations of a set of fundamental solutions, and then choosing the coefficients to satisfy the initial conditions. The analogous step in the present problem is to form a linear combination of the functions (17) and then to choose the coefficients to satisfy Eq. (18). The main difference now from earlier problems is that there are infinitely many functions (17), so a general linear combination of them is an infinite series. Thus we assume that
EV > 2_2 2*/r2 nn x
cnun(x, t) = Y,cne~n Ï à t/L sin—, (19)
where the coefficients cn are as yet undetermined. The individual terms in the series
(19) satisfy the differential equation (1) and boundary conditions (4). We will assume that the infinite series of Eq. (19) converges and also satisfies Eqs. (1) and (4). To satisfy the initial condition (3) we must have
-V—V nn x
u(x, 0) = Y2 cn sin l = f (x). (20)
In other words, we need to choose the coefficients cn so that the series of sine functions in Eq. (20) converges to the initial temperature distribution f (x) for 0 < x < L. The series in Eq. (20) is just the Fourier sine series for f; according to Eq. (8) of Section
10.4 its coefficients are given by
2 fL nnx
cn = L Jo f (x) sin— dx. (21)
Hence the solution of the heat conduction problem of Eqs. (1), (3), and (4) is given by the series in Eq. (19) with the coefficients computed from Eq. (21).
Find the temperature u(x, t) at any time in a metal rod 50 cm long, insulated on the sides, which initially has a uniform temperature of 20° C throughout, and whose ends are maintained at 0°C for all t > 0.
The temperature in the rod satisfies the heat conduction problem (1), (3), (4) with L = 50 and f (x) = 20 for 0 < x < 50. Thus, from Eq. (19), the solution is
(x, t) = ? cne~"2n2“V2500 sin ^, (22)
where, from Eq. (21),
"50 . nnx
5 Ë 50
40 ,, ÷ f80/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 Ó -e—n2n2Ë/2500 sin —. (24)
n ". n 50
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,
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
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.