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Elementary Differential Equations and Boundary Value Problems - Boyce W.E.

Boyce W.E. Elementary Differential Equations and Boundary Value Problems - John Wiley & Sons, 2001. - 1310 p.
Download (direct link): elementarydifferentialequations2001.pdf
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Since f is an odd function, its Fourier coefficients are, according to Eq. (8),
n = 0, 1, 2, • • •;
an = 0,
2 fL nnx
b = — I x sin---------------dx
n L Λ L
2
2 / L I . nnx nnx nnx
= ³\ΟΟ) (sin~l ~Tcos~l~
= — (—1)n+1, nn
n = 1, 2,
Hence the Fourier series for f, the sawtooth wave, is
2L
(—1)
n+1
nn x
sin
n = 1
L
(9)
Observe that the periodic function f is discontinuous at the points ±L, ±3 L, • • •, as shown in Figure 10.4.2. At these points the series (9) converges to the mean value of the left and right limits, namely, zero. The partial sum of the series (9) for n = 9 is shown in Figure 10.4.3. The Gibbs phenomenon (mentioned in Section 10.3) again occurs near the points of discontinuity.
L
0
n
Σ'
L
I 1 I
-3L -2L -L L 2L 3L x
-L
FIGURE 10.4.2 Sawtooth wave.
568
Chapter 10. Partial Differential Equations and Fourier Series
/ y ³/
L
/-2L -L / L / 2L X
\ / /
9
II
n
-L - f
FIGURE 10.4.3 A partial sum in the Fourier series, Eq. (9), for the sawtooth wave.
Note that in this example f (-L) = f (L) = 0, as well as f (0) = 0. This is required if the function f is to be both odd and periodic with period 2L. When we speak of constructing a sine series for a function defined on 0 < x < L, it is understood that, if necessary, we must first redefine the function to be zero at x = 0 and x = L.
It is worthwhile to observe that the triangular wave function (Example 1 of Section
10.2) and the sawtooth wave function just considered are identical on the interval 0 < x < L. Therefore, their Fourier series converge to the same function, f (x) = x, on this interval. Thus, if it is required to represent the function f (x) = x on0 < x < L by a Fourier series, it is possible to do this by either a cosine series or a sine series. In the former case f is extended as an even function into the interval —L < x < 0 and elsewhere periodically (the triangular wave). In the latter case f is extended into —L < x < 0 as an odd function, and elsewhere periodically (the sawtooth wave). If f is extended in any other way, the resulting Fourier series will still converge to x in
0 < x < L but will involve both sine and cosine terms.
In solving problems in differential equations it is often useful to expand in a Fourier series of period 2L a function f originally defined only on the interval [0, L ]. As indicated previously for the function f (x) = x several alternatives are available. Explicitly, we can:
g(x) =
(10)
1. Define a function g of period 2L so that
_{ f (x), 0 < x < L,
j f (—x), — L < x < 0.
The function g is thus the even periodic extension of f. Its Fourier series, which is a cosine series, represents f on [0, L ].
2. Define a function h of period 2L so that
h(x) =
f (x), 0,
0 < x < L , x = 0, L,
(11)
— f(—x), —L < x < 0.
The function h is thus the odd periodic extension of f. Its Fourier series, which is a sine series, also represents f on (0, L).
10.4 Even and Odd Functions
569
EXAMPLE
2
3. Define a function κ of period 2L so that
k (x) = f (x), 0 < x < L, (12)
and let k(x) be defined for (—L, 0) in any way consistent with the conditions of Theorem 10.3.1. Sometimes it is convenient to define k(x) to be zero for —L < x < 0. The Fourier series for κ, which involves both sine and cosine terms, also represents f on [0, L], regardless of the manner in which k(x) is defined in (-L, 0). Thus there are infinitely many such series, all of which converge to f (x) in the original interval.
Usually, the form of the expansion to be used will be dictated (or at least suggested) by the purpose for which it is needed. However, if there is a choice as to the kind of Fourier series to be used, the selection can sometimes be based on the rapidity of convergence. For example, the cosine series for the triangular wave [Eq. (20) of Section
10.2] converges more rapidly than the sine series for the sawtooth wave [Eq. (9) in this section], although both converge to the same function for 0 < x < L . This is due to the fact that the triangular wave is a smoother function than the sawtooth wave and is therefore easier to approximate. In general, the more continuous derivatives possessed by a function over the entire interval —to < x < to, the faster its Fourier series will converge. See Problem 18 of Section 10.3.
Suppose that
1 — x , 0 < x < 1 ,
f (x) = 0, 1 < x^ 2. (13)
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