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

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Once you realize that the maximum error always occurs at x = 0 or x = 2, you can obtain a uniform error bound for each m simply by evaluating |em(x)| at one of these points. For example, for m = 6 we have e6(2) = 0.03370, so |e6(x)| < 0.034 for 0 < x < 2, and consequently for all x. Table 10.2.1 shows corresponding data for other values of m; these data are plotted in Figure 10.2.6. From this information you can begin to estimate the number of terms that are needed in the series in order to achieve a given level of accuracy in the approximation. For example, to guarantee that |em (x)| < 0.01 we need to choose m = 21.

FIGURE 10.2.4 Partial sums in the Fourier series, Eq. (20), for the triangular wave.

FIGURE 10.2.5 Plot of |e6 (x) | versus x for the triangular wave.

10.2 Fourier Series

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PROBLEMS

TABLE 10.2.1 Values ofthe Error e (2) for

m x '

the Triangular Wave

m 2()

(m

e

2 0.09937

4 0.05040

6 0.03370

10 0.02025

15 0.01350

20 0.01013

25 0.00810

em(2)

0.10

FIGURE 10.2.6 Plot of em (2) versus m for the triangular wave.

In this book Fourier series appear mainly as a means of solving certain problems in partial differential equations. However, such series have much wider application in science and engineering, and in general are valuable tools in the investigation of periodic phenomena. A basic problem is to resolve an incoming signal into its harmonic components, which amounts to constructing its Fourier series representation. In some frequency ranges the separate terms correspond to different colors or to different audible tones. The magnitude of the coefficient determines the amplitude of each component. This process is referred to as spectral analysis.

In each of Problems 1 through 8 determine whether the given function is periodic. If so, find its fundamental period.

1. sin5x

3. sinh2x

5. tan n x

7. f (x) =

8. f (x) =

0, 2n — 1 < x < 2n,

1, 2n < x < 2n + 1;

(— 1)n, 2n — 1 < x < 2n,

1 , 2n < x < 2n + 1 ;

2. cos2nx 4. sin n x /L

6. x2

n = 0, ±1, ±2,...

n 0, 1 , 2,

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Chapter 10. Partial Differential Equations and Fourier Series

9. If f (x) = — x for — L < x < L, and if f (x + 2L) = f (x), find a formula for f (x) in the interval L < x < 2L; in the interval —3 L < x < — 2L.

10. If f (x) =

x + 1, —1 < x < 0,

and if f (x + 2) = f (x), find a formula for f (x) in

x , 0 < x < 1 ,

the interval 1 < x < 2; in the interval 8 < x < 9.

11. If f (x) = L — x for 0 < x < 2L, and if f (x + 2L) = f (x), find a formula for f (x) in

the interval - L < x < 0.

12. Verify Eqs. (6) and (7) of the text by direct integration.

In each of Problems 13 through 18:

(a) Sketch the graph of the given function for three periods.

(b) Find the Fourier series for the given function.

13. f (x) = —x, —L < x < L;

\l, —L < x < 0,

0, 0 x < L

14. f (x) =

15. f(x) =

16. f (x) =

17. f (x) =

18. f (x) =

x, —n < x < 0,

0, 0 < x < n;

x + 1, —1 < x < 0,

1 - x , 0 < x < 1 ;

x + L, —L < x < 0,

L , 0 < x < L ;

0, —2 < x <—1,

x , -1 < x < 1 ,

0, 1 < x < 2;

f (x + 2L) = f (x) f (x + 2L) = f (x)

f (x + 2n) = f (x)

f (x + 2) = f (x)

f (x + 2L) = f (x)

f (x + 4) = f (x)

In each of Problems 19 through 24:

(a) Sketch the graph of the given function for three periods.

(b) Find the Fourier series for the given function.

(c) Plot sm (x) versus x for m = 5, 10, and 20.

(d) Describe how the Fourier series seems to be converging.

> 19. f (x) =

> 20. > 21.

f (x) = x, f (x) = x 2/2,

> 22. f (x) =

-1 , -2 < x < 0,

1 , 0 < x < 2;

-1 < x < 1;

-2 < x < 2;

x + 2, —2 < x < 0,

0 < x < 2;

f (x + 4) = f (x)

f (x + 2) = f (x) f (x + 4) = f (x)

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