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

ISBN 0-471-31999-6

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(m n)2

_\ 8/(mn)2, m odd,

| 0, m even.

Finally, from Eq. (14) it follows in a similar way that

bm = 0, m = 1, 2,.

(18)

(19)

By substituting the coefficients from Eqs. (17), (18), and (19) in the series (16) we obtain the Fourier series for f :

f (x ) = 1 -

1

n x

cos

1

+~ cos ?

2

3n x

2 32 2

? cos(mnx/2)

5n x

+

n

E

m

2

1

m = 1,3,5,...

cos(2« 1)n x/2

n

2

n = 1

(2n 1)

2

(20)

0

2

0

2

2

Let

f(x) =

0, 3 < x < 1,

1, 1 < x < 1,

0, 1 < x < 3

(21)

and suppose that f (x + 6) = f (x); see Figure 10.2.3. Find the coefficients in the Fourier series for f.

10.2 Fourier Series

553

EXAMPLE

3

I

y-

1

1 1 1 1 1 1 1 1 ,

?^J 1 cn 1 GO 1 1 3 5 7 t

FIGURE 10.2.3 Graph of f (x) in Example 2.

Since f has period 6, it follows that L = 3 in this problem. Consequently, the Fourier series for f has the form

1st f\

f (x) = f + ^ V -»

2 n= 1

TO

?(?

nnx nnx

a cos + b sin

)?

(22)

3 n 3

where the coefficients an and bn are given by Eqs. (13) and (14) with L = 3. We have

1 f3 1 f1

3/.3f (x) dx = 5X1

dx = -. =3

Similarly,

nnx 1 . nnx

cos dx = sin-------------------

3 nn 3

2 . nn

= sin, n = 1, 2,...

-1

and

b

nn x 1 nnx

sin dx = cos--------------------------

3 nn 3

nn

= 0,

3

n = 1, 2,...

(23)

(24)

(25)

1

Thus the Fourier series for f is

1 CO ^

1 2 nn nnx

f (x) = - + > sin cos

3 nn 3 3

n1

cos(2n x/3) cos(4n x/3) cos(5n x/3)

cos(nx/3) -+-------------------------- --------------- -+

(26)

ao =

1

an =

1

Consider again the function in Example 1 and its Fourier series (20). Investigate the speed with which the series converges. In particular, determine how many terms are needed so that the error is no greater than 0.01 for all x.

The mth partial sum in this series,

, , , 8 m cos(2n 1)nx/2 ^

s (x) = 1 r > --------------------------=------, (27)

m n2 (2n 1)2

can be used to approximate the function f. The coefficients diminish as (2n 1)2, so the series converges fairly rapidly. This is borne out by Figure 10.2.4, where the partial sums for m = 1 and m = 2 are plotted. To investigate the convergence in more detail

554

Chapter 10. Partial Differential Equations and Fourier Series

we can consider the error em (x) = f (x) sm (x). Figure 10.2.5 shows a plot of |e6(x) | versus x for 0 < x < 2. Observe that |e6(x)| is greatest at the points x = 0 and x = 2 where the graph of f (x) has corners. It is more difficult for the series to approximate the function near these points, resulting in a larger error there for a given n. Similar graphs are obtained for other values of m .

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

555

PROBLEMS

TABLE 10.2.1 Values of the Error e (2) for

m v '

the Triangular Wave

m e (2) m v y

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

5 10 15 20 25 m

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 IL

6. x2

n = 0, ±1, ±2,... n 0, 1 , 2,

556

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.

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