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

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\Ë Ääëëëëë/1 n = 8 -

yvvvvvy ³ (/vvvvvv

-2 -1 1 2 x

FIGURE 10.3.3 The partial sum s8 (x) in the Fourier series, Eq. (6), for the square wave.

Additional insight is attained by considering the error en (x) = f (x) — sn (x). Figure

10.3.4 shows a plot of |en (x) | versus x for n = 8 and for L = 1. The least upper bound of |e8(x)| is 0.5 and is approached as x ^ 0 and as x ^ 1. As n increases, the error

4The Gibbs phenomenon is named after Josiah Willard Gibbs (1839-1903), who is better known for his work on vector analysis and statistical mechanics. Gibbs was professor of mathematical physics at Yale, and one of the first American scientists to achieve an international reputation. Gibbs’ phenomenon is discussed in more detail by Carslaw (Chapter 9).

Chapter 10. Partial Differential Equations and Fourier Series

decreases in the interior of the interval [where f (x) is continuous] but the least upper bound does not diminish with increasing n. Thus one cannot uniformly reduce the error throughout the interval by increasing the number of terms.

Figures 10.3.3 and 10.3.4 also show that the series in this example converges more slowly than the one in Example 1 in Section 10.2. This is due to the fact that the coefficients in the series (6) are proportional only to 1/(2n — 1).

PROBLEMS In each of Problems 1 through 6 assume that the given function is periodically extended outside the original interval.

(a) Find the Fourier series for the extended function.

(b) Sketch the graph of the function to which the series converges for three periods.

1. f (x) = 3. f (x) =

5. f(x) =

— 1, —1 < x < 0,

1,

0 < x < 1

2. f (x) =

0, — n < x < 0,

x, 0 < x < n

L + x, —L < x < 0,

L — x,

0 < x < L

0, —n < x < — n/2,

1, —æ/2 < x < n/2,

0, n /2 < x < n

4. f (x) = 1 — x2, —1 < x < 1

0, — 1 < x < 0,

6. f(x) = |x2, 0 < x < 1

In each of Problems 7 through 12 assume that the given function is periodically extended outside the original interval.

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

(b) Let en (x) = f (x) — sn (x). Find the least upper bound or the maximum value (if it exists) of len(x)| for n = 10, 20, and 40.

(c) If possible, find the smallest n for which |en (x)| < 0.01 for all x.

> 7. f (x) = x

f (x + 2n) = f (x) (see Section 10.2, Problem 15)

10.3 The Fourier Convergence Theorem

563

> 8. f (x) = |Ö + 1 ^ ~ x < 1; f (x + 2) = f (x) (see Section 10.2, Problem 16)

> 9. f (x) = x, —1 < x < 1; f (x + 2) = f (x) (see Section 10.2, Problem 20)

,,, , \x + 2, —2< x < 0, r, , r, (see Section 10.2,

> I0. f (x) = X2 — 2x, 0 < x < 2; f (x + 4) = f (x) Problem 22)

> 11. f (x) = j02 0 < x < 0; f (x + 2) = f (x) (see Problem 6)

> 12. f (x) = x — x3, —1 < x < 1; f (x + 2) = f (x)

Periodic Forcing Terms. In this chapter we are concerned mainly with the use of Fourier series

to solve boundary value problems for certain partial differential equations. However, Fourier series are also useful in many other situations where periodic phenomena occur. Problems 13 through 16 indicate how they can be employed to solve initial value problems with periodic forcing terms.

13. Find the solution of the initial value problem

y" + o)2y = sin nt, y(0) = 0, y' (0) = 0,

where n is a positive integer and a? = n2. What happens if «2 = n2?

14. Find the formal solution of the initial value problem

TO

y" + a2y = "Y2 bn sin nt, y(0) = 0, y'(0) = 0,

n=1

where a > 0 is not equal to a positive integer. How is the solution altered if a = m, where m is a positive integer?

15. Find the formal solution of the initial value problem

y" + a2 y = f (t), y(0) = 0, y'(0) = 0,

where f is periodic with period 2n and

f(t) =

1, 0 < t < n;

0, t = 0, n, 2n;

— 1, n < t < 2n.

See Problem 1.

16. Find the formal solution of the initial value problem

y" + a2 y = f (t), y(0) = 1, y'(0) = 0,

where f is periodic with period 2 and

f (t) = X 1—t, 0 <t <1;

J () { —1 + t, 1 < t < 2.

See Problem 8.

17. Assuming that

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