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

ISBN 0-471-31999-6

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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;

[1, 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)

? 23.

? 24.

? 25.

? 26.

27.

f (x) = f (x) =

2 2x,

2 x,

2x 2x2,

0,

x 2(3 x),

2 < x < 0, 0 < x < 2;

3 < x < 0, 0 < x < 3

f (x + 4) = f (x) f (x + 4) = f (x) f (x + 6) = f (x)

Consider the function f defined in Problem 21 and let em (x) = f (x) sm (x ).Plot|em (x )| versus x for 0 < x < 2 for several values of m. Find the smallest value of m for which |em(x)| <0.01 for all x.

Consider the function f defined in Problem 24 and let em (x) = f (x) sm (x ).Plot|em (x )| versus x for 0 < x < 3 for several values of m. Find the smallest value of m for which |em(x)| < 0.1 for all x.

Suppose that g is an integrable periodic function with period T.

10.2 Fourier Series

557

(a) If 0 < a < T, show that

p T I g(x) dx = I g(x) dx.

J0 Ja

p a pa+T

Hint: Show first thaw g(x) dx = I g(x) dx. Consider the change of variable s =

0T

x T in the second integral.

(b) Show that for any value of a, not necessarily in 0 < a < T,

p T p a+T

I g(x) dx = I g(x) dx.

0a

(c) Show that for any values of a and b,

p a+T p b+T

I g(x) dx = I g(x) dx.

ab

28. If f is differentiable and is periodic with period T, show that f is also periodic with period T. Determine whether

F (x) = f f (t) dt

0

is always periodic.

29. In this problem we indicate certain similarities between three-dimensional geometric vectors and Fourier series.

(a) Let vj, v2, and v3 be a set of mutually orthogonal vectors in three dimensions and let u be any three-dimensional vector. Show that

u = ajvj + a^v ^ + a3v3, (i)

where

u vi

a. = L, i = 1, 2, 3. (ii)

i vi vi

Show that at can be interpreted as the projection of u in the direction of v. divided by the length of v..

(b) Define the inner product (u,v) by

fL

(u, v) = I u(x)v(x) dx. (iii)

Also let

(iv)

0n (x) = cos(nnx/L), n = 0, 1, 2,...;

(x) = sin(nnx/L), n = 1, 2,....

Show that Eq. (10) can be written in the form

a to to

(f+n) = 7 (0O,0n) + 12 am (0m ,0n ) + J2 bm (^m ,0n)? (v)

m=1 m=1

(c) Use Eq. (v) and the corresponding equation for (f, ) together with the orthogonality

relations to show that

a = (f 0) , n = 0, 1, 2,...; b = (f ^n) , n = 1, 2,.... (vi)

n (++) n (+ >+)

Note the resemblance between Eqs. (vi) and Eq. (ii). The functions 0n and x+ play a role for functions similar to that of the orthogonal vectors v1, v2, and v3 in three-dimensional

558

Chapter 10. Partial Differential Equations and Fourier Series

space. The coefficients an and bn can be interpreted as projections of the function f onto the base functions ? and ft .

nn

Observe also that any vector in three dimensions can be expressed as a linear combination of three mutually orthogonal vectors. In a somewhat similar way any sufficiently smooth function defined on L < x < L can be expressed as a linear combination of the mutually orthogonal functions cos(nnx/L) and sin(nnx/L), that is, as a Fourier series.

10.3 The Fourier Convergence Theorem

In the preceding section we showed that if the Fourier series

a / m n x m n x \

f + E (am cos + bm sin ) (1)

m = 1

converges and thereby defines a function f, then f is periodic with period 2L, and the coefficients am and bm are related to f (x) by the Euler-Fourier formulas:

1 fL m n x

am = I f (x) cos dx, m = 0, 1, 2,...; (2)

L JL L

1 fL m nx

bm = l J f (x) sindx, m = 1, 2,.... (3)

In this section we adopt a somewhat different point of view. Suppose that a function

f is given. If this function is periodic with period 2L and integrable on the interval [L, L], then a set of coefficients am and bm can be computed from Eqs. (2) and (3), and a series of the form (1) can be formally constructed. The question is whether this series converges for each value of x and, if so, whether its sum is f (x). Examples have been discovered showing that the Fourier series corresponding to a function f may not converge to f (x), or may even diverge. Functions whose Fourier series do not converge to the value of the function at isolated points are easily constructed, and examples will be presented later in this section. Functions whose Fourier series diverge at one or more points are more pathological, and we will not consider them in this book.

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