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

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> 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

r T /> a+T

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

J0 Ja

pa na+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 r a+T

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

0a

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

p a+T r 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 v1, v2, and v3 be a set of mutually orthogonal vectors in three dimensions and let u be any three-dimensional vector. Show that

u = a1v1 + a2v2 + a3v3, (i)

where

u v.

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

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)

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

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

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

(f^n) = 7 (τ0,τ) + 12 am (τς ) + 12 bm (^ς Σ (v)

m=1 ς=1

(c) Use Eq. (v) and the corresponding equation for (f, tyn) together with the orthogonality relations to show that

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

n (Τ,Τ) n (* ,f)

Note the resemblance between Eqs. (vi) and Eq. (ii). The functions το and tyn 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 τ 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 \m 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

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