Bernatz R. Fourier Series and Numerical Methods for Partial Differential Equations

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32 FOURIER SERIES

are orthogonal on the interval (0, c). These functions can be normalized by dividing

a given function by its norm. Because

/ηπχ\\\

/ f

/ηπχ\ /ηπχ\ [c

{—JIr

vi,

H—J

sin

l—)

= \i2

the functions

19.

Τ7.7ΓΤ

1,2,3,...

/ηπχ\

\—)

define an orthonormal subset of the space C

(0, c).

is also an element of C

(0, c), then the corresponding Fourier sine series for

/is

71=1

with b

given by

If b

is defined as

/2 /T17TX\

f(x)\ -

sin

Jo V

c \ c J

^J° f{x)sm(^)dx

(2.23)

then an alternative form of the corresponding Fourier series for

f(x)

/(x)-^6

W-sm^-—J

n=l

EXAMPLE 2.2

In this example, the corresponding Fourier sine series for

f(x) = x

on the

interval (0,1) is determined. Essentially, this requires finding a formula for b

using Equation (2.23). So,

. /ηπχ\ ,

xsm

\—)

y/2

xsin(rarx)dx

y/2(sin(nn)

—

ηποοβ(ηπ))

f-l)

n+1

- (2.25)

'ηπ

FOURIER SINE SERIES ON (0, C) 33

and the corresponding series is

^2 ~ (_l)n+l

x ~ — > vzsminnx)

n—l

~ — > sm(nnx)

π ¿-^ n

v y

n=l

(2.26)

The formula for b

show in Equation (2.25) was actually determined with the

aid of a Maple worksheet.

The "goodness of fit" for two partial sums of the series is shown in Figure

2.4. Note how much better the 10-term partial sum approximates the graph of

Figure 2.4 Plot of

the

function f(x)

—

x (solid

line),

as well as the three-term (dotted), and

10-term (dashed) Fourier sine series partial sums.

f(x)

—

x on the interval than the three-term partial sum. One conclusion that

can be drawn from this graph is that the limit of the partial sum will be actual

value of the function f(x)

—

x for all x on [0,1] except x - 1. Here, the function

value is 1, while each of the sine terms has a value of

zero.

Consequently, the

limit of the partial sums for x -

cannot possibly be 1.

2.5.1 Odd, Periodic Extensions

Because the functions

. /ηπχ\

{

—

)

n = 1,2,3,

are odd

(/(—x)

= —/(#)) and periodic with period 2c (f(x 4- 2c) = /(#))» the

Fourier sine series corresponding to / provides a viable series correspondence for the

odd, periodic extension (with period 2c) of /. For example, if the function f(x) = x

34 FOURIER SERIES

Figure

2.5

Plot

of the odd,

periodic extension

of the

function

f(x) = x. The

original

portion

the function

plotted

as a

solid line.

The odd

extension

to the

interval [-1,0],

well

the periodic extensions

length 2 each, are plot as dotted lines.

is extended

to an odd

function, with period

2, its

graph

on the

interval

[-3,3]

would

be that

shown

Figure

2.5.

The 10-term partial

sum of the

corresponding sine series used

for the

original

function

f(x) — x on the

interval

[0,1] is

plotted

as a

dashed curve

on the

overall

interval

of [-3,3] in

Figure 2.6.

The odd,

periodic extension

of / is

plotted

as a

solid

line

this case. Note that the 50-term partial

sum

serves

as a

viable approximation

the odd, period extension

of/.

Again,

the

locations when the series will definitely

not

converge

to the odd,

periodic extension

are the

periodic, with period

replications

—

That is,

-3,-1,

and 3.

Figure

2.6

Plot

of the odd,

periodic extension

the function

f(x) = x

(solid line)

and

10-term (dashed) Fourier sine series partial

sum.

FOURIER COSINE SERIES ON (0, C) 35

2.6 FOURIER COSINE SERIES ON (0, C)

Similar to the functions sin (!*££), the functions

(

ΤΪ7ΓΧ

J n = 0,l,2,3...

are orthogonal on the interval (0, c) (see Exercise 2.25). Note: The index n begins

with zero in the cosine case because cos(0) = 1 gives a nonzero integral on (0, c).

Consequently, the functions

φο(χ) = — and φ

(χ) = J- cos ( J

form an orthonormal subset of the space C

(0, c).

For an element f(x) from C

(0, c), the corresponding Fourier cosine series is

) ~

Υ^ο

(χ)

n=0

+ E

«Vc

cos

(—)

27)

n—\

With

= / ί{χ)φ

{χ)άχ η = 0,1,2,3,...

Equation (2.27) becomes

/(*) ~ £^f(x))^=dx

^/C

^' ) y/c

i(!jf/w-".)

Ê(!jf/w«(^)*)c-(^)

cos Í j (2.28)

n=l

with

2 f

ΤΙΊΤΧ

= - f(x)cos(—-Jdx n = 0,1,2,3,... (2.29)

36 FOURIER SERIES

EXAMPLE 2.3

Here the Fourier cosine series is determined for the function

f(x)

=xon the

interval (0,1). As with the sine series, this is accomplished by finding values

for the coefficients c

using Equation (2.29). The results are

— - I x

· \dx

1 Jo

/ηπχ\

= -

x cos

—-—

2(-l+

(-!)")

-4

(2n

7Γ

Using these results, corresponding cosine series for x is

1,2,3,...

n = 1,2,3,... (2.30)

-4

*~2

^(2n-l)^

COS((2n

1)7nr)

n=l

v y

Figure 2.7 shows the plots of x as well as the three-term and 10-term cosine

series partial sums for

The 10-term partial sum gives

rather accurate

approximation to the graph of x on the interval (0,1). As with the sine series

case,

Figure 2.7 suggests that the cosine series for x will converge to f(x)

—

for points in the interval (0,1). Note: In the present case, it seems as though

the full series will converge to

f(x)

at the end points as well. In the sine

series example, convergence at

= 1

seems doubtful (refer to Figure 2.4).

2.6.1 Even, Periodic Extensions

Just as it seems reasonable that the Fourier sine series provides a viable correspon-

dence to an odd, periodic extension of

given function

on (0,1), the Fourier cosine

series provides a viable series correspondence for the even, periodic extension of /.

Any function

defined on the interval [0,c] may be extended in an even, periodic

(with period 2c) fashion by requiring f(—x)

= f(x)

and

f(x

4- 2c)

= f(x)

The

even extension of x, with period 2, is graphed in Figure 2.8.

The 10-term cosine partial sum corresponding to the even, periodic extension of

f(x)

—

oven the interval

[-3,3]

is shown in Figure 2.9. The plot provides greater

evidence for the hope of series convergence at points

-3,-1,1,

and 3. Again, this

was not the case for the sine series, as shown in Figure 2.6.

FOURIER SERIES ON (-C, C) 37

Figure 2.7 Plot of the function f(x) = x (solid line) as well as the three-term (dotted) and

10-term (dashed) Fourier cosine series partial sums.

/ \

1 1

0.8

\ 0.6

/ \

* '

/ \

/ * '

/ \ '

/ * '

\ /

1 2

Figure 2.8 Plot of the even, periodic extension of the function f(x) = x. The original

portion of the function is plotted as a solid line. The odd extension to the interval [-1,0], as

well on the periodic extensions of length 2 each, are plot as dashed lines.

2.7 FOURIER SERIES ON (-C, C)

Completion of Exercise 2.27 establishes that the functions

. , v 1 , 1 /ηπχ\ . 1 /ηπχ\

Φθ{Χ) = -7= Ψ2η-1 = —F

<P2n

= —F

COS

ί )

38 FOURIER SERIES

Figure 2.9 Plot of the even, periodic extension of the function f(x) = x (solid line) and

10-term (dashed) Fourier cosine series partial sum.

for n = 1,2,3,... form an orthonormal set on the interval (—c,

c).

Then, for an

arbitrary function / from C

[—c,

c],

the Fourier series corresponding to / is

with

. . ^ Γ 1 . /ηπχ\ 1 /ηπχ\

f{x) ~ C

0O + 2^ \C2n-l-7= Sin ^ J + C

-y=

COS

^ J

Γ i

Co = / -

==f{x)dx

J-c

V2c

1 /

Τ17ΓΧ

{x)

^fc

Shi

\—)

(2.31)

and

1 / 7Ί7ΓΧ \

/(#)—=cos(^ J dx

Substituting for

Co,

_i, and C2

, respectively, in Correspondence (2.31) gives

/(*)

-==f(x)dx I —7=

'Se

;

y x/2¿

/ Γ^ 1 / TÍ7TX \ \ 1 /

ΊΊ7ΓΊΓ

Σ (j-/

(x)

sin

)

7¿

sin

("TV

|(£/(,)-Lcos(^)^)±cos(^)

(

)

/(*)

? + Σ

cos

(^)

+ K sin

(^)l

33)

FOURIER SERIES ON (-C, C) 39

where

In = - / f(x)

COS

( j dx,

1 /" / 7Ί7ΓΧ \

= - I f(x) sin f ) dx n =

1,2,3,..

n = 0,1,2,3,...

and

EXAMPLE 2.4

Find the Fourier series corresponding to the function

/(*) = { 1

-1 <x <0

x 0<x < 1

As with the other Fourier series examples, the primary object is to determine

formulas for the required coefficients. Therefore,

0 = 7 / J\X)dX

f(x)da

= / 1

dx + / (1 - x)dx

1 /* /

T17TX

= - f(x) cos

(^-j—J

= / 1

cos (ηπχ) dx + / (1

—

x) · cos (mrx) dx

(2n - 1)

n =

1,2,3,...

and

1 /* /

7Ζ7ΓΧ

= - / /(x) sin (^—— J dx

= / 1 · sin(mrx)(ix + / (1

—

x) sin(nnx)da

-i + (-i)

ηπ ηπ

(-l)

— n =

1,2,3,...

ηπ

40 FOURIER SERIES

Substituting these coefficient formulas in Correspondence (2.33) gives

/(*)~ΐ +

n=l

2 (-l)

-K-T:

cos ((2n

—

1)πχ)

sin(mrx)

(2n - 1)

ηπ

(2.34)

Figure 2.10 shows the graph of / as well as the 3- and 10-term partial sum

approximation to the Fourier series for /.

MfrvOy^v^

-1 -0.8 -0.6 -0.4 -0.2

Figure 2.10 Plot of

the

function / (solid line) as well as

the

three-term (dotted) and 10-term

(dashed) partial sum approximations of

the

Fourier.

2.7.1 2c-Periodic Extensions

The orthonormal functions making the Fourier series are 2c-periodic (have period

of 2c). It follows that the Fourier series Correspondence (2.34) should give a valid

correspondence to the 2c-periodic extension of /. This implication is evident in

Figure

2.11.

The plot of the 10-term partial sum approximation of the Fourier series

relative to the 2c-periodic extension of / suggests the Fourier series may converge to

/ for x on the interval

[-3,3]

with the exception of x =

-3,-1,1,

and 3. It is evident

in Figure 2.11 that the deviation of the partial Fourier series sum from the function

/ grows larger as one approaches the points of discontinuity in /. This behavior is

known as the Gibbs phenomenon. Consequently, caution must be used, especially

near points of discontinuity, when approximating a discontinuous function / with a

partial Fourier series sum.

2.8 BEST APPROXIMATION

One of

the

main results of

this

chapter is the development of

the

Fourier sine, Fourier

cosine, and Fourier series for a function / belonging to C

(a,

b).

Note the space

(a,

is C

(0, c) for the sine and cosine series, and

(—c,

c) in the case of the

BEST APPROXIMATION 41

\ 0.8

\ 0.6

\ °'

-3 -2 II »

Figure

2.11

Plot of

the

2c-periodic extension of

the

function / (solid) and 10-term (dashed)

Fourier series partial sum.

Fourier series. We have not yet determined that these infinite series actually converge

to the function / at any of the points x in (a,

b).

Convergence of a given Fourier

series to a function / will be established in Section 2.11

In theory, these series actually converge to / at all, or almost all, x in (a, b).

However, because these are infinite series, it is not possible to actually add all the

terms in the series to give an exact replication of /. When actual values are needed in

solutions to PDEs, a finite number of terms in the series, a partial sum of the series,

must be used to represent (approximate) the boundary or initial condition given as /.

The natural question arises: If, in a practical setting, a finite linear combination

of orthonormal functions must be used to approximate /, is the respective Fourier

approximation the best? That is, if

Υ^α

{χ)

=αιφι(χ) 4-α

(^)

-\-α

(χ) (2.35)

n=l

represents an arbitrary linear combination of the first N orthonormal functions, is

there an optimal set of coefficients {71,72, ·. ·,

7N}

such that

n=l

is the best approximation to / on the interval (a, b)l Further, are the optimal

coefficients η

the same as the Fourier coefficients c

that are determined by

= ¡(χ)φ

{χ)άχ

As a way of answering these questions, the following measurement of error E,

called mean-square error, will be used. Given a function / from C

(a, 6), and an