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

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62 FOURIER SERIES

odd function on [—c,

c],

it follows that

/ ΤίΊΤΧ \

= - / F(x)cos( J dx n = 0,1,2,...

c J_

\ c /

is zero for all n = 0,1,2,... Therefore, we know

) = Yl

Sin

f —T I

¿-^

\ΈΧ C J

n=l

with

/717ΓΧ\

= - F(x)sini jdx n = 1,2,3,...

is valid for all x in (0,

c).

The sine series representation for / is valid at x = 0 only

if /(0) = 0. The same is true for x

—

Similar arguments may be used to justify the validity of the cosine series for the

function / defined on the interval

[0,

c].

This process is left as an exercise (see

Exercise 2.32).

EXERCISES

2.1 Show that the set P2, given as the set of all polynomials degree two or less

defined on the interval (a,b), is a subspace of the vector space F(a, b).

2.2 Use proof by induction to show that if f has a finite set of bounded disconti-

nuities given by {χχ, χ<ι,.. .x

} on the finite interval [a,

6],

then f is integrable on

[a,b].

2.3 Prove that if f and g are elements of C

(a, b), then fg is an element of C

(o, 6),

so that fg is an integrable function.

2.4 Provide the details to show that the integral definition given in Equation (2.5)

satisfies the four properties of an inner product outlined in Section

2.1.3.

That is, if

f, g, and h are elements of C

(o, 6), and c is any scalar, show

pb pb

a) / f(x)g(x)dx = / g(x)f(x)dx

Ja Ja

rb nb pb

b) / h(x) [f(x) + g(x)} dx= h(x)f(x)dx + / h(x)g(x)dx

Ja Ja Ja

pb pb

c) / [cf (x)] g(x)dx = c / f(x)g(x)dx

Ja Ja

pb pb

d) / f(x)f(x)dx > 0 and / f(x)f(x)dx = 0 if and only if f = 0

Ja Ja

2.5 Let L

1—>·

W be a linear transformation. The kernel of L is the set of v

such that L(v) = 0, where 0 is the unique zero vector of W. Show that kernel L is

a subspace of V.

EXERCISES

2.6 Suppose

f(x) is a

non-negative function with domain

that may,

may

not, be continuous. Let x* G

be such that

f(x*) < f(x) for

all

Df.

Prove

that

y/f(x*)

y/f(x)

for all

G J9/. Prove the converse as well.

2.7 Let Cp(—1,1)

the inner product space

piecewise continuous functions

on the interval

(-1,1)

with inner product defined as in Equation (2.5).

a) Show that functions

f(x) = 1 and

g(x)

= x are

orthogonal elements

M-i,i).

b) Determine values for

and b so that the function

G C

(—1,1) given by

h(x)

1 4-

ax + bx

is orthogonal to both

f(x) =

1 and

g(x) = x.

2.8 The purpose of

this

exercise is to demonstrate how an orthogonal set of vectors

may be generated from a linearly independent set. This method is commonly referred

to as the Gram-Schmidt process.

a) Suppose

and g are linearly independent vectors of the inner product space

(a, 6), with inner product given by Equation (2.5). Let

= g + aî\.

Under the requirement that

and Î2

orthogonal elements

(a, b),

show that

<fi,g>

Ifil

Let

vector

g be

linearly independent from

the

nonzero, orthogonal

set

{fi, Î2,

■.

■

fn-i} of vectors from C

(a, b). Define

=tr

(

l»g)

(

2>ë)

(fn-l,g)

pip

||f

···

Hf^iP

-1

Show that

is orthogonal to

2.9 Show that for any two functions

and g that belong to the space C

(a, b),

Iff

2 lf(

)9(

)

)Ι(

)Ϋ

dxd

y =

Use this result to establish the Schwartz inequality

K/,0>l<

'\\g\\-(f,9Y

2.10 Suppose

and g are elements of C

(a, b). Prove that

ΙΙ/

ί/ΙΙ<ΙΙ/ΙΙ

+ ΙΙί/ΙΙ·

This

is a

more general statement

the triangle inequality encountered

linear

algebra where vectors from

were considered. [Hint: Begin by showing

ll/

9ll

ll/ll

2(/,9)

lli7ll

and then applying the Schwartz inequality established in Exercise 2.9.]

64 FOURIER SERIES

2.11 Suppose / is an arbitrary element of C

(a, b). Establish the validity of the

following statements:

a) If f(x) = 0, except possibly at a finite number of points in the interval

a<x<6,then ||/|| =0.

b) Given that ||/|| =0, then f(x) = 0 at all x in a < x <

with the possible

exception of a finite number of points.

2.12 Determine /(0+) and

(Q)

for the function

- 1

m - V

2.13 Provide an example of a function that satisfies the following characteristics:

(0)

= 0, but f'(0) does not exist.

2.14 Let

sin - x φ 0

0 x = 0

Determine, if possible,

(0),

/¿(0), /'(()+), and /'(O-).

2.15 Show that the function

x sin - x Φ 0

0 x = 0

is continuous at x = 0, but neither

(Q)

nor /¿(0) exist.

2.16 Suppose / is defined as

x <0

L sinx x > 0

Determine f

(0) and /¿(0). Does /

;

(0) exist? Explain

2.17 Suppose the right-hand derivatives of / and g exist at xo. Prove that the

product function (fg)(x) = f(x)g(x) has a right-hand derivative at x

. [Hint: Use

the definition of right-hand derivative. Look up the proof of the product rule in your

calculus book for an idea of the useful "trick" to accomplish this task.]

2.18 Derive the expression

(u)

= ^

sinI

;

(u φ 0, ±2π, ±4π,... )

for the Dirichlet kernel

DN(U) =

cosrm

n=l

EXERCISES 65

by writing

A = - and B = nu

in the trigonometric identity

sin A cos B = $m{A + B) + sin(A - b)

and then summing each side of the resulting equation from n=\ to n = N.

2.19 An orthonormal set

«5

is said to be closed in the space C

(a, 6), or a subspace

of C

(o, 6), if there are no elements of C

(a, 6) with nonzero norm that are orthogonal

to each element of S. Suppose S is

closed orthonormal set of the space of continuous

functions on the interval a < x < b. Prove that if two continuous functions / and

g have identical Fourier coefficients relative to S, then / and g are identical on the

interval a < x < b. This establishes that the Fourier coefficients for a continuous

function / are uniquely determined.

2.20 Provide arguments similar to those given in Section 2.13 to establish the

validity of a cosine series representation of a function / from the space C

(0, c).

2.21

a) Determine the Fourier cosine series representation for sin x on the interval

0 < X < 7Γ.

b) Provide justification that the series found in part (a) converges to sinx for

all x in

[0,

π].

c) Establish the following summation formulas:

(-l)

_ 1 π

2.22

2.23

1 1

=1 v

^ An

-1 2 ^ An

-1 2 4

n=l n=l

a) Find the Fourier cosine series corresponding to the function f(x) = x on

the interval 0 < x < π.

b) Explain how you know the cosine series converges to / on the intervals

[0,

π]

and [—π, π].

c) Establish the summation result

-i 2

1 _ 7T

(2n - l)

2 =

71=1

a) Find the cosine series corresponding to f(x) = x

on the interval (0, π).

b) Provide arguments as to why the cosine series found above converges to x

on the interval

[0,

π],

c) Establish the following summation formulas:

y, (-1)"

_ 7Γ

V— - —

^ n

" 12 ^ n

~ 6

n=l n=l

66 FOURIER SERIES

2.24 Use the trigonometric identity

2 sin a sin ß

—

cos(a;

—

ß)

—

cos(a + ß)

to show that

0 for m Φ

πιπχ . ηπχ

sin

ax =

\ c

c I -

for m =

where m and n are positive integers.

2.25 Use the trigonometric identity

2 cos a cos β

cos(a

—

/?) + cos(a + )9)

to show that

0 for m Φ

πιπχ ηπχ '

cos

ax =

\ c

c -

for m

—

2.26 Use the trigonometric identity

2 sin a cos β

—

sin(a + β) 4- sin(a;

—

/?)

to show that the functions sin Í

and cos Í 1 are not orthogonal on the

interval

[0,

whenever τηφη.

2.27 Define the set of functions {φ

(χ)} (n

0,1,2,...) as

φο(χ)

1 φ2η-ι{χ) = sin

( J

2η

(x) = cos ^

a) Show

/ φο(χ)φ2η-ι{χ)άχ

and /

(χ)φ2η(χ)άχ = 0

J —c */ —c

forn

1,2,3,....

Then, show that

\\φο\\

φ$(χ)φ$(χ)(1χ = 2c

J — c

-c

b) Show that

*2m-l(^)02n(iP)d^

for anj positive integers m and n

c) Show that

φ2πι-ΐ(χ)Φ2η-ΐ(χ)άχ =

0 f or m

φ η

c f or m

EXERCISES 67

/ φ2πι(χ)Φ2η{χ)άχ = <

d) Show that

0 for m φ η

c for ra

—

As a result of the parts (a)-(d) above, the functions

1 . 1 /ηπχ\ , 1 /ηπχ\

θ(

)

~7K= <P2n-l =

—¡F

Sin 1 02n = "^ COS Í

\/2c x/c \ c / Jc \ c J

/2C V

C V C J

y/C \ C

for n = 1,

2,3,...

form an orthonormal set on the interval [—c, c].

2.28 Given that / is an odd function, show that / f(x)dx = 0.

2.29 Using arguments similar to those given in Section 2.9, show that the Fourier

cosine coefficients a

, corresponding to a piecewise continuous function / on and

the orthonormal functions the interval

[0,

(χ) = —7= Φη{ζ) = \ - COS ( )

are such that

lim a

—

0 n =

0,1,

2,...

n—>oo

2.30 Suppose you are to

find

the Fourier series Correspondence (2.33) for

function

/ on the interval [—c, c].

a) If / is an odd function, show that a

in Correspondence (2.33) is zero for

alln = 0,1,2,3,....

b) If / is an even function, show that b

in Correspondence (2.33) is zero for

alln = 1,2,3,....

2.31 Given the a function / is differentiable at a point Xk in its domain, prove that

both one-side derivatives exist and are equal to

f'(xk).

2.32 Use arguments similar to those presented in Section 2.13 to establish the

validity of the cosine series representation for all x in (0, c) for a function / that is

piecewise smooth on

[0,

and satisfies

/(x,

= Λ£±1±Λΐ=)

for all x in (0, c). Specify necessary condition for the representation to be valid at

x = 0 and x

—

2.33 The purpose of this exercise is to illustrate the difference between convergence

and uniform convergence of a Fourier series to a given function /ona prescribed

interval. In this exercise, the function is f(x) =

1 —

and the interval is 0 < x < 1.

a) Create a Fourier cosine series representation for / on the given interval

using some means of technology. Determine the fewest number of terms

68 FOURIER SERIES

required in the series so that the error in approximating / on

(0,1 )

is less

than e = 0.01 for all x on (0,1). Include a plot of the error on the interval

(0,1).

The ability to determine a single partial series with values within e

of f(x) for all x in (0,1) is defined as uniform convergence.

b) If you attempt to repeat the process of part (a) with a Fourier sine series

instead of the cosine series, you will find that, for a given e such as 0.01,

it is not possible to find a partial Fourier sine series that represent / on

the entire interval (0,1) with error less than e. That is so even though the

Fourier sine series converges to f(x) for all x in (0,1).

To illustrate this and related phenomena, use technology to generate

error plots for 50-, 100-, and 200-term Fourier sine series. Explain what

you see in this progression of plots. In particular, describe what happens

to the location and value of the maximum error. The maximum error

represents what percentage of the difference between /(l) and —/(l)?

This percentage is a characteristic of the Gibbs phenomena.

CHAPTER 3

STURM-LIOUVILLE PROBLEMS

This chapter focuses on the general Sturm-Liouville problem that arises in sepa-

ration of variable techniques in boundary value problems. The section begins by

identifying three basic IBVPs from which Sturm-Liouville problems result. Next,

Regular Strum-Liouville problems are defined, and several important properties are

established. The section concludes with solution examples for certain cases.

3.1 BASIC EXAMPLES

We have seen in Section 1.8 how the particular Sturm-Liouville problem, shown in

Equation (3.1),

X"(x)

+ \X(x) = 0 X{a) = 0 and X(b)=0 (3.1)

evolves in the method of separation of variables for ID heat transfer IBVP given by

Ut — OLU

\ a < x < b, t >0 (PDE)

IBVP {

(

i°) = f(

) (

) (3.2)

u(a, t) = 0 (BCs)

[ u(b,t) = 0

Fourier Series and Numerical Methods for Partial Differential Equations, 69

First Edition. By Richard Bernatz

70 STURM-LIOUVILLE PROBLEMS

The problem stated in Equation (3.1) is more general than that presented previously

because the left- and right-hand boundaries are located at x = a and x — b (a < b),

respectively. Prescribed boundary values for the dependent variable u, as in this

example, are referred to as "Dirichlet" conditions.

For the slightly different IBVP given as

IBVP<

ut = OLU

(PDE)

u(x,0)=f(x) (IC)

(a,t) = 0 (BCs)

(3.3)

the Sturm-Liouville problem

X"(x)

+ XX (x) = 0 X'(a) = 0 and X\b) = 0 (3.4)

results. Recall the BCs given in Problem (3.3) are called "Neumann."

If the BCs prescribed in a similar IBVP are "periodic" in that u(a, t) = u(b, t),

the following Sturm-Liouville problem results

X"{x)

+ XX(x) = 0 and X(a) = X(b) X'(a) = X

(b) (3.5)

3.2 REGULAR STURM-LIOUVILLE PROBLEMS

The three Sturm-Liouville problems presented in Section3.1 are specific instances

of a more general Sturm-Liouville problem given as

f [r(x)X'(x)]' + [q(x) 4- Xp(x)] X{x) =0 (a < x < b)

< a

X(a)+a

X'(a) = 0 (3.6),

{ hX(b) + b

X'{b) = 0

where the functions p, q, and r are independent of the parameter λ, constants a\ and

Ü2

are not both zero, and constants b\ and b

are not both zero.

As indicated in the solution of

the

Sturm-Liouville problem in Section 1.8, a non-

trivial solution X(x) to Problem (3.6) is called an eigenfunction of Problem (3.6),

and a λ value for which a nontrivial solution exists is an eigenvalue of Problem (3.6).

Additionally, if X(x) is an eigenfunction of the ODE, then so is CX(x), where C

is an arbitrary constant. For X(x) to be an eigenfunction for Problem (3.6), we will

require that both X and X' are continuous on the closed interval a < x < b. The

set of eigenvalues for Problem (3.6) is called the spectrum of the Sturm-Liouville

problem. If the spectrum of a Strum-Liouville problem consists of

discrete,

positive

real values λ

, it is customary to list them in increasing order λχ, X

A3,...

A given Sturm-Liouville problem is said to regular when both of the following

conditions are satisfied:

i. The functions

q, r, and r' are continuous, real-valued functions on the closed

interval a < x < b

PROPERTIES 71

ii.

The function p(x) > 0 and the function r(x) > 0 on the closed interval

a < x < b.

If any one of the regularity conditions listed above is not satisfied by a Sturm-

Liouville problem, it is said to be singular. Section 3.3.1.1 and Exercise 3.4 provide

further information about singular Strum-Liouville problems.

The following are known properties about eigenvalues and eigenfunctions of

regular Sturm-Liouville problems:

The spectrum for λ consists of an infinite number of discrete eigenvalues. (Sin-

gular Sturm-Liouville problems may have no eigenvalues, an infinite number

of discrete eigenvalues, or a continuous spectrum.)

Corresponding eigenfunctions X

(x) and X

(x) of distinct eigenvalues X

and λ

are orthogonal with respect to the inner product

p(x)Xm(x)X

(x)dx

where the "weight" function p(x) is that from the ODE in Problem (3.6). (True

for some singular Sturm-Liouville problems.)

All eigenvalues are real. (Not necessarily true for singular Sturm-Liouville

problems.)

Each eigenvalue has a unique eigenfunction. That is, if X(x) and Y(x) are

eigenfunctions corresponding to λ then it follows that Y(x) — CX(x) for

some constant C. (True for singular Sturm-Liouville problems as well.)

If λ is an eigenvalue for a regular Strum-Liouville problem and

q(x) <0(a<x<b) and a\ü2 < 0 b\b2 > 0

then λ > 0. (Not necessarily true for singular Sturm-Liouville problems.)

The first item in the list above is stated without proof in this text because of the

overly complicated nature of the

proof.

See Churchill [11] for details of the proof

for certain cases. Proofs for items 2-5 are given in Section 3.3.

3.3 PROPERTIES

The results concerning eigenvalues and corresponding eigenfunction for regular

Sturm-Liouville problems listed in Section 3.2 are stated formally and proved in

this section.