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

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22 FOURIER SERIES

bj is greater than or equal to the order of all vectors b¿ in the basis. Let m be that

maximum order. Now, consider the polynomial

p(rr) = a

+ a\x

h a

+ a

m+i

m+1

such that

φ 0. With p defined in this way, it follows that p is not in the

set spanned by basis B, because there is no polynomial in B with order m H- 1.

Consequently, there is no finite basis B for P(a, 6), so dim P(a,

is infinite. Because

P(a,

is a subspace of F(a, 6), it follows that the dimension of F(a,

is infinite as

well.

2.1.3 Inner Products

As a means of defining an inner or scalar product, on a vector space V, we begin by

defining a linear form on V. The function L : V

i—►

R is said to be a linear form on

Vif

L(au + /?v) = aL(u) + /?L(v)

for all u and v in V and a and β in R. The function a : V x V

t—►

R is said to be a

bilinear form on V if

α(·,

·) is a linear form in each of its components. That is,

a(w, au + /?v) = aa(w, u) + /3a(w, v)

and

a(au 4- /3v, w) = aa(u, w) + /3a(v, w)

A bilinear form is symmetric on the linear space V if

a(u, v) = a(v, u)

for all u and v belonging to V.

An inner product (u, v) on the vector space V is a bilinear form such that

(u, u) = a(u, u) > 0

for all nonzero elements u of V.

An inner product space is a vector space V on which an inner product is defined.

The usual inner product for the vector space R

is defined as

(u,v) = ((u

),(vuv

,vs))

U\Vi+

U2V2 + l¿3^3

Once an inner product is defined on a vector space V, the notions of length,

distance, and orthogonality may be defined. The length, or norm, of a vector u is

given as

||u|| = v

(u7uT

The distance between two vectors u and v is defined as

dist(u, v) = ||u

—

v||

THE INTEGRAL AS AN INNER PRODUCT 23

Two vectors u and v are said to be orthogonal if and only if

<u,v>=0

A set S = {ui, U2,..., u

} is said to be orthogonal if (u¿, Uj) =0 whenever

ίφό-

A set of nonzero, orthogonal vectors is linearly independent. To show this is true,

suppose

ciui +

h c

= 0 (2.3)

for a set of coefficients

c¿.

Forming the inner product of both sides of Equation (2.3)

with u¿, where i is an arbitrary index between

and n, gives

(u¿,

ciui +

+ · · · + c

) = (u¿, 0)

=> (u¿,CiUi) + (U¿,C

) + ..-(u

U¿> + --(u

) =0

=> ci(u¿,ui> + c

(u¿,u

)

Ci(u¿,u¿)

(u¿,u

) = 0

=Φ ci0 + c

0H Q(U¿,U¿)

0 = 0 (ui>u¿) = Oif

^ j

=Φ Q(UÍ,U¿) = 0

=Φ c¿ = 0 u¿ ^ 0

Because i is an arbitrary index, it follows that all a in Equation (2.3) are zero.

Consequently, set S is linearly independent. This last result means that an orthogonal

set of nonzero vectors form an orthogonal basis for their span. Additionally, if

||u¿||

= 1 for 1 < i < n, the set is an orthonormal basis for the set of vectors

spanned by the set.

2.2 THE INTEGRAL AS AN INNER PRODUCT

The definite integral can be used to define an inner product for real-valued function

on an interval. Recall that any function f that is continuous on an interval [a,

has a

unique definite integral

f(x)dx

Now, suppose f is continuous on (a, 6), and the one-sided limits from the interior of

(a,

are finite at both a and

That is, suppose

lim f(x) = L\ and lim î(x) = L

x—+a+

x-+b~

where L\ and L

are finite numbers. The function f* is defined as

( L\ x

—

f*(x)

f(x) a < x < b

—

24 FOURIER SERIES

is continuous on [a,

and

*(x)dx

ί Π

exists.

Further, we may conclude

pb pb

j f(x)dx = /

(*(x)dx

Ja Ja

The continuity of f on (a,

may be relaxed while preserving the integrability of

f. For example, suppose f has discontinuities at x\ and X2, as shown in Figure 2.1.

Note: The function f has a removable discontinuity at χχ, and

jump discontinuity

atx

a xi X2 b

Figure

2.1

removable discontinuity at x\, and

a jump

discontinuity at

X2.

From properties of definite integrals it follows:

l î(x)dx = / f(x)dx + / f(x)dx + / f(x)dx (2.4)

Ja Ja J

J X2

Because f is continuous on (a, #i), (xi,

X2)»

d (x2

with finite one-sided limits

at end points of each interval, each of

the

integrals on the right-hand side of Equation

(2.4) exist, so that f is integrable on the entire interval (a, b).

2.2.1 Piecewise Continuous Functions

A very important subspace of F(a,

will be defined so that an appropriate inner

product for the real-valued functions F(a, 6)may be defined. Suppose fis continuous

at all but a finite number of points in the interval (a, b). Let this finite set of points

be given as {xi,

X2,

· · ·, x

}, with a < x\ < X2 < · · · < x

< b. An example of

such a function is shown in Figure 2.2. Note: The function / is continuous on each

of the finite-numbered open intervals (a, xi), (xi, X2),..., (x

, b). In addition, if /

has a finite right-hand limit at a, finite one-side limits at each x¿ (1 =

1,2,...,

n),

and a finite left-hand limit at 6, then f is said to be piecewise continuous on (a, b).

The notation

lim f (x) = f(xi+)

THE INTEGRAL AS AN INNER PRODUCT 25

is used to denote the right-hand limit of f at

x¿.

In a similar way,

lim f(x) = f(xi—)

x^xj

is used to denote the left-hand limit of f at x

Figure 2.2 Example of a piecewise continuous function on the interval (a,b) with n

discontinuities.

The set of piecewise continuous functions on the interval (a, b), denoted by

(a, 6), is a subspace of the vector space F(a,6). Certainly the zero function

is an element of C

(a,

because any constant function is continuous (zero discon-

tinuities). Suppose function f, with {x±,X2, · · · ,%n} as its set of discontinuities,

and function g, with continuities at each y in the set {t/i, y<i,..., y

}, are elements

ofCp(a, b). Their sum f + g is an element of C

(o,

as well because its set of

discontinuities is the union of the individual sets of discontinuities, a finite set of

at most n + m elements. Also, cf will belong to C

(a,

because the set of its

discontinuities is the same set as those for f.

2.2.2 Inner Product on C

(a, b)

It is now possible to define an inner product in terms of a definite integral on the

set C

(a, b). Proof by induction is used to establish that all piecewise continuous

functions are integrable (see Exercise 2.2).

Given two functions f and g from C

(a, b), the inner product of f and g is defined

(f,g>= / Î(x)g(x)dx (2.5)

It is left as an exercise to show the definition given by Equation (2.5) is, in fact, an

inner product (Exercise 2.4) by showing this definition satisfies the four properties

of an inner product outlined in Section

2.1.3.

Central to this process is the need to

show that the product of piecewise continuous functions is piecewise continuous, and

therefore integrable (Exercise 2.3).

26 FOURIER SERIES

2.3 PRINCIPLE OF SUPERPOSITION

Recall that a function, or transformation, L from vector space V to vector space W,

is linear if and only if the following properties hold for all vectors u and v of

and

all scalars c in R:

L(u + v) = L(u) + L(v)

L'(cu) = cL(u)

The kernel of a linear transformation L is the set of all elements v of V such that

L(v) = 0, where 0 is the zero of vector space W. It is an easy matter to show that

the null of L is a subspace of the vector space V (see Exercise 2.5).

2.3.1 Finite Case

Recall that the general form of a linear, second-order PDE for dependent variable

u as a function of two independent variables x and y (one of these variables may

represent time) is

+ Bu

4- Cuyy + Du

+ Euy + Fu = Q (2.6)

where the coefficients A —F and right-hand side Q are functions, at most, of x and y.

The expression given in Equation (2.6) can be thought of as a linear transformation

from the vector space V of twice-differentiable functions u(x, y), defined on a subset

of R

, to the vector space W of real-valued functions Q(x,y) defined on the same

subset of R

If Q is identically zero, then Equation (2.6) is a linear, homogeneous PDE. From

the perspective of linear transformations, it would represent the transformation from

the vector space V of twice-differentiable functions, defined on a subset of R

, onto

the zero function 0 of

the

vector space W of real-valued functions on the same subset

(see Figure 2.3). Understand that the solution set of the PDE

+ Bu

+ Cuyy + Du

+ Euy + Fu

—

0 (2.7)

represents the kernel of the linear transformation defined by

L(U)

U{X

y)+B

Jydi

U{X

4ñ

U{X

d d

+D—u(x, y) + E—u(x, y) + Fu{x, y) (2.8)

Because the solution set of the linear, homogeneous PDE represents the kernel of

a linear transformation, it follows that this set is closed under vector addition and

scalar multiplication. That is, any linear combination a\\ii

solutions to L(u) = 0 is also a solution to L(u) = 0. This last statement is referred

to as the principle of superposition. This result is stated formally in the form of a

theorem.

PRINCIPLE OF SUPERPOSITION 27

V W

Figure 2.3 The kernel of L represents the general solution to the linear, homogeneous PDE.

Theorem 2.1 (Principle of Superposition) If each of

the

functions ui, u

, ..., u

solves the

linear,

homogeneous PDE L(u) = 0, then any linear combination

i=l

where each

is a constant, is a solution to the same

linear,

homogeneous PDE. That

is,

(

)

^2°i

\i=l

EXAMPLE 2.1

The linear, homogeneous, ID heat equation is used as an example of a linear

operator. The PDE is

(x,t)

= au

(x,

i) (2.9)

which may be rewritten as

(x,t)

-au

(x,t) = 0 (2.10)

From Equation (2.10), define the linear operator L as

d d

dt-

dx-i

To demonstrate that L is, indeed, a linear operator, the following is sufficient:

Given that f and g are any solutions to Equation (2.9), and a and b are any

scalars,

L(aî + bg)

d(af + bg) fd

(af + bg)

(af + b

{ dx* )

28 FOURIER SERIES

2.3.2 Infinite Case

In Section

2.3.1,

it was shown that a linear, homogeneous PDE may be thought of as

a linear transformation L from a set of suitably differentiable functions to the zero

function of the set of real-valued functions. As such, the general solution to the

linear, homogeneous PDE is identical to the kernel of

the

transformation. As a result,

any finite linear combination of solutions to the linear, homogeneous PDE is also a

solution to the linear, homogeneous PDE by the closure of the kernel under vector

addition and scalar multiplication. The objective of

this

section is to establish that an

infinite linear combination a\U\

Λ-

(I2U2

+ «3^3

l· a

is also a solution

to the linear, homogeneous PDE.

Why is this necessary? The method of separation of variables was introduced in

Section 1.8, and

crucial component of this method is the need to determine a Fourier

series representation of one or more of the initial or boundary conditions. In general,

these series representations require an infinite number of terms in order for the series

to provide an exact replication of

the

boundary or initial condition. Consequently, the

proposed solution for the IBVP will take on a form similar to that shown in Equation

(2.12).

u{x,y) = ^2c

(x,y) (2.12)

n=l

where each function u

(x,y) solves the linear, homogeneous PDE. The general

principle of superposition states the infinite linear combination giving the function

u(x, y) solves the linear, homogeneous PDE as well. That is,

¿ί]Γβ

(*,?/))

=0 (2.13)

A somewhat informal argument will be given

justify the validity of the general

principle of superposition. Let L represent a general linear operator. As such, L is a

finite sum of

terms,

each of which is either a function in the independent variables x

and y, or the product of such a function and a derivative operator in one or more of

d(af) d(bg)

3(f)

+ b

3(g)

(af)

(6g)

ÖL

(

a—- +b

v»

-f

/3(f) 3

"VW-"*?)

aL(f) + bL(g)

32g

9(g) d

(2.11)

PRINCIPLE OF SUPERPOSITION 29

the independent variables. A simple example is

L(u) = f(x,y)u + g(x,y)

Now, let u(x, y) be defined by the convergent infinite series

u(x,y) = y^c

(x,?/)

(2.14)

(2.15)

n=l

For the series to converge to u(x, y) on domain D of

it means that for each point

(x,

y) in D, the following holds

u(x,y) = Jim y^c

(x,y) (2.16)

TV—»oo

*-—'

71=1

That is, the sequence of partial sums for each (x, y) converges to the real number

u[x,y).

The series in Equation (2.15) is differentiable with respect to the dependent

variable x if the partial du/dx exists, the partial derivatives du

/dx exist for all n,

and

du(x,y)

= Σ°η

(x,y)

= lim >^ c

{x,y)

(2.17)

_ ..

_ r/x

n=l n=l

for all (x, y) with the domain D. If this resulting series in Equation (2.17) is

differentiable with respect to x as well, then the initial series in Equation (2.15) is

twice differentiable with respect to x.

For a given linear operator L, as suggested in Equation (2.14), and a sufficiently

differentiable infinite series representation for u, as given in Equation (2.15), it

follows:

L(u) = f(x,y)u + g(x,y)-

= /(x, y) Y^ c

+ g(x, y)w-^2

CnUn

n=l n=l

- /(z,y) Jim V c

+ #(x, y) Jim Vc

N-+00

—'

TV—>oo

—' OX

n=l n=l

(

N λ

[

™oo

\ ^' ^ Σ

CnUn

¿Γ

CnWn

^ n=l

CnUn

\n=l /

Σ»

n=l

= lim

TV—>oo

lim

TV—>oo

n=l

30 FOURIER SERIES

The key components in this result include representing the infinite series as the limit

of partial finite sums, and the application of the principle of superposition to those

finite sums of solutions to L(u)

—

Using the preceding argument as justification, the statement on the general prin-

ciple of superposition is given as the following theorem.

Theorem 2.2 (General Principle of Superposition) Let each function u

(x, y) solve

linear,

homogeneous PDE. That

is,

L(u

(x,y)) = Ofor n = 1,2,3, ... . Then the

infinite series

u(x,y) = Y^CnU

(x,y)

n=l

where each c

is a constant, also solves the

linear,

homogeneous PDE. That is,

L(u)

= L

γ^

{x,

y) I

= o

2.3.3 Hilbert Spaces

The set of functions solving the homogeneous equation L(u) = 0, where L is a

linear differential operator, form a "linear space" because of the linearity of L. That

is,

the sum of two solutions ui and

112,

as well as the scalar multiple cu of solution

u are solutions. The introduction of the inner product and the resulting ability to

measure the distance between elements of the space through the norm provides a

means of defining the convergence of a sequence of elements of the space. The

importance of the general principle of superposition is understood knowing the limit

u of a convergent sequence of solutions {u¿} is also a solution to L(u) = 0. This

fact makes the set of solutions "complete" in that it contains all its limit points. A

complete, normed, linear space is know as a Hilbert space.

2.4 GENERAL FOURIER SERIES

The separation of variables example presented in Section 1.8 include linear, homoge-

neous boundary conditions. Consequently, the general solution to the corresponding

ODE included the functions sin(n7nr), n = 0,1,2,... Because of

this,

a series repre-

sentation of the initial condition f{x) using these functions was sought. That is, the

completion of the problem reduced to determining values for c

such that

f(x) = 2Z

n sm(nnx)

—

1 < x < 1 (2.18)

71=1

Such a representation given in Equation (2.18) is a Fourier Sine series for f(x) on

the interval (-1,1). Fourier sine series will be the topic of Section 2.5. The remainder

of the current section will pertain to general Fourier series.

FOURIER SINE SERIES ON (0, C) 31

Let f(x) be an element of C

(a, b). If coefficients c

can be determined for

real-valued functions φ

(χ) such that

Ϊ{χ) = Σο

{χ) (2.19)

n=l

is true for all but a finite number of x on [a,

6],

then Equation (2.19) is said to be a

Fourier series representation for f(x) on (a, 6). In addition, we say the Fourier

series for / is valid on (a, b).

Prior to determining if such a representation is valid for a given /, c

, and φ

(χ),

the unique (see Exercise 2.19) Fourier series corresponding to / constructed from

is denoted as

/(χ)~Σ<ν^

(χ) (2.20)

n=l

For a given set of functions 0

, there may be various methods for determining the

coefficients c

. The determination of c

is particularly simple, in theory, if the set of

functions φ

{χ) is orthonormal. A set of functions from an inner product space V

is orthonormal if and only if

(φ

(χ),φ

(χ)) =

{ X%ZVn

21)

That

is,

a set of orthogonal functions, each of which has norm one, is an orthonormal

set. For an orthonormal set, it follows that

(φ

(χ),

f{x)) = ( φ

{χ), Σ (X)

\ m=l I

771=1

= Σ °™(Φη(χ),Φπι(χ))

m=l

= ο

(φ

(χ),φ

(χ))

Therefore, each coefficient c

is determined simply by calculating the inner product

of f(x) with the corresponding orthonormal function φ

(χ). When the vector space

is C

(a, b), the inner product is defined in terms of the definite integral, so that

= ¡(χ)φ

(χ)άχ (2.22)

2.5 FOURIER SINE SERIES ON (0, C)

In Exercise 2.24, it is shown that the functions

. /ηπχ\

sini J n = 1,2,3,...