Cain G., Meyer G.H. Separation of Variables for Partial Differential Equations: An Eigenfunction Approach

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for constant and is solved by the d’Alembert solution (1.19) because

0 and

1 are defined on

and odd with respect to

=0 and

x=L

As we remarked in Section 1.4, inhomogeneous boundary conditions can often be made homogeneous

at the expense of adding a source term to the differential equation. This leads to problems of the type

Now a Duhamel superposition principle can be applied. It is straightforward to show that the function

solves our problem whenever

is the solution of

where is a parameter. It follows that if

is of the form

then the problem has the d’Alembert solution

All integrations can be carried out analytically and the resultant solution

uN(x, t)

can be shown to be

identical to the separation of variables solution found in Chapter 7 when an arbitrary source term

approximated by a trigonometric sum

Let us now turn to the general initial/boundary value problem of the form

(1.20)

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where

is a given domain in For convenience we have set

=1 which can always be achieved by

scaling time. We point out that if

and we have a pure initial value problem, then it again is

possible to give a formula for

u(x, t)

analogous to the d’Alembert solution of the one-dimensional

problem (see [5, Chapter 2]), but for a true initial/boundary value problem the existence of a solution

will generally be based on abstract theory. (We note in this context that if

=0 on

∂D,

then, again in a

very general sense, the problem has a lot in common with the ordinary differential equation

We shall henceforth assume that the existence theory of [5] applies so that we can concentrate on

uniqueness of the solution and on its continuous dependence on the data of the problem.

Uniqueness and continuous dependence follow from a so-called energy estimate. If it is a smooth

solution of (1.20) with

0 on

∂D,

then

so that

We now apply the divergence theorem and obtain

(1.21)

For a smooth solution the boundary data

imply

so that

Since we obtain from (1.21) the estimate

(1.22)

where

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and

The inequality (1.22) can be written as

where

is some unknown nonnegative function. This differential equation has the analytic solution

from which we obtain the so-called Gronwall inequality

(1.23)

where

is known from the initial data.

We have the following two immediate consequences of (1.23).

Theorem 1.7

The solution of the initial/boundary value problem

(1.20)

is unique.

Proof. The difference

of two solutions satisfies (1.20) with

F=g=u

This implies that E(0)=0 so that (1.23) assures that

E(t)=

0 for all

. Then by Schwarz’s inequality (see

Theorem 2.4)

from which follows that

Hence

0 in the mean square sense for all

which implies that a classical solution is identically zero.

Theorem 1.7 assures that the separation of variables solution constructed in Section 7.1 is the only

solution of the approximating problem. Similar arguments are used in Section 7.2 to show for a vibrating

string that this approximate solution converges to the analytic solution of the original problem as the

approximations of the data are refined.

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Chapter 2

Basic Approximation Theory

This chapter will review the abstract ideas of approximation that will be used in the sequel. Let

be a

linear space (sometimes called a vector space) over the field

of real or complex numbers. The

elements of

are called vectors and those of

are called scalars. The vector spaces appearing in this

book are

and

with elements etc., or spaces of real- or complex-valued

functions

f, g,

etc. defined on a real interval or, more generally, a subset of Euclidean

-space. In all

spaces denotes the zero vector. The scalars of

are denoted by

α, β,

a, b,

etc.

We now recall a few definitions from linear algebra which are central in our discussion of the

approximation of functions.

Definition Given a collection

of vectors in a linear space

then

is the set of all linear combinations

of the elements of

Note that

is a subspace of

because it is closed under vector addition and scalar multiplication.

Definition A collection

of vectors in a linear space is linearly independent if

A sequence of vectors

is linearly independent if any finite collection

drawn from the sequence is

linearly independent.

The definition implies that in a finite set of linearly independent vectors no one element can be

expressed as a linear combination of the remaining elements.

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Definition Let be a collection of

linearly independent vectors of

with the property

that every element of

is a linear combination of the (i.e., Then the set

is a basis of

and

has dimension

The theorems of linear algebra assure that every basis of an

-dimensional vector space consists of

elements, but not every vector space has a finitedimensional basis. Spaces containing sets of countably

many linearly independent vectors are called infinite dimensional.

2.1 Norms and inner products

Basic to the idea of approximation is the concept of a distance between vectors

and an approximation

g, or the “size” of the vector

f−g

. This leads us to the the idea of a norm for assessing the size of

vectors.

Definition Let

be a vector space. A norm on

is a real-valued function

F:X

→

such that for every

and every scalar it is true that

ii)

F(αf)=|α|F(f);

iii)

F(f+g)≤F(f)+F(g).

Proposition 2.1

F(f)

≥0.

Proof. From ii) with α=0, we know that Thus

F(f+(−f))

≤

F(f)

F(−f)=

F(f).

In other words,

F(f)

≥0.

The value of the norm function F is almost always written as

A vector space

together with a norm

is called a normed linear space. The inequality iii) is commonly called the triangle inequality.

The concept of a norm is an abstraction of the usual length of a vector in Euclidean three-space and

provides a measure of the “distance” ||

−

|| between two vectors Thus in the space

3 of

triples

of real numbers with the customary definitions of addition and scalar multiplication,

the function

is a norm (see Example 2.6a). The distance induced by this norm is the everyday Euclidean distance.

In a linear space

our approximation problem will be to find a member

of a given subspace

that is closest to a given vector

in the sense that ||

f−fM

||≤||

f−m

|| for all We shall be

concerned only with finite-dimensional subspaces

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We know from elementary geometry that in ordinary Euclidean three-space, the closest point on a line

or a plane containing the origin to a given point in the space is the perpendicular projection of onto

the line or plane. This is the idea that we shall abstract to our general setting. For this, we need to

extend the notion of the “dot,” or scalar, product.

Definition An inner product on a vector space

is a scalar-valued function

→

on ordered pairs

of elements of

such that

G(f, f)

≥0 and

G(f, f)

=0 if and only if

ii) denotes the complex conjugate of

];

iii)

G(αf, g)=αG(f, g);

and

iv)

G(f+g, h)=G(f, h)+G(g, h).

We shall usually denote

G(f, g)

by A vector space together with an inner product defined on it is

called an inner product space.

The next proposition is easy to verify.

Proposition 2.2

An inner product has the following properties:

ii)

iii)

Definition Two vectors

and

in an inner product space are said to be orthogonal if

Example 2.3 a) In real Euclidean

-space

Rn,

the usual dot product

where

and is an inner product.

b) In

for and define by where

is the matrix

and

is the usual dot product of and Then is an inner product. First consider

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It is clear that

and if and only if

To see that simply compute both inner products. The remaining two properties are

evident.

c) On the space of all continuous functions (real- or complex-valued) defined on the reals having period

it is easy to verify that

is an inner product.

Theorem 2.4

In an inner product space,

Proof. If then the proposition is obviously true, so assume Let α be a complex

number. Then

Now

Next, let

where

is any real number. Then

This expression is quadratic in

and so the fact that it is never negative means that

In other words,

which completes the proof.

The inequality

is known as Schwarz’s inequality.

Corollary 2.5

Suppose X is an inner product space. Then the function F defined by

is a

norm on X.

Proof. The proofs that

F(f)

≥0 and

F(αf)=|α|F(f)

are simple and omitted. We prove the triangle

inequality.

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Hence

F(f

≤

F(f)+F(g),

and we see that is indeed a norm on

In an inner product space, the norm is called the norm induced by the inner product

Example 2.6 a) Let

be real Euclidean n-space endowed with the usual inner product

(cf. Example 2.3a). Then the norm induced by this inner product is the usual Euclidean norm

b) Let

be the space of all complex valued continuous functions defined of the interval

[a, b]

(cf.

Example 2.3c). Then

is an inner product on

and

is the norm induced by this inner product. Very closely related to this example

is the root mean square of a function on the interval

[a, b]

2.2 Projection and best approximation

In Euclidean space, given a line or a plane

through the origin and a vector the vector in

closest

to is the vector such that is perpendicular to every vector in

. This vector is the projection

of onto

We shall see that the idea of a projection generalizes to the abstract setting of inner product spaces.

Definition Suppose

is a subspace of an inner product space

and suppose The orthogonal

projection of

onto

is a vector such that

r=f−Pf

is orthogonal to every

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In case

has finite dimension, the existence of an orthogonal projection onto

is easy to establish.

Suppose

{φ

, φ

,…, φN}

is a basis for the subspace

. The projection is a member of

and so

and we need only to find the coordinates

αj.

First, observe that a vector is orthogonal to every element

if and only if it is orthogonal to each of the basis elements

φi

. Hence we want

Since

In matrix-vector form

where

Proposition 2.7

The coefficient matrix

is nonsingular.

Proof. Suppose for some vector Then

But the collection is independent and so it must be true that

=β

2=…=

βN.

Hence is nonsingular.

We have thus shown that the system has a unique solution for the coordinates of the

orthogonal projection of

onto

. In other words, the orthogonal projection of

onto

exists and is

unique. We can thus safely speak of

P f

as being

the

orthogonal projection of

onto

with respect to

the inner product

Remark If the basis

{φ

, φ

…,

φN}

is orthogonal (i.e., for

≠

j),

then

is a diagonal

matrix, and the coordinates of

P f

are easily found

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Next we see that the projection of

onto a subspace

is the member of

that best approximates

the sense that of all elements

it is one that makes ||

m−f

|| the smallest.

Theorem 2.8

Let M be a subspace of an inner product space X and suppose

If then

f−Pf

||≤||

f−m

||,

where P f is the orthogonal projection of f onto M.

Proof. Observe that

Then

since because Thus

Hence the orthogonal projection

onto

is a best approximation. We next show that any best

approximation of

from the subspace

must be this unique projection of f onto M.

Theorem 2.9

Let M be a subspace of an inner product space X and suppose

If is such

that

f−h

||≤||

f−m

for all

then h=P f, the orthogonal projection of f onto M.

Proof. Let be an arbitrary element of

. For any real

and any scalar

α,

the vector

tαm

is a

member of

. Thus the function

defined by

has a minimum at

0. But

F(t)

is simply a quadratic function of the real variable

F′

(0)=0 means

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