Gockenbach M.S. Partial Differential Equations. Analytical and Numerical Methods

xx

Preface

2

then derives

the

standard

differential

equations

in one

spatial dimension,

in the

process explaining

the

meaning

of

various physical parameters

that

appear

in the

equations

and

introducing

the

associated boundary conditions

and

initial conditions.

Chapter

3,

which

has

already been discussed above, presents

the

concepts

and

techniques

from

linear algebra

that

will

be

used

in

subsequent chapters.

I

want

to

reiterate

that

perhaps

the

most important

key to

using this

text

effectively

is

to

move through Chapter

3

expeditiously.

The

rudimentary understanding

that

students obtain

in

going through Chapter

3

will

grow

as the

concepts

are

used

in

the

rest

of the

book.

Chapter

4

presents

the

background material

on

ordinary

differential

equations

that

is

needed

in

later chapters. This chapter

is

much easier

than

the

of the

material

is

review

for

many students. Only

the

last

two

sections,

on

numerical methods

and

stiff

systems,

are

likely

to be

new. Although

the

chapter

is

entitled Essential

Ordinary

Differential

Equations, Section

4.3 is not

formally

prerequisite

for the

rest

of the

book.

I

included this material

to

give

students

a

foundation

for

understanding

stiff

systems

of

ODEs

(particularly,

the

stiff

system arising

from

the

heat

equation). Similarly, Runge-Kutta schemes

and

automatic

step

control

are not

strictly needed.

However,

understanding

a

little

about variable step size methods

is

useful

if one

tries

to

apply

an

"off-the-shelf"

routine

to a

stiff

system.

Chapter

8

extends

the

models

and

techniques developed

in the first

part

of the

book

to two

spatial

dimensions (with some

brief

discussions

of

three dimensions).

The

last

two

chapters provide

a

more in-depth treatment

of

Fourier series

(Chapter

9) and finite

elements (Chapter 10).

In

addition

to the

standard

theory

of

Fourier series, Chapter

9

shows

how to use the

fast Fourier transform

to

efficiently

compute Fourier series solutions

of the

PDEs, explains

the

relationships among

the

various types

of

Fourier series,

and

discusses

the

extent

to

which

the

Fourier series

method

can be

extended

to

complicated geometries

and

equations with noncon-

stant

coefficients.

Sections 9.4-9.6 present

a

careful

mathematical treatment

of the

convergence

of

Fourier series,

and

have

a

different

flavor

from

the

remainder

of the

book.

In

particular,

they

are

less suited

for an

audience

of

science

and

engineering

students,

and

have been included

as a

reference

for the

curious student.

Chapter

10

gives some advice

on

implementing

finite

element computations,

discusses

the

solution

of the

resulting sparse linear systems,

and

briefly

outlines

the

convergence theory

for finite

element methods.

It

also shows

how to use finite

elements

to

solve general eigenvalue problems.

The

tutorials

on the

accompany-

ing

CD

include programs implementing two-dimensional

finite

element methods,

as

described

in

Section 10.1,

in

each

of the

supported

software

packages (MATLAB,

Mathematica,

and

Maple).

The

sections

on

sparse systems

and the

convergence the-

ory

are

both little more

than

outlines, pointing

the

students toward more advanced

concepts. Both

of

these topics,

of

course, could easily

justify

a

dedicated semester-

long course,

and I had no

intention

of

going into

detail.

I

hope

that

the

material

on

implementation

of finite

elements

(in

Section 10.1)

will

encourage some students

to

experiment with two-dimensional calculations,

which

are

already

too

tedious

to

carry

out by

hand. This sort

of

information seems

to be

lacking

from

most books

accessible

to

students

at

this level.

xx

Preface

2

then

derives the

standard

differential equations in one spatial dimension, in

the

process explaining

the

meaning of various physical parameters

that

appear in the

equations

and

introducing the associated boundary conditions and initial conditions.

Chapter 3, which has already been discussed above, presents

the

concepts

and

techniques from linear algebra

that

will be used in subsequent chapters. I want

to

reiterate

that

perhaps

the

most

important

key

to

using this

text

effectively is

to

move through

Chapter

3 expeditiously.

The

rudimentary understanding

that

students obtain in going through

Chapter

3 will grow as

the

concepts are used in

the

rest of

the

book.

Chapter

4 presents

the

background material on ordinary differential equations

that

is

needed in later chapters. This chapter is much easier

than

the

material is review for many students. Only the last two

sections, on numerical methods

and

stiff systems, are likely

to

be new. Although

the

chapter

is

entitled Essential Ordinary Differential Equations, Section 4.3 is

not

formally prerequisite for

the

rest of

the

book. I included this material

to

give

students a foundation for understanding stiff systems of ODEs (particularly,

the

stiff system arising from

the

heat equation). Similarly, Runge-Kutta schemes

and

automatic step control are not strictly needed. However, understanding a little

about

variable step size methods is useful if one tries

to

apply

an

"off-the-shelf"

routine

to

a stiff system.

Chapter

8 extends

the

models

and

techniques developed in

the

first

part

of

the

book

to

two spatial dimensions (with some brief discussions of three dimensions).

The

last

two chapters provide a more in-depth

treatment

of Fourier series

(Chapter 9)

and

finite elements (Chapter 10). In addition

to

the

standard

theory of

Fourier series, Chapter 9 shows how

to

use

the

fast Fourier transform

to

efficiently

compute Fourier series solutions of

the

PDEs, explains

the

relationships among

the

various types of Fourier series,

and

discusses

the

extent

to

which

the

Fourier series

method can be extended

to

complicated geometries

and

equations with noncon-

stant

coefficients. Sections 9.4-9.6 present a careful mathematical

treatment

of

the

convergence of Fourier series,

and

have a different flavor from

the

remainder of

the

book.

In

particular, they are less suited for

an

audience of science

and

engineering

students,

and

have been included as a reference for

the

curious student.

Chapter

10 gives some advice on implementing finite element computations,

discusses

the

solution of

the

resulting sparse linear systems,

and

briefly outlines

the

convergence theory for finite element methods.

It

also shows how

to

use finite

elements

to

solve general eigenvalue problems.

The

tutorials on

the

accompany-

ing CD include programs implementing two-dimensional finite element methods, as

described in Section 10.1, in each of

the

supported software packages (MATLAB,

Mathematica,

and

Maple).

The

sections on sparse systems and the convergence the-

ory are

both

little more

than

outlines, pointing

the

students toward more advanced

concepts. Both of these topics, of course, could easily justify a dedicated semester-

long course,

and

I

had

no intention of going into detail. I hope

that

the

material

on implementation of finite elements (in Section 10.1) will encourage some students

to

experiment with two-dimensional calculations, which are already too tedious

to

carry

out

by hand. This sort of information seems

to

be lacking from most books

accessible

to

students

at

this level.

Preface

xxi

Possible course outlines

In

a

one-semester course (42-45 class hours),

I

typically cover Chapters

1-7 and

part

of

Chapter

8. I

touch only lightly

on the

material concerning Green's functions

and

the

Dirac delta function (Sections 4.6, 6.6,

and

7.4.1),

and

sometimes omit Section

6.3,

but

cover

the

remainder

of

Chapters

1-7

carefully.

If

an

instructor wishes

to

cover

a

significant

part

of the

material

in

Chapters

8-10,

an

obvious place

to

save time

is in

Chapter

4. I

would suggest covering

the

needed material

on

ODEs

on a

"just-in-time" basis

in the

course

of

Chapters

5-7. This

will

definitely save time, since

my

presentation

in

Chapter

4 is

more

detailed than

is

really necessary. Chapter

2 can be

given

as a

reading assignment,

particularly

for the

intended audience

of

science

and

engineering students,

who

will

typically

be

comfortable with

the

physical parameters appearing

in the

differential

equations.

Acknowledgments

This book began when

I was

visiting Rice University

in

1998-1999

and

taught

a

course

using

the

lecture notes

of

Professor William

W.

Symes.

To

satisfy

my

per-

sonal

predilections,

I

rewrote

the

notes

significantly,

and for the

convenience

of

myself

and my

students,

I

typeset them

in the

form

of a

book, which

was the first

version

of

this

text.

Although

the final

result bears,

in

some ways, little resem-

blance

to

Symes's original notes,

I am

indebted

to him for the

idea

of

recasting

the

undergraduate

PDE

course

in

more modern terms.

His

example

was the

inspiration

for

this project,

and I

benefited

from

his

advice throughout

the

writing process.

I am

also indebted

to the

students

who

have

suffered

through courses

taught

from

early version

of

this

text.

Many

of

them

found

errors, typographical

and

otherwise,

that

might otherwise have

found

their

way

into print.

I

would like

to

thank Professors Gino Biondini,

Yuji

Kodoma, Robert Krasny,

Yuan

Lou, Fadil Santosa,

and

Paul Uhlig,

all of

whom read

part

or all of the

text

and

offered

helpful

suggestions.

The

various physical parameters used

in the

examples

and

exercises were

de-

rived

(sometimes

by

interpolation)

from

tables

in the CRC

Handbook

of

Chemistry

and

Physics [35].

The

graphs

in

this

book were generated with MATLAB.

For

MATLAB product

information,

please

contact:

The

MathWorks, Inc.

3

Apple Hill Drive

Natick,

MA

01760-2098

USA

Tel: 508-647-7000

Fax: 508-647-7101

E-mail: info@mathworks.com

Web:

www.mathworks.com

As

mentioned above,

the CD

also supports

the use of

Mathematica

and

Maple.

For

Mathematica

product information,

contact:

Preface

XXI

Possible course outlines

In

a one-semester course (42-45 class hours), I typically cover Chapters

1-

7 and

part

of Chapter 8. I touch only lightly on the material concerning Green's functions and

the Dirac delta function (Sections 4.6, 6.6, and 7.4.1), and sometimes omit Section

6.3,

but

cover the remainder of Chapters 1-7 carefully.

If

an

instructor wishes

to

cover a significant

part

of the material in Chapters

8-10,

an

obvious place to save time

is

in Chapter

4.

I would suggest covering

the

needed material on ODEs on a "just-in-time" basis in the course of Chapters

5-7. This will definitely save time, since my presentation in Chapter 4

is

more

detailed

than

is

really necessary. Chapter 2 can be given as a reading assignment,

particularly for the intended audience of science and engineering students, who will

typically be comfortable with

the

physical parameters appearing in the differential

equations.

Acknowledgments

This book began when I was visiting Rice University in 1998-1999 and

taught

a

course using the lecture notes of Professor William W. Symes. To satisfy my per-

sonal predilections, I rewrote

the

notes significantly, and for

the

convenience of

myself and my students, I typeset them in

the

form of a book, which was the first

version of this text. Although the final result bears, in some ways, little resem-

blance

to

Symes's original notes, I

am

indebted

to

him for the idea of recasting the

undergraduate

PDE

course in more modern terms. His example was the inspiration

for this project, and I benefited from his advice throughout the writing process.

I am also indebted to the students who have suffered through courses

taught

from early version of this text. Many of them found errors, typographical and

otherwise,

that

might otherwise have found their way into print.

I would like

to

thank

Professors Gino Biondini, Yuji Kodoma, Robert Krasny,

Yuan Lou, Fadil Santosa, and Paul Uhlig, all of whom read

part

or all of the

text

and offered helpful suggestions.

The various physical parameters used in the examples and exercises were de-

rived (sometimes by interpolation) from tables in

the

CRC

Handbook

of

Chemistry

and Physics

[35).

The

graphs in this book were generated with MATLAB. For MATLAB product

information, please contact:

The

Math

Works, Inc.

3 Apple Hill Drive

Natick,

MA

01760-2098 USA

Tel: 508-647-7000

Fax: 508-647-7101

E-mail: info@mathworks.com

Web: www.mathworks.com

As

mentioned above, the CD also supports the use of Mathematica and Maple. For

Mathematica product information, contact:

xxii

Preface

Wolfram

Research, Inc.

100

Trade Center Drive

Champaign,

IL

61820-7237

USA

Tel: 800-965-3726

Fax: 217-398-0747

E-mail:

info@wolfram.com

For

Maple

product information, contact:

Waterloo Maple Inc.

57

Erb

Street

West

Waterloo, Ontario

Canada

N2L6C2

Tel: 800-267-6583

E-mail:

info@maplesoft.com

xxii

Wolfram Research, Inc.

100 Trade Center Drive

Champaign, IL 61820-7237 USA

Tel: 800-965-3726

Fax: 217-398-0747

E-mail: info@wolfram.com

For

Maple

product information, contact:

Waterloo Maple Inc.

57

Erb

Street

West Waterloo, Ontario

Canada

N2L6C2

Tel: 800-267-6583

E-mail: info@maplesoft.com

Preface

Loosely

speaking,

a

differential

equation

is an

equation

specifying

a

relation between

the

derivatives

of a

function

or

between

one or

more derivatives

and the

function

itself.

We

will

call

the

function

appearing

in

such

an

equation

the

unknown function.

We

use

this terminology because

the

typical

task

involving

a

differential

equation,

and the

focus

of

this book,

is to

solve

the

differential

equation,

that

is, to find a

function

whose derivatives

are

specified

by the

differential

equation.

In

carrying

out

this

task,

everything else about

the

relation, other

than

the

unknown

function,

is

regarded

as

known.

Any

function

satisfying

the

differential

equation

is

called

a

solution.

In

other words,

a

solution

of a

differential

equation

is a

function

that,

when substituted

for the

unknown

function,

causes

the

equation

to be

satisfied.

Differential

equations

fall

into several natural

and

widely used categories:

1.

Ordinary

versus

partial:

(a)

If the

unknown

function

has a

single independent variable,

say t,

then

the

equation

is an

ordinary

differential

equation (ODE).

In

this case, only

"ordinary" derivatives

are

involved. Examples

of

ODEs

are

Chapter

1

classification of

equations

In the

second example,

a, b, and c are

regarded

as

known constants,

and

f(t)

as a

known

function

of t. In

both equations,

the

unknown

is

u =

u(t).

(b)

If the

unknown function

has two or

more independent variables,

the

equation

is

called

a

partial

differential

equation (PDE). Examples include

differential

Chapter 1

ification

of

-al

equations

Loosely speaking, a differential equation

is

an

equation specifying a relation between

the derivatives of a function or between one or more derivatives and the function

itself.

We

will call the function appearing in such

an

equation the unknown Junction.

We

use this terminology because the typical

task

involving a differential equation,

and

the

focus of this book,

is

to

solve the differential equation,

that

is, to find a

function whose derivatives are related as specified by the differential equation.

In

carrying out this task, everything else

about

the relation, other

than

the unknown

function,

is

regarded as known. Any function satisfying the differential equation

is

called a solution. In other words, a solution of a differential equation

is

a function

that,

when substituted for

the

unknown function, causes

the

equation

to

be satisfied.

Differential equations fall into several natural and widely used categories:

1.

Ordinary

versus

partial:

(a)

If

the

unknown function has a single independent variable, say

t,

then

the

equation

is

an

ordinary differential equation (ODE).

In

this case, only

"ordinary" derivatives are involved. Examples of ODEs are

and

du =

3u

dt

rPu du

a

dt

2

+ b dt + cu = J (t).

(1.1)

(1.2)

In the second example,

a,

b,

and c are regarded as known constants,

and

J(t)

as a known function of

t.

In

both

equations, the unknown

is

u =

u(t).

(b)

If

the

unknown function has two or more independent variables, the

equation

is

called a partial differential equation (PDE). Examples include

{Pu 8

2

u

8x

2

+ 8y2 = 0

(1.3)

1

Chapter

1.

Classification

of

differential

equations

Examples (1.3)

and

(1.4)

are of

this

form

(although

the

independent

variables

are

called

x and t in

(1.4),

not x and y). Not all of an,

a

12

,

and a

22

can be

zero

in

order

for

this equation

to be

second order.

A

linear

differential

equation

is

homogeneous

if the

zero

function

is a

solution.

For

example, u(t)

= 0

satisfies (1.1),

u(x,y)

= 0

satisfies (1.3),

2

In

Section 3.1,

we

will give

a

more precise definition

of

linearity.

In

(1-3),

the

unknown

function

is u =

u(x,y),

while

in

(1.4),

it is u =

u(x,t).

In

(1.4),

c is a

known

constant.

2.

Order:

The

order

of a

differential

equation

is the

order

of the

highest deriva-

tive appearing

in the

equation. Most

differential

equations arising

in

science

and

engineering

are first or

second order. Example (1.1) above

is first

order,

while

Examples (1.2), (1.3),

and

(1.4)

are

second order.

3.

Linear versus nonlinear:

(a)

As the

examples suggest,

a

differential

equation has,

on

each side

of the

equals sign, algebraic expressions involving

the

unknown

function

and

its

derivatives,

and

possibly other

functions

and

constants regarded

as

known.

A

differential

equation

is

linear

if

those terms involving

the un-

known

function

contain only products

of a

single

factor

of the

unknown

function

or one of its

derivatives with other

(known)

constants

or

func-

tions

of the

independent variables.

2

In

linear

differential

equations,

the

unknown

function

and its

derivatives

do not

appear raised

to a

power

other

than

1, or as the

argument

of a

nonlinear

function

(like

sin, exp,

log,

etc.).

For

example,

the

general linear second-order

ODE has the

form

Here

we

require

that

a

2

(t)0

in

order

that

the

equation truly

be of

second order. Example (1.2)

is a

special case

of

this general

equation.

In

a

linear

differential

equation,

the

unknown

or its

derivatives

can be

multiplied

by

constants

or by

functions

of the

independent variable,

but

not by

functions

of the

unknown.

As

another example,

the

general linear second-order

PDE in two

inde-

pendent variables

is

2 Chapter

1.

Classification of differential equations

and

(1.4)

In

(1.3), the unknown function

is

u = u(x,

V),

while in (1.4),

it

is

u =

u(x, t). In (1.4), c

is

a known constant.

2.

Order:

The

order

of a differential equation

is

the order of the highest deriva-

tive appearing in

the

equation. Most differential equations arising in science

and

engineering are first or second order. Example (1.1) above

is

first order,

while Examples (1.2), (1.3),

and

(1.4) are second order.

3.

Linear

versus

nonlinear:

(a)

As

the

examples suggest, a differential equation has, on each side of

the

equals sign, algebraic expressions involving the unknown function and

its derivatives, and possibly other functions and constants regarded as

known. A differential equation

is

linear if those terms involving the un-

known function contain only products of a

single

factor of the unknown

function or one of its derivatives with other (known) constants or func-

tions of

the

independent variables.

2

In linear differential equations, the

unknown function and its derivatives do

not appear raised to a power

other

than

1,

or as the argument of a nonlinear function (like sin, exp,

log, etc.).

For example, the general linear second-order ODE has

the

form

Here

we

require

that

a2(t)

f:.

0 in order

that

the equation truly be of

second order. Example

(1.2)

is

a special case of this general equation.

In

a linear differential equation,

the

unknown or its derivatives can be

multiplied by constants or by functions of

the

independent variable,

but

not by functions of

the

unknown.

As another example,

the

general linear second-order

PDE

in two inde-

pendent variables

is

a

2

u a

2

u a

2

u

au

(x,

y)

ax

2

+a12(x,

y)

axay

+a22

(x,

y)

ay2

au au

+ al (x,

y)

ax

+a2(x,

y)

ay + ao(x, y)u =

f(x,

V)·

Examples (1.3) and (1.4) are of this form (although the independent

variables are called

x and t in (1.4), not x and

V).

Not all of

au,

a12,

and

a22

can be zero in order for this equation to be second order.

A linear differential equation

is

homogeneous

if

the zero function

is

a

solution. For example,

u(t)

==

0 satisfies (1.1),

u(x,y)

==

0 satisfies (1.3),

-::------

2In Section 3.1, we will give a

more

precise definition of linearity.

Chapter

1.

Classification

of

differential equations

and

u(x,t)

=

0

satisfies (1.4),

so

these

are

examples

of

homogeneous

linear

differential

equations.

A

homogeneous linear equation

has

another

important property: whenever

u

and v are

solutions

of the

equation,

so

is

cm + ftv for all

real numbers

a and ft. For

example, suppose

u(t)

and

v(t)

are

solutions

of

(1.1)

(that

is,

du/dt

—

3w

and

dv/dt

=

3i>),

and

w

=

au

+ ftv.

Then

Thus

w is

also

a

solution

of

(1.1).

A

linear

differential

equation which

is not

homogeneous

is

called

inho-

mogeneous.

For

example

is

linear

inhomogeneous,

since

u(i)

=

0

does

not

satisfy

this equation.

Example

(1.2) might

be

homogeneous

or

not, depending

on

whether

/(*)

=

0 or

not.

It is

always possible

to

group

all

terms involving

the

unknown

function

in

a

linear

differential

equation

on the

left-hand side

and all

terms

not

involving

the

unknown

function

on the

right-hand side.

For

example,

(1.5)

is

equivalent

to

"Equivalent

to" in

this context means

that

(1.5)

and

(1.6) have exactly

the

same solutions:

if

u(t)

solves

one of

these

two

equations,

it

solves

the

other

as

well. When

the

equation

is

written this way, with

all of

the

terms

involving

the

unknown

on the

left

and

those

not

involving

the

unknown

on the

right, homogeneity

is

easy

to

recognize:

the

equation

is

homogeneous

if and

only

if the

right-hand side

is

identically zero.

(b)

Differential

equations which

are not

linear

are

termed nonlinear.

For

example,

is

nonlinear. This

is

clear,

as the

unknown

function

u

appears raised

to

the

second power. Another

way to see

that

(1.7)

is

nonlinear

is as

follows:

if

the

equation were linear,

it

would

be

linear

homogeneous,

since

the

zero

3

Chapter

1.

Classification of differential equations

3

and

u(x,

t)

==

0 satisfies (1.4), so these are examples of homogeneous

linear differential equations. A homogeneous linear equation has another

important

property: whenever u

and

v are solutions of

the

equation, so

is

au

+

(3v

for all real numbers a and

(3.

For example, suppose u(t)

and

vet) are solutions of (1.1)

(that

is, du/dt = 3u

and

dv/dt

= 3v),

and

w =

au

+

(3v.

Then

dw d

ill

= dt

[au

+

(3v]

du

dv

=a

dt

+(3dt

= a3u +

(33v

= 3(au +

(3v)

=3w.

Thus w is also a solution of (1.1).

A linear differential equation which

is

not homogeneous

is

called inho-

mogeneous.

For example

~~

= 3u + sin

(21ft)

(1.5)

is

linear inhomogeneous, since u(t)

==

0 does

not

satisfy this equation.

Example

(1.2) might be homogeneous or not, depending on whether

J(t)

==

0 or not.

It

is

always possible

to

group all terms involving

the

unknown function

in a linear differential equation on the left-hand side

and

all terms

not

involving

the

unknown function on the right-hand side. For example,

(1.5)

is

equivalent

to

~~

- 3u = sin

(21ft).

(1.6)

"Equivalent to" in this context means

that

(1.5)

and

(1.6) have exactly

the

same solutions: if u(t) solves one of these two equations,

it

solves

the

other

as well. When

the

equation

is

written this way, with all of

the

terms involving

the

unknown on

the

left and those

not

involving

the

unknown on

the

right, homogeneity is easy

to

recognize:

the

equation

is

homogeneous if

and

only if

the

right-hand side

is

identically zero.

(b) Differential equations which are

not

linear are termed nonlinear. For

example,

(1.7)

is nonlinear. This is clear, as

the

unknown function u appears raised

to

the

second power. Another way

to

see

that

(1.7) is nonlinear

is

as follows:

if

the

equation were linear,

it

would be linear homogeneous, since

the

zero

Chapter

1.

Classification

of

differential equations

Thus

w

does

not

satisfy

the

equation,

so the

equation must

be

nonlinear.

Both linear

and

nonlinear

differential

equations occur

as

models

of

physi-

cal

phenomena

of

great importance

in

science

and

engineering. However,

linear

differential

equations

are

much better understood,

and

most

of

what

is

known about nonlinear

differential

equations depends

on the

analysis

of

linear differential equations.

This

book will

focus

almost

ex-

clusively

on the

solution

of

linear

differential

equations.

4.

Constant

versus

nonconstant

coefficient: Linear

differential

equations

like

function

is a

solution. However, suppose

that

u(t)

and

v(t}

are

nonzero

solutions,

so

that

4

have constant

coefficients

if the

coefficients

ra, c, and k are

constants

rather

than

(nonconstant)

functions

of the

independent variable.

The

right-hand

side

f(t)

may

depend

on the

independent variable;

it is

only

the

quantities which

multiply

the

unknown

function

and its

derivatives which must

be

constant,

in

order

for the

equation

to

have constant

coefficients.

Some techniques

that

are

effective

for

constant-coefficient problems

are

very

difficult

to

apply

to

problems with nonconstant

coefficients.

As

a

convention,

we

will

explicitly write

the

independent variable when

a

coef-

ficient

is

a

function

rather than

a

constant.

The

only

function

in a

differential

equation

for

which

we

will

omit

the

independent variable

is the

unknown

function.

Thus

is

an ODE

with

a

nonconstant

coefficient,

namely k(x).

We

will only apply

the

phrase

"constant

coefficients"

to

linear differential

equations.

4 Chapter

1.

Classification of differential equations

function is a solution. However, suppose

that

u(t) and v(t) are nonzero

solutions,

so

that

rPu 2 rPv 2

dt

2

+ u = 0,

dt

2

+ V = 0,

and

w = u +

v.

Then

rPw 2

rP

2

dt

2

+ w =

dt

2

[u

+

v]

+ (u +

v)

rPu d

2

v 2 2

=

dt

2

+

dt

2

+ u +

2uv

+ v

(

rPu

2)

(d

2

v

2)

=

dt

2

+ u +

dt

2

+ V +

2uv

=

2uv.

Thus w does not satisfy the equation, so

the

equation must be nonlinear.

Both

linear

and

nonlinear differential equations occur as models of physi-

cal phenomena of great importance in science and engineering. However,

linear differential equations are much

better

understood, and most of

what

is

known about nonlinear differential equations depends on the

analysis of linear differential equations. This book will focus almost ex-

clusively on the solution of linear differential equations.

4.

Constant

versus

nonconstant

coefficient:

Linear differential equations

like

have

constant

coefficients

if

the

coefficients m,

c,

and k are constants

rather

than

(nonconstant) functions of

the

independent variable. The right-hand side

f(t)

may depend on

the

independent variable; it

is

only

the

quantities which

multiply the unknown function

and

its derivatives which must be constant,

in order for

the

equation

to

have constant coefficients. Some techniques

that

are effective for constant-coefficient problems are very difficult

to

apply

to

problems with nonconstant coefficients.

As

a convention,

we

will explicitly write the independent variable when a coef-

ficient

is

a function rather

than

a constant. The only function in a differential

equation for which

we

will omit

the

independent variable

is

the unknown

function. Thus

d [

dU]

--

k(x)-

=

f(x)

dx

is

an

ODE with a nonconstant coefficient, namely

k(x).

We

will only apply the phrase "constant coefficients"

to

linear differential

equations.

Chapter

1.

Classification

of

differential equations

It is

important

to

realize

that,

since

a

differential

equation

is an

equation

involving

functions,

a

solution

will

satisfy

the

equation

for all

values

of the

inde-

pendent

variable(s),

or at

least

all

values

in

some restricted domain.

The

equation

is

to be

interpreted

as an

identity, such

as the

familiar trigonometric identity

5

5.

Scalar

equation

versus

system

of

equations:

A

single

differential

equa-

tion

in one

unknown

function

will

be

referred

to as a

scalar equation (all

of the

examples

we

have seen

to

this point have been scalar

equations).

A

system

of

differential

equations consists

of

several equations

for one or

more unknown

functions.

Here

is a

system

of

three

first-order

linear, constant-coefficient

ODEs

for

three unknown

functions

xi(t),X2(t),

and

xs(t):

To

say

that

(1.8)

is an

identity

is to say

that

it is

satisfied

for all

values

of the

independent variable

t.

Similarly,

u(t)

—

e

3t

is a

solution

of

(1.1) because

this

function

u

satisfies

for

all

values

of t.

The

careful

reader

will

note

the

close analogy between

the

uses

of

many terms

introduced

in

this section ("unknown

function,"

"solution," "linear"

vs.

"nonlin-

ear"

) and the

uses

of

similar terms

in

discussing algebraic equations.

The

analogy

between

linear

differential

equations

and

linear algebraic systems

is an

important

theme

in

this book.

Exercises

1.

Classify

each

of the

following

differential

equations according

to the

Gockenbach M.S. Partial Differential Equations. Analytical and Numerical Methods

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