Abel Integral Equations: Analysis and Applications

Lecture Notes in Mathematics Edited by A, Dold, B. Eckmann and E Takens

1461 Rudolf Gorenflo Sergio Vessella

Abel Integral Equations Analysis and Applications

Springer-Verlag Berlin Heidelberg New York London Paris Tokyo Hong Kong Barcelona

Authors

Rudolf Gorenflo Fachbereich Mathematik Freie Universit&t Berlin Arnimallee 2 - 6 1000 Berlin 33, Federal Republic of Germany Sergio Vessella Facolt& di Ingegneria Universit& di Salerno 84100 Salerno, Italy

Mathematics Subject Classification (1980): 45E10, 45D05, 44A15, 65R20 ISBN 3-540-53668-X Springer-Verlag Berlin Heidelberg NewYork ISBN 0-387-53668-X Springer-Verlag NewYork Berlin Heidelberg

This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. Duplication of this publication or parts thereof is only permitted under the provisions of the German Copyright Law of September 9, 1965, in its current version, and a copyright fee must always be paid. Violations falI under the prosecution act of the German Copyright Law. © Springer-Verlag Berlin Heidelberg 1991 Printed in Germany Printing and binding: Druckhaus Beltz, Hemsbach/Bergstr. 2146/3140-543210 - Printed on acid-free paper

Preface Abel's integral equation, one of the very first integral equations seriously studied, and the corresponding integral operator (investigated by Niels Henrik Abel in 1823 and by Liouville in 1832 as a fractional power of the operator of anti-derivation) have n e v e r ceased to inspire mathematicians to investigate and to generalize them. Abel was led to his equation by a problem of mechanics, the tautochrone problem. However, his equation and slight or not so slight variants of it have in the meantime found applications in such diverse fields (let us mention a few from outside of mathematics, arisen in our century) as inversion of seismic travel times, stereology of spherical particles, spectroscopy of gas discharges (more generally: " tomography" of cylindrically or spherically symmetric objects like e.g. globular clusters of stars), and determination of the refractive index of optical fibres. More pertinent to mathematics think of particular (inverse) problems in partial differential equations (e.g. heat conduction, Tricomi's equation, potential theory, theory of elasticity - we recommend here the books of Bitsadze and of Sneddon) and of special problems in the theory of Brownian motion. Of course, these variants of Abel's original equation comprise linear and nonlinear equations, equations of first and of second kind, systems of equations, and the widest generalizations consist in simply retaining in the kernel of the integral equation its integrability in the sense that this kernel is "weakly singular". There are several good books on fractional integration and differentiation (investigating the Abel operator and its inverse) on different theoretical levels and on Volterra integral equations of which Abel equations are a particular type. Let us just cite those of Oldham and Spanier, McBride, Linz, Nishimoto, and the monograph of Samko, Kilbas and Marichev. In addition, there is an ever-growing literature of research results, on applications and on numerical methods. See, e.g., the book of Craig and Brown on Inverse Problems in Astronomy that devotes many sections to applications of Abel integral equations and their numerical treatment, and the recent conference report edited by Nishimoto. We neither try to be exhaustive nor to give a balanced presentation of the various questions involved. We also do not compete with the quoted books, but rather strive for contrast. So, what are our intentions ? And what types of readers do we have in mind ? We want to stimulate the flow of information between at least three sorts of people. (i) Theoretical mathematicians also interested in applications and in application-relevant questions. (ii) Mathematicians working in applications and in numerical analysis. (iii) Scientists and engineers working outside of mathematics but applying mathematical methods for modelling and evaluation. In the past there often has been an astonishing lack of this flow of information. This lack becomes conspicuous when one studies basic papers on stereology written independently from each other by biologists, chemists, metallurgists, physicists, geologists, and by

IV

authors unaware of research publications of other disciplines whose methods they could have used instead of re-inventing them. We treat the elementary theory and describe in detail many applications in the first part, and present the harder topics, such as ill-posedness and the behaviour of Abet integral operators in various function spaces, on a higher level of theory later (thereby trying to exhibit the relevance of the results, in particular of the stability estimates, for the applications). In two aspects we have deliberately limited our scope. One is the theory of generalized Abel equations (they are particularly well treated in the book of Meister), the other one concerns discretization methods for numerical treatment. For the latter we recommend the recent comprehensive monograph of Brunner and van der Houwen. We have restricted ourselves to a survey on discretization schemes, stressing, however, the point (in the literature too rarely given due observance) that, in numerical evaluation of error-contaminated measurements, because of ill-posedness it is often better to use a crude low-accuracy method, thereby taking account of available extra information on the shape of the solution. What are the prerequisites to read the book ? The first three chapters should be accessible to every student of mathematics, physics or engineering after two years of study at university, and also to mathematically inclined students of other natural sciences. For the rest we suppose some familiarity with basic functional analysis (like theory of integration, /2 and Sobolev spaces, linear operators, and Fourier transforms). How is the book organized ? Subdivision is into Chapters, paragraphs and sections. Thus, by 6.4.3 we mean the third section of the fourth paragraph of Chapter 6. The theorems, lemmas and formulas are numbered within each paragraph (ignoring subdivision of paragraphs into sections that, by the way, not always is made). Theorem 6.4.3 is the third theorem of the fourth paragraph of Chapter 6, and (6.4.3) is the third enumerated formula of the same paragraph. Some topics that we did not want to treat comprehensively have been delegated to Appendices to Chapters marked with capital letters, an appendix standing on the same structural level as a paragraph. References to the literature are made in a self-explanatory way by giving a shortened form of the names of authors and the year of appearance. We are indebted to the Italian Centro Nazionale delle Ricerche and to the Freie Universit/~t Berlin for making possible several mutual visits of the authors for work on this book, and we highly appreciate the readiness of many individuals for discussions, correspondence and for critically reading parts of the manuscript. In particular we mention Prof. Carlo Pucci who also offered us excellent working conditions in the Istituto di Analisi Globale e Applicazioni of CNR in Florence, Italy, Prof. Robert S. Anderssen, Prof.

Dang Dinh Ang, Prof. Gottfried Anger, Prof. Mario Bertero, Prof. S. Campi, Dr. Paul Eggermont, Prof. Marco Longinetti, Dr. Rolando Magnanini, Prof. Erhard Meister and Prof. Giorgio Talenti. For the tedious work of typing the manuscripts our thanks are due to Mrs. Silvia Heider-Kruse and Mrs. Monika Schmidt, for typing preliminary versions to Mrs. Angelika Hinzmann and Mrs. Ursula Schulze. And for the important work of carefully proof reading the whole manuscript, checking many of the calculations, and drawing figures we are indebted to cand.math. Vera Lenz~ Dipl.-Math. Andreas Pfeiffer and cand. math. Uwe Schrader.

Rudolf Gorentio and Sergio Vessella

Introduction Chapter 1 1.1 1.2 1.A 1.B

Basic Theory and Representation Formulas The Abel Integral Operator Solution Formulas Appendix: Existence and Uniqueness in L t Appendix: List of Solution Formulas

Chapter 2

Applications of Abel's Original Integral Equation: Determination of Potentials General Considerations Abel's Mechanical Probiem Throwing a Stone The Oscillating Pendulum Tile Inverse Scattering Problem for a Repelling Potential

2.1 2.2 2.3 2.4 2.5

Chapter 3 3.1 3.1.1 3.1.2 3.2 3.2.1 3.2.2 3.2.3 3.2.4 3.2.5 3.3 3.3.1 3.3.2 3.4 3.A

Chapter 4 4.1 4.2 4.2.1 4.2.2 4.3 4.A Chapter 5 5.1 5.2

8 8 11 17 22 26 26 27 29 30 31

Applications of a Transformed Abel Integral Equation Spectroscopy of Cylindrical Gas Discharges Modelling by an Abel Integral Equation Complications Arising in Practice Stereology of Spherical Particles The Problem Formal Solutions Is the Formal Solution Correct ? Calculation of Moments Final Remarks Inversion of Seismic Travel Times General Considerations The Flat Earth Model Refractive Index of Optical Fibres Appendix: Linear Generalized Abel Integral Equations

35 35 35 37 39 39 41 43 46 48 50 50 51 56 61

Smoothing Properties of the Abel Operators Continuity Properties of the Abel Operator in LP Spaces Continuity Properties of the Abel Operator in Some Spaces of Fractional Order HSlder Continuous Spaces Sobolev Fractional Spaces Compactness of Abel Operators Appendix: Proof of a Lemma

64 64

Existence and Uniqueness Theorems The Linear Case A Nonlinear Abel Integral Equation

83 83 91

69 69 72 78 81

vii Chapter 6 6.1 6.2 6.3 6.4 6.5 6.6 6.A 6.B

Relations between the Abel Transform and Other Integral Transforms Relations of Abel Operators with Abel Operators A Brief Account on Generalizations of Abel Operators Relations between Abel Operators and tile Fourier Transform Relations between the Abel Operator and the Mellin Transform Some Relations between Abel Operators and Hanke[ Transforms Some Relations between the Abel Operator and the Plane Radon Transform Appendix: Generalized Abel Equations: Survey of Literature Appendix: A modified Abel Transform

95 95 98 100 107 113 115 121 123

Chapter 7 7.1 7.2 7.3 7.4

Nonlinear Abel Integral Equations of Second Kind Introductory Remarks Linear Abel Integral Equations of Second Kind Analysis-Motivated Investigations Applications-Motivated Investigations: Problem Formulations, Newton's Law of Cooling Applications-Motivated Investigations: Survey of Literature A Very Brief Survey of Literature on Numerical Methods

129 129 129 132

7.5 7.6

139 146 152

8.1 8.2 8.2.1 8.2.2 8.2.3 8.3 8.3.1 8.3.2 8.3.3 8.3.4

Illposedness and Stabilization of Linear Abel hltegral Equations of First Kind General Topics in Ill-Posed Problems Preliminary Discussion of the Stability of Abel's Equation Mechanical Problem Inversion of Seismic Travel-Times Other Examples and Instability Properties Stability Estimates for Solutions of Abel-type Integral Equations Auxiliary Lemmas /]'-bounded First Derivative of the Solution LP-bounded Second Derivative of the Solution Discrete Data

154 154 158 158 164 165 168 168 171 176 179

Chapter 9 9.1 9.2 9.3 9.4

On Numerical Treatment of First Kind Abel Integral Equations General Considerat ions Quadrature Methods Evaluation of Measurements A Numerical Case Study

182 182 184 185 191

Chapter 8

References

195

Subject Index

210

Introduction

In 1823 N.Ho In the v e r t i c a l

Abel

considered

(x,y)-plane

the graph of an i n c r e a s i n g under

constant downward

to fall, t(y)

in order

that

of the initial

the f o l l o w i n g

(see Eig. function

acceleration

height

of mechanics:

1) find a curve C that is

x = ~(y),

its falling

problem

g

y 6 [o,H],

a particle

time

equals

along w h i c h

must be c o n s t r a i n e d a prescribed

function

y.

4 Jl / / I j

y--

B

--

f i

/

i

/ / /

/

/

/

I f

X

0

Fig. I In absence

of friction

the p r o b l e m

is reduced

to that of solving

the e q u a t i o n (I)

~

(y - z) -I/2 u(z)dz

= 2/~ t(y) ,

y 6 [o,H]

o

where

u(z) Abel

treated

= /I + ~ solved

a more

'2

(z)

this e q u a t i o n

general

equation,

in

lAb,

1823]

replacing

and in

[Ab,

1826].Infact,he

(y - z) -I/2 For

these

reasons

by

an e q u a t i o n

(y - z) ~-I

of the

form

x

(2)

jS u

:=

I

f

(~)

0

F

where

F is the E u l e r ~quation

(for a p.

the end

(2)

history

91 - 96],

is one

1977],

19th

~-I

(x - t)

function,

u(t)dt

first

equations

[Wo,

century)

= f(x) ,

is c a l l e d

of the

of i n t e g r a l

[La,

of the

gamma

O < ~ < I

an A b e l

integral

In h o n o u r

equations

of type

integral

equations

the r e a d e r

1965]).

O < ~ < I,

may

equation.

ever

consult

treated

[Di,

of V. V o l t e r r a

1981,

(towards

(2) or of the m o r e

general

type (3)

(Asu) (x)

:=~K(x,t)• o

are o f t e n

We r e f e r and,

"singular

called

more

of type equations

(x - t) 1-s or

Abel

integral

interest

essentially

(C)

of A b e l

other

Properties functions

(D) (E)

that

Such


0

Fig.

2.2.1

Shape

of a hill

x

28

Parametrizing as arclength, strictly exist

wards

= y(O)

increasing,

and

again

So we have

to look

the d i f f e r e n t i a l m A"(t) governing

=

the

s(O)

altitude

is finite)

by

x=x(s),

y=y(s)

see t h a t on t h e hill, inverse

with

V = V(s)

function

s

is

s = s(V)

does

= O.

the p a r t i c l e

a maximal

(if T ( E ) / 2

the h i l l

= O. T h u s

increasing,

we expect

at t = T(E)/2

of

= O, w e

V(O)

is s t r i c t l y

Intuitively ches

the p r o f i l e

x(O)

to g l i d e

y -- y ( E ) ,

returning

for an e x p r e s s i o n

upwards, and

then

until

to the o r i g i n

giving

T(E)

it r e a -

to g l i d e

down-

a t t = T(E).

from V(s).

From

equation dV(~)/d~

--

the m o v e m e n t

time

t, w e o b t a i n

tion

the

, ~(O)

A s'(O)

:

Vo>O

,

of the p a r t i c l e ,

s = A(t) b e i n g its p o s i t i o n a t A w i t h s' (t) a n d s u b s e q u e n t i n t e g r a -

by multiplication

constancy of energy,

(2.2.1)

O,

:

(A, (t))2

namely m 2 = E = ~ vo

+ V(A(t))

,

hence (2.2.2)

t =

(m/2) I/2 ~

(E-V(~)) - I / 2

do

O a s l o n g a s V(s)

! E

.

If V(s) < E for a l l flatter

and

flatter

the p a r t i c l e Remember V = V(s).

the h i l l

is of f i n i t e

upwards,

and

remains

moving

that we have

If s(E)

(2.2.3)

O < s O, w(u)

= I

-

V(u)/E,

is d e c r e a s i n g ,

increasing,

V(u)

w

-

,

increasing,

O <w ~Xma x _

I 2 b

to b > b

u 2 v -I

=

v

-1/2

du

g(w)

=

d--w

v(u)

decreasing

(if w e r e s t r i c t

"

but always

measurements

min

. > O). mln

using

(2.5.10)

we

immediately

arrive

at an Abel

tion

(2.5.11)

x f g(w) d w = B(x), w = O (x-w) I/2

O<x_<Xma

x

,

integral

equa-

34

where us

the

that

square

u

From

root

(2.5.12)

we

obtain,

to c a l c u l a t e

this

denominator

to

1 d - ~ dw

g(w)

g(w)

From

the

= u o corresponds (2.5.10)

How

in

w

y ~ x=O

v

I -I = ~ v

w

g(w)?

I/2

Here

we

have

V(O)

= V(~) What

dence

of

used =O

the

facts

implies

(2.5.10)

we

obtained

V

r,

namely

on

V

(2.5.14)

= E

=

obtain

the

(2 g (w)

u

= Vvw,

dependence

whence

u

= 0

of

-i)

w

- w

implies

w

v on

w:

. dw

= 0 and

that

I.

? A parametric (see

gives

-i/2

~ O

that

v(O)

have

= w gives

dv I -I/2 d--w + 2 w

relation, we

= exp

x

B (x) dx I/2 (w-x)

from

= v(w)

for

(2.b) ,

w (2.5.13)

zero

.

using

V(r)

differential

= x

being

again

representation

(2.5.9),

of

the

(2.5.10)

and

use

repelling

field

depenr =

I/u)

(1-v(w)) , r : - -

~w ~ (w) I

for

O < w < Xma x

=

2 b

Remark:

It

min

should

be

noted

strong near r=O, then even ~ w h e n whole

range

distance r < r*, be

r* > O but

then

the We

take

shall

of

identifying

parameter many

b

to

the

the

if v e r y

repelling

if

the

the

a fixed

energy

method

. The

value

of

impact

gives

reason

information

is

for the

parameter

E > O, t h e r e no

still energy

may

information

that

r*

in

in

the

be

a minimal

on

V(r)

for

= inf{rolb

~O}

may

smaller E

is v e r y

b

values

order

to

of come

r,one nearer

field.

discuss

the

the

r ~ r*

a greater

not

refer

for

that

obtain

of

We

to ~

for

To

source

O

such

only

positive.

must to

from

that

varying

in

detail

specialized

dependence many

centers.

of

particles

these physical the

and

related

literature

deflection

are

angle

in a p a r a l l e l

problems. for

the

@ on beam

problem

the shot

impact against

C h a p t e r 3: A p p l i c a t i o n s of a T r a n s f o r m e d Abel integral E q u a t i o n

We c o n s i d e r here several problems leading to Abel's

integral equa-

tion in the form b S x

(3.a)

u(t) dt ~t2_x 2

- f(x)

Here 0 < x < t < b < ~

. Appropriate

(3.b)

u(t)

=

(3 .c)

u(t)

= ~

2

solution formulas are

d ~ x f (x)dx dt t 2 ~ _ t 2

{

f(b)

,

~ f' (x)dx

}

t xI

i~ Fig.

H~rmann, integral ments,

see

ma p h y s i c s of t h e s e

performance. It is of

recent

many

interest

investigating

papers

to n o t e

radially

see

to h a v e

through have

various

reviews

tube

been

been

exactly

symmetric

of m e a s u r e m e n t

section

the

first of

the g r o w i n g devoted

numerical

[Go,1979]

that

cross

the c o n f i g u r a t i o n

decades

research

with For

seems

for m o d e l l i n g

in

deal

Discharge

[H~,1935],

equation

and

3.1.1

line

"side-on"

to this

(computer) ,1982]

the

same m o d e l l i n g sky,

and

e.g.

Abel's measureof p l a s -

topic ; m o s t

procedures

[Br

in the

to use

importance

and

objects

x

and

~hapter is

their

9.

s u i t a b l e for

globular

clu-

sters of stars w i t h r e s p e c t to t h e i r s m e a r e d - o u t d e n s i t y ( s e e [ B r a , 1 9 5 6 ] ) and [ C r a - B r , 1 9 8 6 ] ) . R e t u r n i n g to our p r o b l e m of f i n d i n g the i n t e n s i t y g(r) for O < r < R, we

imagine

(see F i g u r e

a detector 3.1.1)

moving

which

outside

measures

the

the p l a s m a

parallel

to the x - a x i s

integral

9(x) (3.1.1)

of lar

the

G(x)

intensity

to the a x i s

=

J _ g(r)dy, y = - y (x)

g along

parallels

of the c y l i n d e r .

O f = -~

O<x O

depend

continuously

Zma x is the m a x i m a l

the c o m p l i c a t i o n s

geophysical

ray does

increasing

x=X x

~

z=zp

Fig.

lowest

point

z = z

3.3.q

A ray trajectory

from which

on it t u r n s

upwards

again

reaching

the sur-

P f a c e at a d i s t a n c e tion

from x=O

cial

explosion)

face

points

Knowing

x=X

(this

from

moment

the m o m e n t s

its

source.

is p r e c i s e l y at w h i c h

Knowing known

rays

the m o m e n t

in the c a s e

come

up a g a i n

A

v(z)

this

of course,

also

problem, we

first

for O < z < Zma x w i t h

rive~ a relation b e t w e e n

rays.

sur-

A

where,

To a n a l y z e

gral

of an a r t i f i -

at v a r i o u s

can be m e a s u r e d , thus y i e l d i n g a t r a v e l time T = T(X). ^ this f u n c t i o n T(X) for O < X <X, one w a n t s to d e t e r m i n e v=v(z)

for 0 < z < z,

velocity

of its e m a n a -

equation.

We must

take

of

v(z)

z is n o t k n o w n

as g r a n t e d

the p r o p e r t i e s

the f u n c t i o n s investigate

the v a l u e

and

the e x i s t e n c e

described

T(X),

the p r o p e r t i e s

a priori.

and

in fact of the

then

an A b e l

of a deinte-

trajectories

of

53

Consider x = x(t), T being

a definite

z = z(t), the

O < t O a n d V(Zp)

exist Z=Zp).

parameter

p

= O and,

dt

by

i(x,z)

(3.3.2),sin

i(x,z)=pv(z).

,

_

I

/ 1 _ ( p v ( z ) ) 2 ' dz

v(z)/1-(pv(z))

finiteness

values

= f P O

purpose

of t h e r a y w i t h

dz ~-~ : v ( z ) c o s

of b o t h

Z

(3.3.5)

t

equations

i(x,z),

x(O)

values

z = z(t)

as l o n g a s O < z < zp -

d_xx =

We h a v e

taken

finite

of d i f f e r e n t i a l

condition

that,

p.

such

dx d--[ = v ( z ) s i n

(3.3.4)

with

now

to o u r v a l u e

Z

P pv(z) dz

, t

/I- (pv(Z)) 2

we use = I/p,

= P

the a s s u m p t i o n and we

find

P O

dz V(Z) /I- (pv(z))

(3.3.1).

In p a r t i c u l a r

we h a v e

55

1-(pv(z)) 2 = So both

(1-pv(z)) (1+pv(z))

improper

For

further

(3.3.6) Note

and

(pv' (Zp) ( Z - Z p ) + O ( Z - Z p ) )

convergent,

x

and

P

is c o n v e n i e n t

t

P

(1+pv(z)) .

are

finite.

to substitute

= I/v .

u = u(z) that

are

treatment,it

u

that

tive,

integrals

:

decreases

u' (z)

from

v' (z)

= -

u ° : I/v o downwards,

2 < 0 for

all

z ~O

remaining

posi-

•

(v(z)) With

this

Theorem

new

variable

we can

If there

3.3.1

is

as

result

a v a l u e Zp f o r which U ( Z p )

then the ray with parameter travel

state

the surface a g a i n a t

preach~

(where O < p < u o)

= p x

= X

Zp X = 2 S

(3.3.7)

z p dz

0

We

thus

functions It m a y

have

need

happen

X or T first However,

we

X = X(p),

not that

as

can

p begins

as

See,

to d e c r e a s e

the

= T(p)

so-called

dz

/(u(z) ) 2_p 2

functions

e.g.,

the

of

p,

however,

discussion

from

b e g i n s to d e c r e a s e

introduce

T(p)

(u(z) 0

T = T(p)

be monotonic.

P

rays

2_p2

delay-time

function

dz

O

is m o n o t o n i c .

Indeed,

assuming

sufficient

smoothness,

Z

P ~' (p)

: - 2 f

P

O this

and

from

(3.3.7)

T' (p)

hence

locally

we

dz

/(u(z) ) 2_p 2 find,

formally,

dT dx = up~ - p ~

- X(p)

dT dX = d-p - p u ~p

+ T'(p)

,

these

[Ga,1971].

penetrating

- p X(p)

/(u(z))

in

I/v o = u o downwards,

for

Z

= 2 f

From

finite

)2

, T = 2

/(u(z) ) 2_p 2

increase@then

(3.3.8)

which

with

t i m e T, and

= - X (p) .

deeper.

56

dT p = ~-~ ,

(3.3.9)

SO t h a t p c a n be d e t e r m i n e d Introducing using

the

now

integral

as

the

the function

= 2 Pf

of the

z = z(u)

representation

T(p)

slope

in

inverse

(3.3.7)

/u2-p 2

dz(u) du

u z(u)

du

travel-time

curve.

to u = u(z)

and

we get

du and,

integrating

by parts,

Uo U

O

(3.3.10)

T(p)

The

inverse

problem

of m e a s u r e m e n t s (this c a n n o t tions

X(p),

= 2 S P

/u2_p 2 is the

(X,T), w e h a v e

be measured T(p),

hence

If w e n o w h a v e

following.

directly) by

T(p)

Given

sufficiently

to d e t e r m i n e for e a c h p a i r from

(3.3.7)

(3.3.8).

This

many

pairs

its p a r a m e t e r

yields

p

the func-

T(p) .

for u ° ~ p ~p*, w e c a n

solve

the A b e l

integral

equation U O

S P for

z = z(u),

u z(u)

= ~I T(p)

du

/u2_p 2

where

u O ~ u ~ p*,

and obtain

u(z)

as function

inverse

to

z = z(u) . Noting

now

t h a t a t the d e e p e s t

point

z = z

of a ray w e h a v e

i=~/2,

P hence

by

(3.3.2)

u = ~ = p,

we

see,

using

the m o n o t o n i c i t y

of

the

V

functions v(z)

u = u(z)

= I/u(z)

(3.3.11) 3.4.

U(Z*)

Refractive

In r e c e n t and more will

and

certainly

the c e n t r a l

of O p t i c a l

years,optical as a means

continue.

are needed

the r e f r a c t i v e

that we can

u(z)

and hence

Fibres

fibres

the

produced

by deposition

rotating

silica

tube).

(or g l a s s

of t r a n s m i s s i o n

In t h e m a n u f a c t u r e for m e a s u r i n g

i n d e x n as f u n c t i o n

axis,

obtain

where

= p*

Index

important

tive methods

z = z(u),

for O < z < z*

fibre being

their

of t h e assumed

of s u c c e s s i v e

of

fibres) of

such fibres

optical

radius

r

become and

trend

non-destruc-

quantities,

of d o p e d

more

this

(the d i s t a n c e

as a l o n g

layers

have

signals,

circular

i.e. from

cylinder

silica within

a

57 We describe in some detail a method to determine the refractive index under the assumption that this index in the fibre

(compare Marcuse

less restrictive assumptions) discussed by Shibata

only slightly

varies

(1979)). A more complicated method

with-

(under

gives more information and is treated and

et alii

(1979),see also Anderssen and Calligaro

(1981). This "Japanese method" uses the photoelastic effect by illuminating by laser light the fibre which by elastic stress is optically anisotropic.

The phase difference

(retardation)

splits of the laser ray is measured, equations for the refractive than in Marcuse's method,

index. The restrictions are much less severe

the mathematics

even discontir~ousrefraction functions,

of the two orthogonal

and again one arrives at Abel type is much more tricky ; however

indices can be determined

which are often relevant in practical

(for example step

situations).

However, we refer the reader to the quoted papers for this intricate method,and are now going to describe Marcuse's simple model. He puts the fibre into a homogeneous liquid whose refraction index n

matches that of the fibre surface so that there is no jump of the c index n at the fibre boundary. He assumes that within the fibre,the refraction index n(r) varies

only slightly

(depending on r) and as already said n c = n(a). Thus a

ray is bent only by a small angle, after leaving the fibre

~cross-section

of fibre

t

i

~x

,,

,,

" .....

;

~

>

x

Di

>

/ a

/

I

\

n=nc

/ n(r) unknown

\ incident parallel light with constant intensity Fig. 3.4.1

L i E= plane of observation

58

the

ray

is a g a i n

to t h e ing

at height

fact

that

y = O and of

the

of

t.

straight.

direction

on y

There

e m i s s i o n of ^ h e i g h t y(t)

t,its

of

(see

course,

Fig.

it

3.4.1

hit

do not E at

where of

a

the

cross

is a s s u m e d

each

different is t h e

= y(x)

other

points.

radius

n(r),

within

(consult

that

of

latter

the

be

observation

can

for

be

orthogonal

each

ray

determined

is a k n o w n

plane

E of

so n e a r

emanat-

from

fibre

reaching

E and

However,

L should

be

the

fibre,

L the

linear

the

function

observation

the

before

O ~ r ~a,

varies

fibre

be

the

text-books

on

Outside

the

fibre,the

fibre,

with

fibre.

A typical

O < t < a

a slope

(thus

that

that

of

is n o t different

different

so l a r g e

distance

that

E from

rays

a 6

80

I --~ ]J II vll ~ C0[O,a]

--~

I ---~

[v]t

(4.3.3)

< 2 -

By (4.3.3)

Cl[O,a] mC#[O,a],

and

([v].~) ~

hence u ECl[O,a]

~

I __l

[unk- u]l _< 2 L ~ Hunk - u licO[o,a] Therefore

since

(unk) k converges

converges

to u in Cl[O,a].

to u in C°[O,a],the

Now we present some compactness I (A u)(x) Theorem 4.3.2:

= ~

L e t be K C C ~ ( T )

A

: LP(o,a)

sequence

(unk) k

theorems for the Abel operator

~ K(x,t)u(t) O (x-t)1-s

dt

with 0 ~

5.1.3

I O, a n d w e

exists

a positive

the p r o p e r t y Now

= I

define,

with

as

=~

f o r a n y c 6 (O,1]

gral

equation

(7.3.1),

of the g a m m a for

B = B(s)

r' (c~+1) F(a+1) function

r' (1) F(1) for p o s i t i v e

sufficiently

small

of the e q u a t i o n

argu-

positive

u(1+B)

= I +

B = B(g) , the f u n c t i o n

0r(s+1) (1+~)s B

Then

-

$ ~ O.

IF (s+1) (1+g) s-Sy f(s,y)

u' (I)

see t h a t

solution

B(s) ~ O

and

~+B for

0 < y < s

for

y { s

for

Y ! 0

the f u n c t i o n namely

y(x)

~+B .

: c x

, x ~ O,

solves

the

inte-

139

x

(7.3.11) which

= ~I

y(x)

thus

has

In fact,

0~

infinitely

(x-s) ~-I

many

f(s,y

(s))ds,

x>_O,

solutions.

writing x

z(x)

(7.3.12)

f (x-s)~-I

I

f ( s , y (s))ds

,

=r-~o we find

x

Since

( x - s ) ~-I

I f = r--ELY o

z(x)

c s

~ s

f(s,csa+B)ds

.

(use L e m m a

, it f o l l o w s

7.2.1)

that z(x)

X f O

= s(1+S)

S+B

= cs(l+~)x

S (X-S) ~-I

c s

F(B+I)F(s)

ds

=

s+B

- C(I+6)x

F(s+I)F(B+I)

F(~+B+I) =

But U(I+B)

Looking

back

1 + ~ means

F F (( ~ ~+ +B I ) +FI()I + B )

at

shows

(7.3.12)

F(~+B+I) =

that

1 + s,

(7.3.11)

and

we

has

see

that

infinitely

= y (x) .

z(x)

many

solu-

tions.

7.4.

Applications-Motivated Problem

Formulations,

In

Mann

1951

conduction

reduced

this

ary x = O, Their

and W o l f

on a h a l f - l i n e

and a n o n l i n e a r

Investigations ; Newton's published

results

[Pa,1958],

were

[Le,1960],

treating

heat

given

at time

t = O

integral

x = 0,

equation

t > O.

They

on the b o u n d -

generalized

[Ke-O1,1972]

of

, [O1-Ha,1976],

solutions

(see

and

a

[Lu2,1985],

treatment

[Mi-Fe, 1971] , [ 0 1 - S p , 1 9 7 4 ] , In o r d e r

to h a v e

solution

ways ~

(see

[Ro-Ma,1951],

shall give a review of

of

at h a n d

[Bru,1982],

[Ke,1982],

[Li,1985],

[Lui,1986], specific

a general

in the q u a r t e r

of C a n n o n ' s

[Ba,1982],

[Lu2,1986]).

q u e s t i o n s , we

[Gr,1982], [Li,1969],

For

refer to

a general [Mi

,1971],

[Ca,1984].

of the N e u m a n n

equation 5.2.2

in v a r i o u s

next paraqraph). Furthermor~num~ricalmethodshavebeendevelop-

[Lui,1985],

orientation

and

Abel

[Ha-Lu,1986],

[Lu,1983],

5.2.1

paper

condition

at the b o u n d a r y

later

[Gr,1985],

the h e a t

pioneering

condition

to a n o n l i n e a r

ed for a p p r o x i m a t i o n

of

their initial

t > O.

tnese w o r k s i n t h e

ness

of C o o l i n g

x > 0 with

radiation

problem

Law

book

theorem

initial-boundary plane

on e x i s t e n c e value

and u n i q u e -

problem

x > O , t > O, we c o n d e n s e

[Ca,1984]

into

our T h e o r e m

(N) for Theorems

7.4.1

140

(N)

D e t e r m i n e u(x,t)

in x>O, for

u t : Uxx

If

t > O,

for

t >O,

u(x,O)

= f(x)

for

x>o.

continuous

use

the

for t > O,

for x>O.

: f(x)

kernel

K(x,t)

we r e q u i r e

: g(t)

lira u(x,t) t~O

(7.4.1)

x > O,

= g(t)

lira Ux(X,t) x~O

shall

SO t h a t

Ux(O,t)

f and g a r e

We

t>O

I

-

functions

(Green's

2 x (-~-{),

exp

x 6 IR,

functions) t >0,

g4~t

(7.4.2)

N(x,~,t)

Theorem

7.4.1

be c o n t i n u o u s satisfy

: K(x-~,t)

Let the

and l e t

+ K(x+~,t) , x,

functions

f with

suitable

g(t)

E~ 6 ]R , t > O

for t >_0

constants

and

f(x)

C l , C 2 E [O,~)

.

for x > _ o

and ~ 6 [O,1)

a growth condition If(x) l < C I e x p ( C 2 x 1+a)

Then t h e

.

function t

(7.4.3)

u(x,t)

f o r x > O, t > 0 i s This solution nonnegative

is

: f N(x,~,t) f(~)d~ O

solution

of t h e

unique within

constants

- 2 ~ K(x,t-'[)g(~[)d7 O

Neumann p r o b l e m

the

class

(N).

of s o l u t i o n s

v satisfying

with

and C 4 a ~ r o w t h c o n d i t i o n

C3

Iv(x,t) I _< C 3 e x p ( C 4 x 2 ) . One

now arrives

ary x = O,

t >O

at an A b e l

instead

equation

of the v a l u e s

of

g(t)

second

kind

of Ux(O,t)

if at

the b o u n d -

a radiation

condi-

tion (7.4.4)

Ux(O,t)

is p r e s c r i b e d ,

connecting

-Ux(O,t))

the b o u n d a r y

with

Remark: u(x,t)

We

denotes

(by p r o p e r

imagine density

choice

= F(t,u(O,t)), the o u t w a r d temperature

here

the

of h e a t

of u n i t s

and

t >O, flux

(or the

inward

flux

u(O,t).

following

(which

Ux(O,t)

situation

is energy)

zero-temperature

to be m o d e l l e d :

and point)

simultaneously temperature.

141

However,

we could

also

imagine

u(x,t)a~thedensii~

of a d i f f u s i n g

material

substance. Putting (7.4.5)

~(t)

= u(O,t),

(7.4.6)

g(t)

= F(t,%0(t)) , t > O ,

and

into

formula

inserting

~(t)

which gral

(7.4.1)

of second

(7.4.7)

(7.4.3)

w(t)

and

t >0

(7.4.6)

into

=

1

exp(-

(7.4.3)

gives

is c o n t i n u o u s

Remark:

(7,4.7)

on y .

Alternanively

(D) D e t e r m i n e

u(x,t)

is

The

function

u s u(x,t)

is

as an A b e l

inte-

~(t)

F(T,~('~))

= u(O,t)

in t h e w h o l e

d'c

whose

insertion

quarter-plane

x>O,

equation

a linear

if

integral

F(t,y)

depends

equation.

solution

formula

for

(D) m a y b e used.

in x >0, for

t > 0

SO t h a t

x > O,

t > O,

u(O,t)

=

k0(t)

for

u(x,O)

=

f(x)

for x>O

t>O, .

we r e q u i r e

lira u(x,t) x-*O

= ~(t)

for

t > 0

l i m u(x,t) t~O

= f(x)

for

x>O

solution

- ~

integral

(7.4.7)

and ~ a r e c o n t i n u o u s

and conditions

-~)f(i~)d~

a nonlinear

U t = Uxx

f

itself

for the determination of u (x,t), the

problem

If

reveals

in t > O .

Otherwise

the D i r i c h l e t

obtain

kind

if $(t)

nonlinearly

7.4.1,we

t - 2 S K(O,t-T)F(T,~(T))dT O

(7.4.2)

for the determination of t h e u n k n o w n via

of T h e o r e m

= S N(O,~,t) f(~)d~ O

w i t h x = O in

equation

t>O,

formula

is

of v a l i d i t y )

(see C h a p t e r

4 of C a n n o n ' s

book for the Qetails

142

(7.4.8)

u(x,t)

t ~K = - 2 ~ ~ O

(x,t-T)$(T)dT

+ ~ G(x,~,t)f(~)d~ 0 with G(x,~,t)

L e t us n o w

treat

The more customary at

appropriate

The problem

u(O,t),

(NH)

rod

(x >_ O)

there

(O,t) x we assume,

to v a n i s h .

Hence

D e t e r m i n e u(x,t)

in

for

u t = Uxx - Ux(O,t)

u(x,O)

I and

t > 0

x >O,

treated

heat

in

analogously:

[Ma-Wo,1951].

conduction is taking

temperature

I. N e w t o n

t > 0 to be p r o p o r t i o n a l inside with

Mann

faced

: c(1-u(O,t))

with a given constant

in w h i c h

x = O,

following

x > O,

discussed

is c o n s t a n t

= c(1-u(O,t))

we a r e

heating.

inverted.

briefly been

temperature

is - u

simplicity

u(x,O)

has also

can be

cooling

to be

a t the b o u n d a r y

of o u t s i d e

that

have

of w h i c h

of N e w t o n i a n

the p r o b l e m s

of N e w t o n i a n

signs

a semi-infinite

radiation

difference

For

problem

places

to the l e f t

assumes

in d e t a i l

of h e a t i n g

Consider place,

: K(x-~,t)-K(x+{,t)

boundary

a nonnegative

and Wolf,

with

the

the

to the

temperature constant

initial

following

c.

temperature

problem.

So t h a t t >0,

for t >O

c >_ O ,

for

= 0

x > 0 ,

and f u r t h e r m o r e

Inserting the l i n e a r

for

lira u(x,t) t~O

= 0

l i m u(x,t) x~O

exists

F(t,z)

x>O,

and i s

= -c(]-z)

and

second kind Abel integral

continuous

f(~)

= O into

for

t > 0

.

(7.4.7), w e a r r i v e

at

equation

t (7.4.9)

If outside the

~(t)

(NH) by

f

~(T)

~

0

Vt-~

is a g o o d m o d e l

inward

solution

: 2c g[ _ ~

of

radiation

(7.4.9)

for

the

, t >_O

(Newtonian)

proportional

should

dT

reflect

process

of h e a t i n g

to the d i f f e r e n c e

properties

from

of t e m p e r a t u r e s ,

of the p h y s i c a l

process

143

which the

can

be

inside

observed

temperature

function

strictly

if c > O.

In t h e

In order solve

the

and

technique.

at

trivial

to s h o w

are The

that

~(t)

(7.4.10)

is

expects

c = O we

(7.4.9)

should

= O towards should

indeed

reader

intuitively.

x = O,

~(O)

~ does

to the

result

from

case

equation

left

one

the boundary

increasing

integral

Details

which

using whom

we

be

= I

~(t)

= O for

in t h i s

way,we

the Laplace assume

~(t),

a continuous

l i m ~(t) t-~o

have

behave

Namely:

to b e

all

t > O.

explicitly

transform familiar

method.

with

this

(see[Ab-St,1972])

= -2c V~

t _ c 2 ~ e x p ( c 2 s ) e r f c (c'v~) d s O

~

where (7.4.11)

erfc(r)

-

2

~ exp(-s2)ds,

r 611R ,

r

is t h e

complementary

Obviously

~(t)

case

c = 0 we have

will

assume, the We

can

r = c~,

error function. is c o n t i n u o u s

~0(t)

global

simplify

= 0 for

for all

behaviour

t >_O, %0(O)

t > 0

. But

of %0 c a n n o t

by getting

rid

of t h e

be

= O,

and

in t h e

if c > O, w h i c h seen

constant

trivial

we henceforth

immediately. c.

Substituting

we get c~

~(t)

and

by

a second

{

(7.4.12)

for

t {O

or

= 2__ccg ~ - 2 V[

substitution

~(t)

~(s)

s hO,

[ O

r exp(r2)erfc(r)dr,

s = cg~,we

find

: ~(c~)

s s - 2 S r exp(r2)erfc(r)dr

= 2

~

o

respectively.

F©r an i n v e s t i g a t i o n

~' (S)

of

= 2

the

growth

properties

of ~, w e

_ 2 s exp(s2)erfc(s)

co

2

(I - 2s e x p ( s 2)

S e x p ( - r 2 ) dr) • s

differentiate

144

By

the

inequality

(7.1.13)

e x p ( s 2)

of

[Ab-St,1972]

~ exp(_r2)dr

!

we

have

I

s

for

s ~ O

,

s+~s2+4 g

hence ~' (s)

> _~2 (I -

(7.4.13) It

s+

~' (S) > 0

follows

that

~(s)

2 s

~n-

for

and

)

2 +-4 ~s

s > O.

~(t)

are

strictly

increasing

for

s > 0 and

t > O

respectively.

Therefore assume

the

~(t)

tends

contrary. (i)

In c a s e

t_>t o ~(t) (7.4.9)

(i) >b.

I < ~

there Let

(taking

to a

There

two

+ ~

,

exist

of

(ii)

which

is n o t In c a s e

compatible (ii)

we

- b

with

have

show

and

the

for

t >O)

- b-

dT

t S

CZ

= 2_~c ( ~

from

>O

_ c

To

that

Z =

I we

O < ~ < ]

b C (I,~)

deduce

~(t)

< 2_~c g~

H C (O,~].

cases:

numbers

t > t c and

account ~(t)

limit

are

t

the

O

£ (O,~)

integral

such

that

equation

t o Ct-
O,

and

(7.4.9)

implies

~(t)

> 2_~c g ~

- CZ

_

which

again We

another

now

representation

converging integral

by

infinite

2C

(I-~)

the of

series

representation

H

VZ o

contradictsthe derive

t S

_ c

V~

d~

Ct-~

~

~

as

t ~ ,

assumption. infinite the

series

solution

which

(7.4.10).

is,

technique

~ of in

some

(7.4.9), sense,

described namely

in

§ 7.2

a rapidly

complementary

to

the

145

By f o r m u l a

(7.2.11)

~(t)

and t Of

: d

22

d

(7.2.10)

EI/2

2c

t

eo

~

~/~

Convergence

Now application

Thanks

that

of L e m m a ~(t)

entiation,

X

- 2c V~

~

d X dt n= 0

~(t)

=

:

and c o r r e s p o n d i n g l y ,

~(s)

:

Z n:O

Theorem

the r e s u l t s

7.4.2:

boundary v a l u e u ( o , t ) 0

,

n+l

as a theorem.

equation

(7.4.9)

of t h e ~ e a t i n g t

= ~(t)

continuous

has as s o l u t i o n

problem

_ C 2 5 exp( c 2 s ) e r f c ( c

= I - E I/2 ~(t)

1/2

(-s)

V~

The f u n c t i o n

s

integration

t~

> 0

(-1) n 3 F (7 + 2)

The i n t e g r a ~

= 2__~C~

n/2

interchange

(-c ~ )

= 1-El/2

We c o l l e c t

n+l

s = c ~

oo

(7.4.15)

ds

we o b t a i n

(-I) c n 3 F([ + 7)

I - El/2 with

and

we can

= F(I/2)

X n=O

(t-s)

(-1)n cn 5 F(~n +7)

n

(7.4.14)

1/2

yields

to f~st c o n v e r g e n c e and u s i n g

(t_s) n/2

F ( 7 + I)

summation

7.2.1

2__ccV~ ds

V~

n

(_l)ncn t f F ( ~n + I) 0

dt n=O

is so f a s t

(-I) n c n

n:O

co

d

(_c(t_s) I/2)

f

%/~ dt O

we h a v e

the

(NH),and we have

~)ds

o

(-cg~)

for t >_0.

and s t r i c t l y

increasing

f o r t >_0, and

146

~(O)

: O,

~(t)/~

Remarks:

~k(t)

•

infinite

iteration

(7.4.17)

hand

Lemma

The the

7.2.1 This

limit

Survey

(RD)

(-1)

n+1

n I -~ + t~

n

c n 3 r IT + 7)

(7.4.14)

t ~

.

for ~(t)

according

k 6 [IN

'

can be obtained

by a Picard

to

,can b e u s e d

relation

(%) ~k-1 Vt-T

dT

td_:ea~cmlate the

method

has

problem

by

been

integral

generalized

on the r i g h t -

and

successfully

[Ma-Wo,1951].

2c/g~ as t ~ 0

~(t)/g~

, k £ ~ .

can also be

seen

from

(7.4.14).

- Motivated

Investigations:

of L i t e r a t u r e

shall

give

published

an o v e r v i e w since

we have

reformulate

Determine

1951

of

important

contributions

on the r a d i a t i o n - d i f f u s i o n

treated

the

linear

case.

For

the p r o b l e m .

u(x,t)

f o r x > O, t > O

so t h a t u t = Uxx u(x,O) Ux(O,t)

If

,

o

: O

representation

in § 7.4 w h e r e ience,we

in

iteration

7.5. A p p l i c a t i o n s

results

as

sums

(7.4.9)

to the n o n l i n e a r

series

We

~(t)--.I

t ~ k ( t ) : 2_~c g~ - c S g~ V[ 0

side.

applied

to

~o(t)

t~O,

as

k-1 = Z n:O

series

applied

I

Again

2__cc

The partial

(7 4.16)

of the

~

for

x>O,

: f(x) : F(t

for x > o

l i m u(x,t) t~O

,

, u{O,t))

f and F a r e c o n t i n u o u s , lira U x ( X , t ) x~O

t >o,

for t > O

we r e q u i r e

: F(t,u(O,t))

: f(x)

for x > O

.

that for t > O

.

,

with

selected

problem described the r e a d e r ' s

conven-

147

To h a v e

a correct

u ( x , t ) a s the densityof for example) inward

(i),

an extensive

distributed

flux

problem

visualization

-Ux(O,t)

has

along

at time

a solution

quantity

t. F r o m

and

(i)

F

(ii)

the associated integral

halfline

Theorem

given

(ii)

is h a p p e n i n g , c o n s i d e r

(of a s u b s t a n c e

the p o s i t i v e

u(x,t)

(iii) ~ e l o w

of w h a t

7.4.1

by f o r m u l a

or of energy,

and having

we can

(7.4.8)

at x = O

deduce

that

the

if the c o n d i t i o n s

a r e met.

is a continuous

function

on [ 0 , ~ )

equation

x

~

,

namely

(7.4.7)

~2 (7.5.1)

~(t)

_

__I g~t t ] o

I

has a u n i q u e c o n t i n u o u s f satisfies

(iii) This

solution

growth

f exp(O

~-6) f ( { ) d ~

F(T,~(T))

d~,

t>O

,

Vt-T

solution, t h e growth c o n d i t i o n

is u n i q u e

within

the c l a s s

of Theorem 7.4.1

of

solutions

v satisfying

the

condition lv(x,t) I _< C 3 exp

In

the

cases

to be

f and F or d e s c r i b e In

1951

Mann

listed

their

They

assumption

which

we k n o w

they

discussed

First linear

problem

Newton's

further

tive

constant

tion,

still

c. T h e n

their

essential

hypotheses

(B) G(1)

=

(C) G(y)

is strictly

They (7.5.2)

.

arrive

density

for

and F ( t , y ) = - G ( y )

for O < x < ~ , relaxed

detail

O O.

the

function

of ~. We

exposition

G

suggest and

they

that

content

deduce

the

ourselves

function

G is

is n o n - d e c r e a s i n g

(t-T) -I/2

look

with

iipsahZZz-a0ntZn-

t ~.

t > O, ~(t) ~ I

1951

further

the r e a d e r

results.

(A) ,(B) , (C),

(7.5.2)

[O,T],

the property

{or ~ po&Z.£iue 6, then ~(t)

Roberts

for

and Mann

by a function

. They

thus

replaced

K(t-~)

could

(t-'i) -I/2

reflecting

extend

for

the

t ~ O,

in the

the e s s e n t i a l

theory

to m o r e

gene-

equations. Padmavally

(7.5.])

denoted

on

properties

limit

on a n y b o u n d e d

equation

for

important

same y e a r

integral

0 h a v i n g

for all

equation

properties

< I

to

uniform

integral

excellent

in a d d i t i o n

from

= O and

< ~5(t) < ~3(t) < ~1(t)

of a c o n t i n u o u s

assumptions

expected)

generates ~n(O)

property

~o(t) < ~2(t) < ~ 4 ( t ) < . . . . . .

~0us

condi-

~ C[O,T]

¢~ 0

which

a Lipschitz

by (Bv) (t)

where

satisfies

with

f(x)

the o u t s i d e

natural

investigated = O

density

properties

for

for

the p r o b l e m O < x 0

just

of x = O.

to the The

She

left

data

a discrete integral

assumes

set and

equation

(7.5.4)

problem

that

}(t)

in any

of %(t)

finite

x=O

(or t e m p e r a t u r e )

of the f u n c t i o n s

the d i s c o n t i n u i t i e s

is b o u n d e d

at the e n d p o i n t

density

consist

into a c c o u n t

g and

in t > O

interval

~.

form

O < t < T. Her

is

~(t)

= _~I ~ 0

and by a lot of h a r d vely

(or t e m p e r a t u r e )

, a n d v to the o u t s i d e

of P a d m a v a l l y ' s

furthermore

properties,take

g(~(T),%(T)) gt-~

analysis

aT

she o b t a i n s

,

the

following

results

(intuiti-

expected). (a)

%7 (t) ~ %2(t)

If

corresponding with

f(x)

= %2'

(b)

If

(c)

If ~(t)

(d)

If

then

> 0 and _O

~(t)

are

diffusion

= -g(u(O,t) , %(t))

m < }(t)

for

< M

the

problem

(RD)

where

~ = ~I'

x {O,

t { O.

for

0 < t < T,

.

non-decreasing

O
0

0 .

exists

and i s

finite,

tihen ~(t) ~ l

aS t ~

.

t~

In took

1960 L e v i n s o n ,

problem

(RD)

F(t,y) data

further

are

the

assumed

by a p r o b l e m

of

superfluidity

theory,

with

f(x)

The

motivated

= O

for

= ~(y-~(t)),

functions that

x > O,

~ a n d f,

~ is s t r i c t l y

t >O,

-~
O,

of

~*(t)

.

of w h i c h

for t > O

- ~(t),

period increasing

.

Olmstead

n o t be the

= S(y)

has

~(t)

strictly

S ( y 2 ) - ¢ ( y l ) Z k(Y2-y I) f o r

! 2M

(the o u t s i d e

that

a positive

periodic

in the f o l l o w i n g

for x > O

a and

is

that

Starting from 1972 , K e l l e r ,

stant

exist

continuous

(A) a s s u m e

J~there

I ~ ( t ) - ~ * ( t ) l ~ O as t~. i ~ ( t ) i ~ maxl~(t) i f o r t ~ 0

of p a p e r s

are

a unique

there

0 < t < ~.

and that

series

a uniform H61der con. L e t }(y) be s t r i o t [ y

and,

more

generally

151

The In

data

are

the f u n c t i o n s

[Ke-O1,1972]

as x ~ ,

as w e l l

a condition

of this

paragraph

§ 7.4).

The m a i n

totic

behaviour

the a s y m p t o t i c extensive In

(')

(")

G(O)

they

("')

of the

have

relaxed

Handelsman

function

G

in the p r e c e d i n g

and Olmstead

. We do n o t

the h y p o t h e s e s

continuously differentiable f u n c t i o n G-I(Y). OO

("),

of a n o n -

("') .

and has a w e l l - d e f i n e d

'

"

The f u n c t i o n g(t)

is locally A sufficient

The

topic

t = f O

in the p a r t i c u l a r t ~ O

are,

remark:

~ exp(-

, with

to h o l d

is that

~-{)d~

,

t hO

,

a c o n s t a n t M. f(t)/t

and ~(t)

are

integrable. compactness

arguments

The

in the way

is the a s y m p t o t i c s

(g(s)-G(~(s)))ds case

x = 0

the a n a l y s i s - o r i e n t e d the n o n l i n e a r i t i e s

with

(~u(O,s)) ~x

the b o u n d a r y

Final

(''')

by applying

dealt

E(t)

across

and 0 ~ g ( t ) £ M

for

is d o n e

f f(~) O

to a

sequence

iterates.

Another

which

I 2V~£3/2

bounded and l o c a l l y

proof

of P i c a r d

integrable

condition

non-negative,

here

it is a s s u m e d

from Cannon's

of K e l l e r ,

or t ~ O

behaviour

= O,

where

[01-Ha,1976]

can be c o n s i d e r a b l y

interest

solution

is inverse

.

the q u o t a t i o n s

as t ~

tables

G(y)

as in

which

and

[Oi-Ha,1976]a

negative

f,G,~

f(x) ~

of

,

O is the n e t

inward

flux

ds

.

applications-motivated

they

(as t ~ )

are c a r r i e d

ones

are very

described distinct

out,

completely

in 7.3. from

investigations

The

each

independent

conditions

other.

reported from

concerning

152

7.6,

A Very In

Brief

1982

in 1951

Survey

and

1985,Groetsch

bz M a n n

and]Wolf,

wise

linear

that

if ~ 6 C2[O,T]

satisfies imate val h.

of L i t e r a t u r e

ansatz

provided I/g~T

uniformly

for

investigations

relaxed

are

In

with

to the

solution

~(t) . He and G

L < I/g~T

then

solution

is of the o r d e r constant

this

is v e r y

restriction

posed

an e q u i d i s t a n t

(but fixed)

to the e x a c t

the L i p s c h i t z whether

the problem,

tested

constant

T o > O. C o n v e r g e n c e

further

can

(see

showed 7.5)

the a p p r o x -

in e a c h

of the

inter-

steplength

restrictive

be

piece-

removed

and

or

desirable.

1969

equations

and

T > 0 is a r b i t r a r y

condition

Methods

numerically

for an a p p r o x i m a t i o n

solutionsconverge

His b o u n d

treated

He a n a l y z e d

where

a Lipschitz

[To,T]

on N u m e r i c a l

Linz

analyzed

of the

product

integration

methods

for

integral

form X

u(x) under

the

Lipschitz Typical

essential

1982

+ ~ p(x,t)K(x,t,u(t))dt 0

condition

continuous

forms

In

= g(x)

with

of p(x,t) Kershaw

that

respect

are

K is c o n t i n u o u s to its

third

(x-t) -I/2" and

treated

by

, and

argument

in p a r t i c u l a r u.

t(x2-t2) -I/2"

the p r o d u c t

trapezoidal

rule

integral

equations v

u(x)

where

0 < a < I and

rem on

= g(x)

+ ~

K satisfies

the e x i s t e n c e

? 0

K(x,t,u(t)) (x-t) I-~

certain

of a s o l u t i o n

dt,

conditions.

by a p p l y i n g

He a l s o

Banach's

gives

fixed

a theo-

point

principle. J.J. imating

te R i e l e

solutions

in

1982

described

of e q u a t i o n s

a

of type

X

u(x) where He

u(x)

thus

of the

linear

u(x)

Kershaw

interval very

In

with

consideration at the o r i g i n

and

recent

functions

of p o s s i b l e

X and

~.

non-differentiability

x = O. equations

in a

. years

Abel

by m e t h o d s

of R u n g e - K u t t a

Schlichte.

See [Lui,1986],

smooth

dt

te R i e l e considered their i n t e g r a l

O <x < a

and nonlinear

[Lu2,1985],

+ f K ( t , u ( t ) ) ( x - t ) -I/2 0

+ x I/2 ~(x)

specific

solution

Linz, finite

= X(x)

took

= g(x)

several

integral type

have

contributions

equations been

given

[Ha-Lu,1986], [Lu 2,1986].

They

to the

by m u l t i s t e p by Hairer,

[Lu ,1983],

have

treatment methods Lubich

[LuJ1985]

succeededin

and and ,

generalizing

of

153

the Dahlquist

theory u(x)

corresponding A-stability the

linear

= u(O)

special

and related

Volterra

integral

equation.

x + f f(t,u(t))dt, O

to t h e l i m i t i n g

case

concepts

s = I

are

of A b e l ' s

integral

studied by Lubich

equation.

in p a r t i c u l a r

test equation u(t)

where

of the

= f(t)

+ ~

t 0f

(t-s) s-1

u(s)ds,

t >0,

0 < s < I. The

monograph

reader

interested

of B r u n n e r

in n u m e r i c a l

and van

der Houwen

methods (1986) .

should

consult

the

for

Chapter

8.1.General

Topics

In C h a p t e r erator

6 we o b s e r v e d

to give

linear Let

other

Hadamard's

mappings

(8.1.1)

the p r o b l e m

We

a more

examples. for

say that

For

these,

normed

linear

of

of the o p e r a t o r

definition

between

X and Y be two

operator.

that

inverse

definition;

of of F i r s t

Kind

Problems

H e r e , we d e v e l o p

and d i s c u s s

a formal

We r e f o r m u l a t e

linear

The

in L 2 - n o r m .

posedness

of

in I l l - P o s e d

is i l l - p o s e d .

tinuous

8: I l l p o s e d n e s s a n d S £ a b i ± i z a t i o n Linear Abel Integral Equations

inverting exists,

general

our

notion

aims,it

we r e f e r

normed

con-

of

ill-

is less

important

in the c o n t e x t

spaces.

spaces

the p r o b l e m

op-

is not

to the b i b l i o g r a p h y .

of w e l l - p o s e d n e s s

linear

the A b e l

but

and

let

of s o l v i n g

A:

X ~ Y be a

the e q u a t i o n

Au = f

where

f E Y is g i v e n

three

conditions

(i)

The

equation

f (that (ii)

The

(iii)

The

and

u £ X is u n k n o w n , i s

well-posed

if the

following

hold. (8.1.1)

has

is the o p e r a t o r

equation

(8.1.1)

at least

one

solution

for g e n e r a l

data

A is s u r j e c t i v e ) .

has

at m o s t

one

solution

(that

is A is in-

jective). solution

side We

say

f

that

There

(that

u of is

A

the p r o b l e m are m a n y

(8.1.1) d e p e n d s c o n t i n u o u s l y on the r i g h t - h a n d -I : Y ~ X is a c o n t i n u o u s o p e r a t o r ) . (8.1.1)

physical

is i l l - p o s e d

problems

if it is not w e l l - p o s e d .

that, in m a t h e m a t i c a l

formulation,

are i l l - p o s e d . When in o t h e r case,

condition words,

(8.1.2)

that

the d a t a

for e x a m p l e , I

has

fined by

(i)

is n o t

satisfiedtthe

of the p r o b l e m

for the

simple

linear

X = ~2

,

are

space

Y is

"too

incompatible.

large",

This

is the

system

xI + x2 = I x I - x2

O

2x I + x 2

3

no solution.

Here

Y = ~3

, and

A

: ~2

~ ~3

is de-

155

(ii)<x1>

A ( x I ,x 2) : In some is t o o

sense~there

x2

"too m a n y

data"

for h a v i n g

existence,

i.e.,

Y

large.

When

condition

for the u n i q u e the

are

-

linear

(ii)

is n o t

determination

satisfied

of the

there

solution.

are,

often,

Consider,

too

few data

for e x a m p l e ,

system xI + x2 + x3 = I xI - x2 + x3 = O

Here

X = 393 , Y = IR2

a n d A:

IR 3

~ 392

is d e f i n e d

(i 1 A ( X l , X 2 , X 3)

by

1

I I

lJLX21 \x3/

and the

system

When

condition

pathological

the

a unique

is n o t

Then

solution.

satisfied,we

it m a y h a p p e n

The

have,

that,

error, t h e

solution

c a n be d e t e r m i n e d

to t r e a t

a physical

satisfy

(iii).

In

fact,

accurately,therefore

we

extract

generally,

if its

problem,the

any useful

small.

a more

the d a t a

only with

problem

in a p r a c t i c a l

cannot

s p a c e Y is t o o

when

It is i m p o s s i b l e

doesn't known

(iii)

situation.

u p to a s m a l l error.

does not have

are known

a very

large

formulation data

are never

information

about

solution. The problem

fulfil

the

proved

in 4.3.

of i n v e r t i n g

condition

(iii).

The Abel

(8.1.3)

continuous

operator,

then

Id =

operator

I -F(~)

J~u(x)

a compact

the A b e l

Tc p r o v e J~:

this

La,

1968;

cannot th.

A direct b y the

be

3.6,

(J~) w o u l d

proof

following

Example

be

f

(x) n

istby

doesn't results

the

formula

continuous. (see

in L P ( o , I )

If it w e r e

[Ta-La,

would

be

1968;

compact.

LP(O,I) is not of finite dimension (see[Ta-

65]).

of the n o n - c o n t i n u i t y

of the

With -

be

compact

example.

8.1.1:

spaces

the c o m p a c t n e s s

~ LP(o,I)

cannot

sphere

the case because pag.

in L P ( o , I )

'

(j~)-1

theorem 7.2, p.298]),that is the u n i t But

LP(o,I)

? u(t)dt 0 (x-t) I-~

therefcre (j~)-1

operator

this we use

sin ~ n x (zn) ~

,

O


n-~o~ limll (js)-1 _

lim II (j(x)-1 f n t L P ( 0 , 1 ) n-~co

to L P ( o , I ) .

In fact,

f n LiL I (0,1)

2 =

lim

liunii I

n~

L

In the arises:

study

How

we m u s t can

not

be

boundedness

to use

informations

problem

equation

of the p r o b l e m

ation, ator.

and,

informations becomes

in s o m e

Arsenin

fact

[1977]),

a continuous invertible to give

map

The

a precise

final

this

sense,

they

as g o o d

step,

this

crucial

method question

3]).

guarantee

of the

of AIK

In

many the

X. T h i s

of g e n e r a l

even

in X,

of the m o d u l u s

in an e x a m p l e

below)

for the

in C h a p t e r

9.

the a

situ-

continuous

topology

oper-

(see T i k h o n o v -

set

in X,

is

if A is a c o n t i n u o u s important

of the o p e r a t o r

estimates

applications,

consists

the

of the

solutions

the

is a h a p p y

and

of c o n t i n u i t y

stability

to a p p r o x i m a t e

cases,

These

f r o m the

set of p o s s i b l e

it is v e r y

as p o s s i b l e

so on.

than

K is a c o m p a c t

case, set

and

the

solution,

directly

true"

that

space

, where

In any

and K is a c o m p a c t

to find

"more

situation

of the

solution,

available

problem,

[Pu,1959],

derivatives

are

[1959,§

see

X ~ Y is an i n v e r t i b l e

to K.

evaluation

the p h y s i c a l

of the

by a t h e o r e m

inverse

from A(K)

that

generally

subset A:

case,

(we i l l u s t r a t e

a constructive consider

the

operator

(AIK)-I , i.e. solution

when

in this

are

on the data)

of a p h y s i c a l

solutions,

properties

solutions

a compact

of

question

dependence

formulation

or of some

(see P u c c i

on the

in p a r t i c u l a r , In

sign

important

(iii)?

informations

the

or c o n v e x i t y

a very

continuous

condition

[Pa,1975]

physical

solutions

all

example

of e n e r g y ,

of a p r i o r i

priori

the

problem (i.e.

is the m a t h e m a t i c a l

For

the m o n o t o n i c i t y types

stability

satisfying

careful us.

.

of an i l l - p o s e d

(8.1.1)

suggest

-

-

to r e s t o r e

in a p r o b l e m When

>

(0,1

for

the

in f i n d i n g

problem.

We

157 Example X and

Y

8.1.2:

are

normed

Let

X = cO([o,I]),

linear

spaces

with

f u(t)dt

,

Y

=

the

{u 6 C 1 ( [ 0 , I ] ) ,

sup-norm.

Let

u(O)

A be

= O].

defined

by

X

(Au) ( x ) =

O

< x
O

or

u'(x)

< O

(iv)

u"(x)

or

u"(x)




We

equation

of t h e s e

are

of s p e c t r o s c o p i c

solution

conditions" are

shall

for simimeas-

u of t h e e q u a t i o n

O

.

For (n+1)! F(~+n+1)

u n of the e q u a t i o n

n+a x

,

there

the f o l l o w i n g

.

O

spaces.

of the A b e l

(ii). M a n y

in the c a s e

that

the f o l l o w i n g The

> 0

instability

from

we prove

function

.

u(t) d t - f(x) (x-t) 1-a

o n e of

in L P - s p a c e s .

Example

the

xS 0

in 8.3

the

different

precisely

(8.2.13)

satisfies

estimates

illustrate

bounds

l a r to the e x t r a urements.

on f in the a f o r e m e n t i o n e d

stability

some a p r i o r i

bility

u = f

O<x1

isn't ones:

sta-

167

J~un is

(use E u l e r ' s

= fn

beta

integral,

Un(X)

We

see a l s o E x a m p l e

1.1.1)

= (n+1)x n

have u n >- O, u'n >- O, u"n >- O

and for n ~

I 1-c~--(n+!) :

n

P

II fnl~P(o,1 ) =

I

F(a+n+1) [p(n+a)+1] !/p by S t i r l i n g ' s Furthermore

P P

formula. (for all

I ~ q !~) n

II UnllLq(O, i ) Now

iS

n~.

for

Therefore, n o n e

bility -fn

f II ~O n LP(o, i )

for the Abel

(nq+1) I/q n~

if

I }

If' (%)IdT

Using Young's inequality for convolutions,we

obtain

II21 j II f'll h 3/2 L (O,1) Putting together To prove

these estimates we obtain

(8.3.8).

(8.3.9),we observe that Ju(x) ~ < lu(x)-u(x-h) I + II ull

-

L~(O, I-h)

Now U (X) --U ( x - h )

Y

I

"{

f' (T)dT

2

I x h{1

+ --

o

Vx-T

_I }

f' (T)dT

Vx-h-T

U~i~g the same arguments as for I2, we obtain lu(x)-u(x-h) I < 4}{ f'll L~(O,I )

hi/2

W This proves 8.3.2.

(8.3.9).

LP-bounded First Derivative of the Solution

Theorem 8.3.1:

If s 6 (0,I), p 6 [1,+~J, u 6LP(o,I),

and

(8.3.10) we h a v e

Jau = f

u' 6LP(0,1)

172

(8.3.11)

II Ullp

If

uEw@'P(o,I)

furthermore

c~ a {11 u' II 1+'c~ + II flip 1+c~ }

_< C l ( a )

for

a value

@610,1)

1

II flip1+---&

and a v a l u e

p E [I,+~)

then

II Ullp 0 .

I < p < +~

, leaving

to the r e a d e r

of p = + ~. Take u

(x) = .[p(n- I) + I] I/p n

T h e n all u

const

=

We c o n s i d e r treatment

!

n

x

n

for n = 1,2,3,...

n

6 ]K and I/p J~'u (x) = n!

n

F(~)

[ p < n - I) + I ]

x

n+c~

n F ( c ~ , + n + 1)

We o b t a i n

II Unll p

=

I [ p(n-

~

I) + I

pn + 1

] I/p

aJ

I

~

the

174

llJ~Unllp

because,

by S t i r l i n g ' s

F r o m these

asymptotic

=

n! nF(e)F(e +n+

formula,

n-~o lim TIJ &U n lip The

importance

in m a n y p h y s i c a l bound

]I/p r(~)n I + ~

I

I n

get 1

~, (i

of T h e o r e m

problems

[m(n-1)+I p(n+e)+

n! F(e + n + I)

relationships,we II Unll

I)

=

+~)

8.3.1

(F(e))l

consists

formulated

in r e s t o r i n g

as an Abel

equation

the s t a b i l i t y with an a priori

(see 8.2)

(8.3.16)

E

]lu' llLP(o,1 ) ~

for some

p 6 [I, +~].

By the a p r i o r i restored.

bound

Let us c o n s i d e r

conditions

(a) and

(b)

and %:he e s t i m a t e for e x a m p l e

(see 8.2.1

dx N O W if we data fi'

-

mut U i = j1 + ¢12, i = 1,2,

11 u 1 - u 2 IIL~

Now by c o n d i t i o n ,

MI/3

Therefore,since

lIL~

(O,1)

- ml

2m 2

=: 0

-

Theorem (8.3.17)

mI

+ II ¢1 - ¢ 2

M I/3 C { ~ 2/3 mI

8.3.2:

+

II fl - f2

i£t u 6 LP(o,I) A u(x):=

the

to the

we have 12/3

L (0,1)

we have

!


= f w j ( x ) f ( x ) d x w i t h wj ~ O 0 a r e to b e c o n s i d e r e d . We w a n t

portant

the a p p r o x i m a t e

properties

non-negativity, dependent A

the e x a c t

solution

~ to share

one

solution

u is k n o w n

to have, e.g.

monotonicity,

functions

is i n j e c t i v e

convexity,

unimodality.

u n for n = 1 , 2 , . . . , N

these

are

linearly

or m o r e

and

With

functions

independent

as well)

of the

smoothness,

linearly

with

~ =

~ n=1

the c o e f f i c i e n t s

cn un c

n

take

,

to be d e t e r m i n e d .

Correspondingly

N

(9.3.5)

~ =

We wish,

of course,

Our problem

to

the

data

in

~ n=1 ~u,

c

n

f

n

~f

.

c a n n o w be f o r m u l a t e d

such

a way t h a t

the

as o n e of o p t i m i z a t i o n :

extra

conditions

in-

fn = A U n ( b e c a u s e

N

(9.3.4)

im-

u satisfies

to

fit are

187

(approximately) f u l f i l l e d for ~, t h e f i t t i n g being achieved by minimizing an appropriate measure of deviation° The unconstrained Gauss ,~east squares f i t c o n s i s t s in minimizing the quadratic

form J Q(~) = 9 =1~ yj (aj- < ~ j , ~ > )2, c 6 ~ N

(9.3.6)

where the positive weights yj are prescribed

,

(all = I in the simplest

case). Here c denotes the col l~n vector with components ci,c2,...,c N. The integer number N may be smaller or larger than or equal to J, the particular case N = J meaning interpolation The quadratic

(Q(~) = O).

function Q(~) has a unique minimizer ~ if and only if

its second degree part is strictly positive definite. and sufficient condition for positive-definiteness

To find a necessary

we write Q(~)

in matrix

vector notation with aI

lit> c2

a2 as data vector, ~ =

as vector of coeffi-

N

aj

c ients,

<sl,fl > <sl,f2>

M

=

I

<s2'f1>. <s2'f2>

"'" ...

<s1'fN> 1

\ < s j , f l > <sj,f2 >

• ..

<sj,fN >

as Gram-matrix,

and the diagonal matrices

F = diag(yi,Y2,...,yj)

as weight matrix, F I/2 = d i a g ( ~ ,

7V~2. . . . .

~j)

We denote transposes of vectors and matrices by the superscript T. Note that all quantities are assumed to be real. With these notations a straightforward and changes in orders of summation, yields (9.3.7) Q(~)

= (F I/2 M ~)T

The second-degree

calculation

(FI/2 M ~) - (F~) T M ~ + (r~) T

term

(F I/2 S ~)T(FI/2 M ~)

using

(9.3.5)

188

is p o s i t i v e We also

definite

see that

only if J ~ N , cients

i.e.

this necessary

functionals

(9.1.2)

none

(9.1.5)

none

of

zeroes

To apply

And

have

fn(Xj)

Gorenflo

spectroscopy

flo and Kovetz

thus

the e~act

for

(9.1.4)

and

for the corresponding

= 0 f o r n = 1,2 .... ,N, h e n c e

of b a s i s

1966,and

Minerbo

polynomials

functions

at a

u was

(large)

problem)

and Levy,

choice

In the p r o b l e m

set of p o i n t s

obtain ~

for

sense:

of l a r g e r

treated

to b e e v e r y w h e r e

finite

they

1969,take

M

by a s u i t a b l e

(in a c o l l o q u i a l

low accuracy,

known

u n or fn is

a Gram matrix

multiplied

"smoothness"

oscillations.

solution

~(x) ~ 0

be met are coeffi-

if the a ~] s are point f o r e q u a t i o n s (9.1.1),

in s u c h a w a y a s to y i e l d

Disadvantages:

optimization

can

M.

achieving

to u n w a n t e d

By forcing

to a q u a d r a t i c

(A u n) (xj)

(9.1.5)

suppressed).

N leads

=

and Kovetz,

equation

a n d N small,

oscillations

tive.

condition

see t h a t

then

a good choice

of r a n k N.

values

we

= f(xj))

in the m a t r i x

this method

It s h o u l d b e m a d e

function

sufficient

r a n k N.

less data values~ne~.there

furthermore

(

essential.

the

and

M has

of the x. s h o u l d b e e q u a l to O, w h e r e a s 3 of t h e m s h o u l d b e e q u a l to a. O t h e r w i s e

j we would

a line

if t h e m a t r i x

if t h e r e a r e n o

to b e d e t e r m i n e d .

evaluation

index

if a n d o n l y

by Goren-

non-nega(this l e a d s

"approximately"

non-

negative. Today general

it is w e l l - k n o w n

sense

of the word)

kind

of a p p r o x i m a t i o n ,

wise

linear

(or r o o f

ansatz.

that piecewise are better.

namely

This

We

polynomials recommend

approximation

means

taking

("splines"

a very

of u b y c o n t i n u o u s

the f u n c t i o n s

in t h e

robust

u n as h a t

piecefunctions

functions):

With O = t I < t 2 < t 3 < ... < t N = a take Un(t)

= 0

for

t < tn_ I

and

t > tn+ I

t-tn- I (9.318)

Un(t ) - tn_tn_1

for

tn-1 < t < t n

for

t

tn+ I -t Un(t)

if n £ { 2 , 3 , . . . , N - I } [O,a]

in c a s e s

[tj,tj+1], k#n

.

-

tn+1-t n

and omit

in t h e s e

n = I a n d n = N.

j = 0, I,...,

N-I,

definitions

Every

and all

< t < t n - n+1

everything

u n is l i n e a r

U n ( t n)

in e a c h

= I, w h e r e a s

outside subinterval

U n ( t k)

= 0 for

189

/

U4

O=t I

t2

t3

t4

tN_ I

t5

tN=a

~g. 9.3.1

Because

of m e a s u r e m e n t

for the a p p r o x i m a t e Numerical However,

solution.

experience the m a i n

u n is the e a s e constraints

shows

they

offer

with

cannot

So,

that

advantage%

in f o r m

now coincide

errors,we

why

expect

should

order

we use h i g h e r

of a c c u r a c y

degree

these

splines

of d e g r e e

I are

of u s i n g

the h a t

functions

as b a s i s

in i n c o r p o r a t i n g

of l i n e a r

the v a l u e s

a high

customary

inequalities ~(tn).

shape

splines

good

functions

conditions

for the c o e f f i c i e n ~ c

as

n which

We n o w have:

N

Nonnegativity is

equivalent

of

~ =

~

c u

n=1

n n

to

(9.3.9)

cn

for

Z 0

Monotonic i n c r e a s e

n

=

1,2 .....

of ~ i s

N

.

e@uivalent

to (9.3.10)

Cn+ I - c n ~ O

C o n v e x i t y of ~ i s Cn+2-Cn+

(9.3.11)

equivalent 1

Xn+2-Xn+ I The as may

use

be

rization. it for h a t

of such

seen

functions

Cn+t-c

Another

n

> O -

for

constraints possibility

in m i n i m i z i n g as b a s i s

.

to

Xn+l-X n

inequality

in 9.4.

It c o n s i s t s

for n = 1 , 2 , . . . , N - I

n = 1,2,...,N-2

has

some

the q u a d r a t i c

functions

u n)

.

regularizing

is a d i s c r e t e

Tikhonov

function

?

enough.

effect regula-

(we d e s c r i b e

190

~(~)

(9.3.12)

(with

or w i t h o u t

positive

= QI~)

+ ~ n=1 z

additional

parameter

T T ~ + I -tn/

inequality

I. For

any

I >O

constraints)

the f u n c t i o n

with

QI(~)

a suitable

is p o s i t i v e

defi-

nite. Still means

effect, acts

another

that

that

is,

there

is an o p t i m a l

of r e g u l a r i z a t i o n

In 9.4 we

shall

basis

esting

present

functions

We c o n c l u d e variants

have

this for

non-negativity by F.M.

first

the e x a c t

of the

by g i v i n g

the

deriving

function

arrives

f(x)dx

data at

space ~

In his

case 1977.

study

account

These

in w h i c h

of

two

(9.1.5)

variants

tests, shown

(representing

the

an e l e m e n t

study

interin c a s e

have

to w o r k

been well.

set of c a n d i -

by a maximum

the v a l u e s

O = x ° <x I


aj

~(t)

To o b t a i n

solutions

~ 0

are not

and

as n e a r

... < x N = a

values they

replaced

to be c o m p a t i b l e

must

be m o d i f i e d

by c o m p a t i b l e

as p o s s i b l e

.

values,

to the o r i g i n a l

with

by a the

one),

he

of m i n i m i z i n g

log ~(t)

t dt

under

(or the m o d i f i e d

method, L a r k i n

de-finitely

the m e a s u r e d

technique

the p r o b l e m

As asecond

where

solution

a f ~(t) O produce

case

equation

solution.

he d e t e r m i n e s

conditions

preprocessing

modified

in this

Natterer,

form

a non-negative special

a short

spectroscopic

1969 and, by n u m e r i c a l

solution)

x. 3+I S x. 3 After

Compare

used.

likelihood estimation t e c h n i q u e . are

of N w h i c h

This

a regularizing

is as follows:

Ia% an a p p r o p r i a t e for

has

of a n u m e r i c a l

of the e x a c t

Larkin,

variant

been

treating

of k n o w n

value

results

by i~cr~iz~on". itself

parameter.

paragraph

proposed

dates

of d i s c r e t i z a t i o n

as a k i n d

these

The

is the " r e g ~ i z ~ o n

method

the c o a r s e n e s s

for

proposes

0 < t < a

smoother

than

the r e s t r i c t i o n

ones,

that ~ should

respectively).

an a n s a t z

~(t)

=

(h(t)) 2 so that

. the

method, he n o w m i n i m i z e s the e x p r e s s i o n a f (h' (t))2 t at 0

ones

usually

found

by his

first

191

under

the r e s t r i c t i o n s

motivated)

boundary h' (0)

The

Euler-Lagrange

problem

9.4.

W.

zikoll

ies u s i n g Without

u produces

= h(1)

= 0

technique

for w h i c h

A Numerical

that

aj a n d

the

(physically

he g i v e s

Case in h i s

.

leads

an

to a n o n l i n e a r

iterative

method

generalized

eigenvalue

of c o m p u t a t i o n .

Study diploma

the q u a d r a t i c

thesis

has

optimization

going into~edetailsof

used

optimization

will

display

a few typical

tion

(9.1.5)

with

carried

method

computation

algorithms

of Cryer,

results

out numerical

of the p r e c e d i n g (let us

1971,

obtained

the

stud-

paragraph.

just m e n t i o n

a n d of E c k h a r d t , for

case

t h a t he 1974),we

spectroscopic

equa-

a = I, n a m e l y

I 2 f u(t) t d t t:x / t 2 _ x 2

(9.4.1)

the v a l u e s

conditions

- f(x),

0 <x < I ,

with f(x)

(9.4.2) The

exact

= ~6

solution

(9.4.3)

u(t)

= ½

is strictly monotonically concave

being

= n/(N-1)

n

the v a l u e investigate

j/J, n o i s e

lation

on f, m e a n i n g

f but

rather

(9.4.4)

was

from I/~

evaluation

= ~

I,

(9.3.8)

have

been

taken,

the

the

,

functionals

for

by

of

j = 1,2 ..... J

inexact

.

measurements

superimposing

functionals

of f + %o w h e r e ~(x)

f r o m 0 to

< t < I

taken.

influence

simulated that

I/2 to 0 as t runs

functions

= f((j-1)/J)

the

I ,

for n = O , I , 2 , . . . , N - I

J = 11 w a s

points

O