Applications of the Hilbert problem to problems of mathematical physics

T

APPLICATIONS OF THE HILBERT

PROBLEM TO PROBLEMS OF

MATHEMATICAL PHYSICS

PROEFSCI-IRIFT

TER VERKRIJGING VAN DE ORAAD VAN DOCTOR IN

DE TECHNISCHE WETENSCHAP AAN DE TECHNISCHE HOCESCHOOL rE DELET, KRACHTENS ARTIKEL 2 VAN

HEI KONINKLIIK BESLUIT VAN 16 SEPTEMBER 1927, STAATSBLAD NO 310, OP GEZAG VAN DE RECTOR MAGNIFICUS DR O. BOTTEMA, H000LERAAR IN DE

AFDELINO DER ALOEMENE WETENSCHAPPEN, VOOR EEN COMMISSIE UIT DE SENAAT TE VERDEDICEN OP

WOENSDAG 5NOVEMBER 1958 DES NAMIDDAGS TE 4 UUR DOOR

JOHAN ADOLF SPARENBERO

Rau de nagedachtenis van mijn Ouders

Ran Paula

Content s

Introduction

Chapter le The Hubert problem I

1 .1 The formulae of Plemelj I

1.2

The Hubert problem for an arc 3

1 .3 Singular integral equations 5

Chapter

2.

The Wiener-Eopf type integral equation 7

2.1

Transformation of the equations 7

2.2

Solution of the homogeneous integral equation

of the second kind 8

2.3

The irthornogeneous equation of the second kind

and the equation of the first kind 10

2.14 Examples

12

Chapter 3. A shrink-fit problem for a half infinite

range of contact 114

3.1

Formulation of the problem, determination of

Green's function for the tube

ILl-3.2

Solution of the integral equation for the

contact pressure 17

3.3

Numerical calculation of the shrink-fit

stresses 21

3.14 The limiting case of an infinitely thick tube 22

3.5

Discussion of the results 23

Chapter Li.. The homogeneous first order

integro-differential equation of the Wiener-Hopf type 25

14.1 The general equation, case 1 25

14.2 Case 1 with a0 0, a = O and b1 # 0 28

14.3 The general equation tase 2 30

Li.Lt Example 32

Chapter 5. A Wiener-Hopf type integro differential

equation with fourth order derivatives 35

5.-1 Fourier transformation of the equation 35

5.2

The asymptotical expansion of E(A)

₃₉

Ohapter 6. The finite dock Li.6

6.1 Formulation of the problem Li.6

6.2

Derivation of the integral equation Lj.8

6.3

Determination of '(x) 50

6.L- Calculation of (x) 52

6.5

Construction of the solution for prescribed

incoming waves 5L.

6.6

Numerical calculation of the reflected and

transmitted wave for the case that no

breaking occurs 56

/

Chapter

7.

On the inf'ience Of the cross-section form

of a ship on the added mass for higher order

vibrations 59

7.1 Formulation of the problem for the case of a

strip 59

7.2

Derivation of the integral equation

6i

7.3

The cases a = O and a - O 63

7.14 The case a - 65

7.5

The case of finite a 67

7.6

The added mass for the strip 70

7.7

The added mass for a vibrating infinite

cylinder 70

IN TR ODUCTIOI

Problems arising in mathematical physics,can in general

'ce classified from different points of view. From the

physical formulation a classification to several branches of physics and mechanics is obvious. From a mathematical

point of vìew, problems, originating in different parts of

physics,can sometimes be submitted to a uniform treatment by the same mathematical method. The problems dealt with in

this thesis have in conwion, that they all can be formulated

in terms of the Hilbert problem. This problem can be treated by the use of sectionally holomorfic functions (ref.1),which

concept is based on a set of formulae derived by Plemelj.

These functiors are regular in the whole complex domain with the exception of a discontinuity on a curve. In chapter 1 we give a short survey of the notations and results of this

theory.

The chapters

_2-5

are concerned with Wiener-Hopf type

integral and integro-diff'erential equations. Usually these equations are solved under the condition that there exists some strip of convergence for the Fourier-transformations (ref. 2,3).

Using

however, the theory of sectionally hob-morfic functions it becomes clear that a strip is not essential

and we need only to demand convergence on a line. In chapter 3

we shall discuss an application of the theory on a problem of shrink-fit stresses. The stresses are calculated by a method which is equivalent to the procedure for obtaining approximate

solutions described in an article by W.T. Koiter (ref. 24). In

the next chapter we discuss the homogeneous Wiener-Hopf type

integro-differential equation with first order derivatives.We

conclude the treatment of Wiener-Hopf type equations by

conside-ring an integro-differential equation with a fourth order

deri-vative which occurs under the sign of integration. This equation is a result of consideraequations about the anomalous skin

-effect of electrons in a metal (ref. 5) and is a generalisation

of an equation discussed by Reuter and Sond.heimer (ref. 6).

In the chapters 6 and 7 we discuss two problems on the motion of water with a free surface. First the two dimensional problem of the reflection and transmission of progressing waves, when a part of the surface of the water is fixed by introducing on the surface a rigid strip of infinite length. This is the so called finite dock problem. A proof of the existence of the solution

has been given by H. Ruhm (ref. 7), while Mac Carny (ref. 8)

dis-cusses the pressure under the dock. Here we have to solve a HUbert problem for a function which possesses some prescribed discontinuity on the strip. The second problem of this part

considers the forces exerted by the water when the strip executes

are compared with the forces at a half ininersed cylinder which executes a vibration with shear deformation. This has

been done in order to obtain a check on the three dimensional correction coefficient for the added mass, used in naval

architecture (ref.

9).

The question arose whether this

correc-tion coefficient, derived from the added mass of a vibrating ellipsoid of revolution, would be accurate enough for ships of shallow draught. We have to consider a singular integral

ecuation which can be reduced, by the theory of sectional

holomorfic functions, to an integral equation of the Fredhoim trpe.

Chapter

I

TEE HILBERT PROBLEM

The formulae of Plemelj, the Hubert problem and singular

integral equations are discussed thoroughly in ref. i For

direct reference, however, we shall state some results. We

shall not enter into details but consider the theory to an

extent necessary for understanding the applications.

-1 .1 THE FORMULAE OF PLEMELJ

Let L be a smooth arc, in the complex plane, defined by

x=x(a),

_{y=y(s), SaSSb}

(i.i.i)

where s is a parameter and x(s) and y(s) have continuous first ordex derivatives which do not vanish simultaneously. Also we assume L to be simple; this means that never x(s1) = x(s2) and

zb

z=c

4_ (to)

.1 .1 Arc of discontinuity for s sectionally holomorfic function.

y(s1) = y(s2) for values

8a i s2 S-e.

On this arc we consider a function ç9=cp(t) which satisfies the Hölder condition

k(t2)-ç(t1 )I <A t2-t1 I , (i .1 .2)

where t=t(s) is a point of L which corresponds to the

para-meter value sand A and /L are positive constants. Then we form

the Cauchy integral

2.

From this definition we see that (z) Is an analytic function

in the entire complex domain with the exception of the arc L.

At L the values of (z) exhibit a jlxnp by passing from one

side of L to the other. Introducing on L a positive direction for increasing values of the parameter s we call the left hand side of L the positive "+" side and the right hand side the negative side. The limiting values of (z) are denoted

respectively by (t) and _(t) (fig. i.i.i). The values of

(z) are continuois up to L, with exception of the ends of L

for which q'(t) # o and satisfy, as Is proved in ref. 1, the following relations of Plemelj

- jt0) = (t0) (1.1 .L1.)

and

_j_

jr

c(t)

+ _(t0) _{- t_t0} dt.

The integral is to be taken in the sense of Cauchy

Idt

= C

L-dt, (i .1 .6)

where C is a part of L with ends t1 and t2 in such a way that

< o < and 1t2-t01 = k0-t1

I.

Formulae (i .1 .L.) and (i .1 .5) can be verified directly in the case that ç(t) represents the values on L of a function

p(z) analytic in a neighbourhood of L. In this case we may

deform L slightly (fig. 1 .1 .2) in order to calculate for

instance the limit value

(t)

dt + q (t0).

= t_t0

/

r'i,1.i.2.

Deformation of L when (z) is analytic.

- r

lnG(t

¶(z) = exp

- (t-z) dt. (i .2.Lia)

However by multiplying this function by an arbitrary ratioii

function P1(z), which may possess poles only at the ends of

L, we do nôt disturb relation (1.2.2). Hence

¶(z) = E exp

-dt P1 (z) (1 .2.L4b)

is a more general solution of (1.2.2).

In order to deal with the inhomogeneous equation,we write (1.2.1) in the form

_(t) (t)

_____ - w(t) - (1.2.5)

Analogous

= _{dt -} (-t .1 .8)

Subtracting these formulae yields (i.i.L), adding yields

(-i I .5).

Now we Con-e to the definition of a sectionally holomorfic function. A function will be called sectionally holomorfic

when it is holomorf Ic in each finite region which does not

contain points of some smooth line L, while it is continuous up to L with possible exception of the ends of L.

It is proved in ref. I that (z) is such a function.

I 2 THE HILBERT PROBLEM FOR AN ARC

The problem is to find a sectionally holomorfic function

(z) which satisfies the relation

- c(t) (t) = g(t) (1.2.1)

on a smooth arc L, where G(t) and g(t) are given functions , which satisfy the H6lder condition and &(t) # O on L.

First we consider the homogeneous equation

= G(t) ¶(t). (1.2.2)

We solve this by trying to find a sectionally holomorfic

function ¶(z) without zero's in the whole complex domain.

Then in

F(Z)

is also sectionally holomorfic and has to

satisfy

Ein (t)1 - Ein r(t)_ = -in G(-t) (1.2.3)

on L. Comparing (1.2.3) and (1.1.Li) we find as a solution of

L1..

Hence again by (i .1 .Li.) we obtain

ç

f

g(t)

t

P ' ' " ₆

Zj

-2ici ¶_(t)(t-z) d + 1.2.

where it is assumed that W(z) is chosen In such a way that

the integral converges, P (z) is again an arbitrary rational function with the same retriction as P (z) in (i .24b).

In ref. I lt is shown that (i .2.6) aid (1 .2.Lb)

are

the

general solutions of the probleme (1.2.1) and (1.2.2) when

the behaviour at infinity is prescribed to be algebraic.

When the arc L becomes an infinite line, for instance

parallel to the real axis,the sectionally holomorf Ic function '(z) in (i .2.L1.) is cut into two separate functions ¶(z) and.

F_(z) which are analytic in the half planes S

and S,

situated at the + and - side of L. Assuming tat the integral

in (1.2.L.) is convergent for this line we see that the

solution of (1 .2.2) yields the "factorisation" of a function G(t) defined on L

G(t) = ¶_(t)/ ¶(t) (1.2.7)

into two functions W(t) and _(t) which are boundary values

of functions W (z) and. W_(z) regular and without zero's in

and S respetively and continuous up to L. It Is clear

that we have to take z in S or S_ when we calculate _W+(z)

or ¶_(z) with the use of (1.2.L).

The next step done in (1 .2.6) Is to form, under the assumption of convergence, the integral

(t) _{dt, z in S. (i .2.8)}

This means for an infinite line L that we "split" the Í'uncti g(t)/'_(t) defined on L

=

e(t) - e_(t)

(1.2.9)

into two functions e(t) and e_(t) which are boundary values

of functions

8

(z) and 8_(z) regular in S

ana

S

arid

continuous up o

L.

Sometimes it.is necessary to multiply G(t) by a simple

function in order to produce convergence of the integral in

(1.2.L). This will be demonstrated later on (para. 2.3).

The solution of (1.2.1) for an infinite line can now be

described very briefly as follows. First, factorize G(t),this

yields (1.2.5). Second, split the right

hand

side of (1.2.5).

Third, compare the parts analytic in the same half planes

with the result

W(z)

1 .3 SINGULAR INTEGRAL EQUATIONS

At last we discuss in this chapter the relation between

singular integral equations and the Hubert problem. In

chapter 7 we shall have to treat a singular integxal equation of the form

i(-t0) + k(t-t0)1 (t)dt = f(t0), (1.3.1)

where (t) is the unknown function and k(t) and f(t) are

given functions, k(t) is bouzided at the interval Jt < 2.

First we consider the t'dominantt' part of this equation,defned

as

1

,-T 1' (t-t

dt = r(t0), (1.3.2)

o

which is closely related to the theory of sectional holomoific functions. IntroducIng

+1

«z) = iT

I

?i:t:

dt (i .3.3)

-.i

we find from (1.1.5) and (1.3.2)

+ _(t) = r(t). (1.3.L)

Substitution of G(t) = -1 in (i .2.L.) shows that we may take as a solution ¶(z) of the homogeneous part of (i .3.L.)

¶(z)

(1_z2)2

The solution of (i .3.U) becomes by using (i .2.6)

«z)

₂

f

r(t)(-t2)

dt

+ 2iri(1-z )

where we have fixed the values of (z) by taking

T(t) ₌

_-I(i-t2)I,

t 1. Then by (1.i.L.)

(t )= ₂

7f(t(1;t2)

dt + AI. (1.3.7) ° ti(1-t)2 ¿

-Here we have chosen for the arbitrary polynomial P(z) the

arbitrary constant

A

i f

ç(t)dt, (i .3.8)

in order that (1.3.7) satisfies (1.3.2).

(i

6,

Now we return to (i .3.1) which we write in the form

t! dt =

r(t0)- f

(t)

k(t-t0)dt.

(i .3.9)

Then apparently each solution of

+1

+1 fç(t)k(t_r)dt(1_r2)ciT

1i(i_t)*

(T-t)

where (t ) is a solution of the dominant part and B is sorne

arbitray Sonstant, satisfies also (i..9). This can be veri-fied by devidin,g both sides of (1.3.10) by (t-t ) and

inte-grating from -1 to +1 with respect to t0. Instead of (1.3.10)

we may consider +1 B ri(i-t3

(t01t0)+[ t(t) K(t,t0)dt

(1.3.11) where í'(t0) = o, and +1 k(t-T) (i 2) _dT. (i .3.1 2) K(t,t0)=

i(1_t)(1_t2)*_/

(T-t0)

This integral equation is of the second kind and the known

function and the kernel are quadratic integrable. Hence the

theories on the Neumann expansion of the solution and the

replacement of the kernel by approximating kernels of a

simpler type can be applied.

(i .3.10)

Chapter 2.

THE WIENER-HOPF TYPE INTEGRAL

E Q U A T I O N.

We now shall give a treatment of Vìener-Hopf trpe intel

equations, which resembles the classical procedure. It

deviates however at one point since we shall not demand a strip of convergence of the Fourier integrals to be used,but

only a line. This is an inmediate consequence of the fact

that we start from the concept of the. sectionally holomorfic function.

2.1 TRA.NSFORMATION OF THE EQUATIONS

The following three equations are considered , the

homogeneous equation of the second kind

f(x) -

k(x-) f()d

O,

o

the inhomogeneous equation of the second kind f(x)

-

f

k(x) f()d

=

and the equation of the first kind

fk(x) f()d

=

where k(x-) is the kernel of the equations, h(x) is a

function known for X > O and f(x) is the unknown function. The equations are valid for all values

_{oo < X <}

l-oo hence we shall have to cieterminein equation (2.1.3) also the values of h(x) for x(o.

We introduce the notation Cor the Fourier transformation of a function f(x)

and for the one sided transforms

ff(x)e1 _{dx = F(X)}

(2.1 .L)

-1

f(x)ei

₌ o I _[f(x)e1 (

= F(A).

,rj

-c

In order to apply a 1ourier - transformation to (2.i.i),

(2...2) anci (2.1.3) it is sufficient to assume that

e' k(x), e

h(x) and e f(x)belong to L (-, +oo). On these assumptions the Fourier transform K(A) of k(x)

exists for Im A

= ¡

and the one sided transforms of k(x) and

r(x) denoted by H(A), F(A) and H(A), F_(A) exist for

Im A

= ji

and are regular in S with Im A

> ji

resp. S_ with Im A <

ji

Sy the convolution theorem equations (2.1.i), (2.i.2) and

(2.

.3) are equivalent to the following Hubert - problems

i«Xfl F(A)

= O,

F_(A)+I-

fE

K(A)F(A)=U(A)+H(A),

F(A)

K(A) - H_(A) =

holding on the line L of infinite length

where F_CA), F

(A)

and H

(A)

are unknown

we consider F_tA) - H_(A7 as one unknown

vVe

assume that the known functions

K in (2,1,7), (2,1,8) and (2,1,9) satisfy tion (, 2) on L, in which case we can

of chapter 1.

with Im A =

functions.In (2.1 .8) function.

())

and H

(X)

the Hölder

condi-apply the theory

2.2 SOLUTION OF THE HOMOGENEOUS INTEGPL EQUATION OF THE

SECOND KIND

We first consider the ifilbert problem (2.1.7) which

corresponds to the homogeneous eçuation (2.1 .1) and assume a

strip ¡9 of convergence for the integral (2.1 .Ll) for k(x).The

case of a line of convergence will be discussed at the end of

this paragraph. In ¡3 we choose a

line

L parallel to the real

axis in the A plane, on which no zero of

i-TE

K(A) lies.

The line L with a positive direction (viz. Re A -+ +) defir

the half planes 8 and S_.

Of course we have no knowledge a priori whether the

Fourier-transformation of (2.1 .1) actually holds on L.However,

if

the transformation holds on some line L

in ¡3

we may

de-termine this line afterwards and construct the solution to

w_(A) +

_K(Afl

(Xa)n

(A-b)

where a is a point in 2 and b a point in S_, then ( 2.2.1) is satisfied. ive can write down at once the solution of

(2.2.3) with the aid of (1.2.4) and the discussion above

(1.2.7) 'n in[i- K(7fl IIJ exp - .- ¡ " _{diP(A),A in S.f.} -

ii

(r-X)

Hence also the functions

F(A)=(X-b)

¶(A),

F_(A)=(A-a) ¶_(X) (2.2.5)(2.2.6)

are analytic in 2 and S respectively and are solutions of

the Hubert problem (2.1 .7)

We wish to obtain functions F (A) and F(A) which are

Fourier-transforms, this restricts P(A), because for Re A - ± F (A) and F_(A) must tend to zero.

Now suppose that, for the case of a strip

_ß

of convergex

of the transform of the kernel k(x), the function

i-f

K(X)1 has zero's A1; A2;.... in the S part of , arranged according to increasing imaginary parts. We investigate the functions

F(A)

F(A)

(A-A

F1JA)_

(x-A (2.2.7)(2.2.8)

'

1' \ 1'

cc

These functions are regular in which are situated each on one parallel to L but lies between

F (A) is holomorfic in S

homorf Ic in S1 follows rorn

(2.2.3)

(2.2.L)

the regions S + and S

side of a lin L , which is

the zero's A anL A2. That

obvious and 1chat F (A) is

equation (2.1.7) bcause

In order to be able to apply the theory of para. 1 .2. it

is necessary that the integral in (.2.) converges for the

line L of infinite length. This is the case when

lin in {i K(Afl

= 0.

(2.2.1)

Re A-+±00

On first sight this seems to be true because K(A) is a

Fourier-transform and hence K(A) - O for Re A -+

.

It

re-mains however possible that

urn lni-.#T

K(A)}- um ini-4

K(Afl=-2rsii (2.2.2)

Re A-+ Re A-+-cc

where n is an integer. If we consider instead of (2.i .7) the problem

lo.

F(X) and i-r K(Afl must have the saine zero's in the

part of ß. So it is possible, by moving u L parallel to the

real axis, to construct new functions F

_+(x)

and F

which are of lower order for JA! an. which satisfy

equation (2.1 .7) on new lines L1 , L2 ...

In the case that F(A) and F(A) are not

Fourier-trans-forms, it is possible that F , F or F , F .... will be.

I-f-. Ir

¿t.

¿-,-.

So the most direct way to obtain ail solutions is to take the line L above the zero with the largest imaginary part within

the strip of regularity of K(A) and to evaluate f(x) by the

inverse transformation. However it may be more convenient

to take for L, if possible, the real axis, in view of the

evaluation or approximation of the integrals in (2.2.Li) and

to translate L afterwards into the correct position. Vhen the transformation of k(x) is only permitted on a

line L, while {i-'T7z K(Afl has no zero on L we can use the

above theory. If in this case -T K(Xfl does possess zero's

on L we can also solve the problem, this will be discussed

in para. L..L1..

2.3 THE INHOIiOGENEOUS EQUATION OF THE SECOND KIND AND THE

EQUATION OF THE FIRST KIND

VVe are now in a position to solve the inhomogeneous

equation (2.1 .8). It is assumed that the line L, on which we consider the Hubert problem, is within the strip ß and above the zero of ti-.1r K(Xfl with the largest imaginary part within ß.

First we consider the case that the integer n defined in

(2.2.2) is positive or zero. We select a suitable solution '

the homogeneous equation (2.1 .7) from the set provided by (2.2.5) and (2.2.6) by taking for P(A) sorne polynomial of degree n without zero's on L. Denoting this solution by Y(A) and y(A) we may assume

hrn y(A)

= 1 .

(2.3.1 )

Re A±o

-We can rewrite (2.1 .8) in the form

IF_(Ä)-H_(A)}

(A)

H(X)

Y(A)

- Y(A) - Y(A)

This equation has by (1.2.10) the solution

Y(A)

F(A)=

- 2ai

/

Y_()(-X)

d, A in

d, A in S_.

In this case (n o) the solutions (2.3.3), (2.3.Li.)

can be

interpreted as Fourier-transforms and we find the solution of (2.1.2) by the inverse transformation.

For n < O we cannot find functions (A) and *'(X) with

the property (2..1) and in general we arinot interpret in

this case (2.3.3) and (2.3.Li-) as a Fourier transform. Only for special functions H(7?) (2.3.3) and (2.3.L.) will tend to

zero for Re A

±,

We now shall treat equation (2.1.9) which reads

F(A)

K(X)-H(X) =iç(A). (2.3.5)

This equation differs from (2.1.8) in a rather significant

way, viz, the function K(A) which is here the factor of

F+(A) tends to zero when Re A -+±oo. This mei that we

can-not use (i .2.Li.) because the integral will can-not exist for the

line L which is infinite. We assume the existence of a

function K*(X) with the properties

um K(A)/ K*(X)

= -i , (2.3.6)

Re A -± and

IM(A)i I = IK(A)/K*(A)_1 I < e. < i, (2.3.7)

for A on L. Further we assume that the function K*(A) can be "factorized" by inspection in the following way

K*(X) = K(A)/K*(A), _(2.3.8)

where K(X) and K*(A) are functions which behave algebraicly at IAl

-

and are regular and without zero's in S and S_ respectively. Then we consider the equation

*[ x(A) M(X) - x_(A) = H(A) K*(X)

which is of the type (2.1.8),

while the integer is zero. Hence the problem can be treated in the way. The solutions of the original equation (2.3.

b e corne

F(X)=X(X)/K(A)

, F_(X)=X_(A)/K(A). _(2.3.10) When e, defined in (2.3.7) can be made sufficiently small we can use this method for obtaining approximate solutions.

This will be demonstrated in chapter 3, where we discuss a

shrink-fit problem.

(2.3.9)

n (2.2.2)

indicated 5) then

1 2.

2.L- EXAMPLES

First we consider an equation which is solved in ref. 2 by the method of Viener and Hopf

f(x) = m

[e_1X

f()d.

(2..1)

We assume O < m < . After transformation we find

(p') ₂ + F (X) o. (2.L.2)

+ (i+X)

The zero's of

- 2 are X1 = - (2m-1 ), X2= +(2m-1 ),

(i+X )

where we assume Im X > O. The strip of convergence for the

Fourier-transformatign of the kernel is -1 < Im X < + i In this strip we choose the real axis as the line L, on which w have to solve the Hubert problem (2.L.2). On this line

1

-

>0 and hence the number n in (2.2.2) is zero. The

1 +X integral in (2.2.L) becomes +oo inh- 2m +,

f= ln (77-X) inh-

_i+2

/

-r _7? in

iiÄI

f

H-

ln (77-X)2m

(i+î

₎

- m

-

j (1+772)(i+772_2m)

This last integral can be calculated by residues, where we

have to take care that our contour does not enclose the branch point 77 = X. Substitution of the results in (2.2.L) yields

(X+i

= (-X1 . p(X) , X in S

F(X)

₌

_(X-i)

. p(X) , X in

These two functions do not tend to zero for Re X -

±o

and.

hence cannot be interpreted as Fourier-transforms. However,

by translating L upwards over a distance between Im X2 and

(7+i' p(Â

F(A)

₌

(X-1)(A-2 - (X-i)

It is obvious that P(x) must be a constant. Br the inverse

transformation we

find

the solution of (2.L.1).

f(x) = C fcos ((2m-1) x)

x>O,

(2rn-1) _(2.L.E

f(x)

=Ce

X

,x<O.

As a second example we consider the equation co

f(x)-m

f

f(E)

2 = h(x), m

. (2.L.7)

i+(x-)

This equation cannot be solved with the method of Wiener and Hopf because the Fourier-transform of the kernel converges

only on the real axis.

Fourier-transformation yields the equation

+

F(X)=H(X)

+ H_(X). (2..8)

The function I-mze is positive on the real axis and

hence n (2.2.2) is zero. The solution of the homogeneous part of (2.L..8) is obtained from (2.2.L) and this solution

can not be interpreted as a Fourier-transform. However, by

taking P(A) 1, we may obtain the solution of the

inhomogeneous equation by the procedure outlined in para.2.3. In the present case

Y(X)=exp -

T1;f

_,

A in

s.

(2..9)

00

Then the solutions (2.3.3) and (2.3.Li.) may be interpreted as

a Fourier-transform. The solution of (2.L-.7) can now be

evaluated by the inverse transformation.

A SHRINK-FIT PROBLEM FOR A HALF

INFINITE RANGE OF CONTACT

We consider an in'inite elastic tube which is shrunk onto a semi infinite rigid shaft. In dealing with this problem we aim exclusively at the contact pressure between shaft and tube, for which an integral eauation of the Wiener-Hopf trpe is established. The integral representing the contact pressux is approximated numerically by a method equivalent to the one developped in ref. L1., which rests on approximating the kexl of the governing integral equation.

3.1 FOR1UITION OF TLIE PROBLEM, DETERMINATION OF GREEN'S

FUNCTION POR TI TUBE

Let be cylindrical coordinates such that the

axis_coincides with the axis of the tube (fig.3.1 .1). Let a

and b

be the inner ari outer diameter of the tube,respectiy. Assume that the uniform radial shrinkage of the tube is

The stress distribution in the tube for the case to be considered here is then governed by the following boundary conditions:

Chapter

3

where Or() and

T are the normal and the tangential

stresses_rspectivey and u is the radial displacement. ie

seek o (a,x) for O < x < appropriate to the stress

distriEution governed by (3.1.1), (3.1.2) and (3.1.3). To

this end we note that this normal stress roust satisfy the

integral equation

for''

(3.i.L1.)

for

r=b,

-

00

X < + co,

Trx = O, 0r = O, for

<O,

Trx=OOr=O

where () is a Green function which will be defined presently.

for

Fig.3. .1 Infinite elastic tube ahrk ontn a semi-infinite zigid shaft.

The function () is the radial displacement of the

inner wall of the tube corresponding to the following singular loading conditions:

for

co<<+oo5 T

=0,cr =0,

rx r -

co < X < + co,

T_rx 0,cr ò(x), (3.1.6) r for

T)1

0r' x'

°o

0,

(5.1.7)

in which á'(x) is the delta function of Dirac. The rotational

synmetry of this problem suggests an approach by means of

Love's stress function (ref.

io)

p(r,x) which satisfies the

differential equation

2

22

d2ç1

(_+1__+)

rp =0.

Or

ròr

0x

The associated stress field is given by

Ox

= -4: [

(2- -Ox 2

:

Or 2 Ox (s.l .5) (3.1 .8) (3.1 .9)

16. = - [vLi - -_]

ÇD (:,3),

òx rôi' o2 - -. --- [ (i-v)

- .-]

(r,x), Ox i iA -e (r,x)d, bL1

< Im A

<

co

exists and (3.1

.8)

is carried into

(!

+ i - A ,A) =

o.

&r

rÒr

(3.1 .9)

where

t'

designates Poisson's ratio, which we take equal to

0,25 in numerical calculations.

The corresponding radial and axial dispiscements appear as

- - - _(i+v)

o2)

(3.1.10)

u(r,x)=-E

--ô rO x = (1±v)[(1_2v) + + ] (3.1.11) a Òr

ròr

Thus we need to determine a function p which meets

(3.1 .8) and is such that the stresses 3.1

.9)

conform to

(3.1.5), (3.1.6) and (3.1.7). The desired kernel in (.i .1i) is then obtained from

- -

- - - (1+v)

o2()

u (X) = u(a,x) = - _{E - -}

-o _{òrÒx a,x}

We now establish with the aid of the Fourier

-transform. Let

e1X(,)

be absolutelr integrable with

respect to X in the interval

( , +)

l'or < Im A <

then the Fourier transform

+00

(3.1 .1 2)

(3.1 .13)

(3.1 .1)4)

The general solution of (3.1 .1)4) admits the representetion

= . (A)K(A) + A2(A)AK1 (Ai) +

+ B1(A)I0(X) + B2(X)2cI1(A), (3.1.15)

Bessel-functions of first and second kind, respectively, Throughout this work we shall use the definitions for these functions

given in ref. 11. The arbitrary functions

A (X),

A2(2L),B (X)

and B (X) are to be determined consistent with the transorms

of th boundary conditions (3.1.5) and (3.1.6). This process

yields four linear equations in the four unknowns A1 , A2, B1

Thus (,A) is comletely determined.

2 _{Upon in9ersion of and substitution in (3.1.12)}

we obtain after some computation

-(i-v2D

f

N(,X)

(3.1.16)

u0()_

nE

_XD(,,X)

00 with =

X[K1

(À)I0(À)

+ K0(X)I1 (2)]2 -[2(1v) + X

[(K1(x)I1(X)- K1(X)I1(X)]2-

, (3.1.17) 2Lb = [ 2(1-v)

+][K1(x)I0(A)+K0()I1(x)]2

+ Aa + X { 2(1-v)

+][K1 (A)I0(A)

+ K0(A)I1 (AS)]2

-Ab

- [K0(xS)I0(x) -

K0(A)I0(AS)]2

-[ 2(1-V) +

A] [

2(1V)

+ Ag]

Aa Ab

[K1 (XS)i (Xi)-

K1

(AS)]2

[(Iu)

¡

_{+ .}

(3.1.18)

S

3.2

SOLUTION OF THE INTEGRAL EQUATION FOR THE CONTACT

PRESSURE.

At this stage it is convenient to introduce the dimension-less variables

2

x=/,

b=S/,

p(x)_ 2(1-v )

(3.2.1 ) Then (3.1

.L4) can

be written

i 8. co =

i_

rk(X)

_P()

where k(x)= NÇ1 ,b A cIA. 00

The integral equation (3.2.2) is of the Wiener-Hof type.

We first investigate whether the Fourier transform K(A) of

the kernel k(x) possesses a strip of regularity in the con)ex A-plane. From (3.2.3) we obtain

j NÇ1 .ò.X

K(A)

= X D(1 ,b,X

Using known expansions of the modified Bessel-functions, we

find and i b2(i+v)

+ (1-v)

-

2(1-v2)

(b2-1)

K'(0)

= o

I«X) =

+ o(A2)

as Re X

'±co

(3.2.2) (3.2.3) (3.2.L) (3.2.5) (3.2.6) (3.2.7)

The function K(X) has no poles on the real axis of the A -plane, because otherwise a cosine loading of the innerwall of the tube would cause infinite displacements. A point of singularity which can be expected, however, in view of the

branch points of K (A) and K (X) Is the origin A = O. This,

however, Is not thS case. Coisider for instance a typical

part of the function N(1 ,b,A),

K1(X)10(Ab) + K0(Ab)11(A). (3.2.8)

Using here a representation of K0(A) and K (X) in the

neighbourhood of the origin (ref. 11), we 'ind that the

logarithmic singularities of K (X) and K1 (A) cancel each

other. Hence K(A) possesses a Strip of règularity around

the real axis. This means that the kernel k(x) decreases exponentially for x -. ± co. Because the lefthand side of

(3,2.2) is a constant for

X >

O we take Im A > O and sufficiently small when we apply a Fourier-transformation to this equation. Then we obtain the Hilbert problem

P(A) K(X) - O(A)

- rj

A

valid on a line L just above the real

axis

in the A-plane.

This equation is of the form (2.3.5) and will be treated

by

the sanie procedure. First of all we have to find the function

K*(A). Because k(x) is an even function, it is clear that

K(A) is also even. We now consider in correspondence to ref.L. T (A2)

2

1

2 (3.2.10)

(A +s )2 T2(A )

where s is a rea constant and T (A2) and T2(A2) are

polynomials in A

with the same cerm

of the highest degree.

From this we see that y*(A)_

TT

+ 0(IA!3) as Re A _+±co, (3.2.11)

where fIni AJ < s. Hence by (3.2.7) the leading

term of

the

asymptotic expansions of K(A) and K*(?) is the

srne.

Next we

choose s and the coefficients of T1 (A )

and

T2(A ) so that

IM(A)-i I

= ¡K(A)/K*(A) -1 I < e < 1, (3.2.12)

where e is some prescribed positive quantity. That this is

possible follows from the theorem of Tschebyscheff

on

the

approximation of

continuous functions by rational functions (ref. 12, pages 55 and 65). Then we have to solve in

accordance to para. 2.3.

x(X) ií(A) - x_(A)

'

Krn(A). _(3.2.13)

A

The homogeneous part of (3.2.13) yields as a solution which satisfies condition (2.3.1 )

W(A)=

exp - d77, A in S.

_(3.2.1L)

Herewith the inhomogeneous equation has the solution

K*(77)

x(A)=

()-A)

di7, Am S.

(3.2.15) and the solution of (3.2.9) becomes

P(A)

=

x(x)/K(x),

(A) = x(A)/K*(A). (3.2.16)

20.

?tg.3.2.I. Contour 01 integration.

indicated in fig. 3.2.1 . Because K*(77) and ¶_(77) are analytic in S_ we find with (3.2.16) Km(0) (X) X

-

+ +

¶(0)

A . hence Ka(o)

w(X)

p(x) -

₍₀₎

A i(A)

(.2.1

8)

Because of the occurrence of _W+(A) in the integrand, this

integral is too involved for numerical calculations. It can

however, be expected (ref. 14) that, if e is sufficiently

small, we obtain a good approximation p*(x) of p(x) when we

put w(A) and

¶_(A)

equal to unity for all A, this yields

-iAx

p*(x) =..i± K*(o)

_f

e dA

2ii

-A Even when we choose more specii'ally

K*(0) ₌ K(0) we can easily show

p(x) - p*(x) 0(x°),

x -

O and p(x) - p*(x) - o

X -'

00. Consicer

p(x)_p*(x)

= 1 Ka(0) r

(x)

2i -X

K(X)

E

_w(o)

1}d7t. (3.2.19) (3.2.20)

(3.2.21

(3.2.22)

(3.2.23)

Then it can be shown by estimating (5.2.1L) that the function

between brackets

under

the integral sign in (3.2.23) is

absolutely integrable. This implies (3.2.21) ana (3.2.22).

5.3

N1JhERICJL JJALCULATIOI OF TH

s:-INK-FIT

STRESSES

As has been suggested in para. 3.2. we shall use instead

of the exact function

K(A) (3.2.1)

an approximate function

y*(A)_

(A2+s2)

b0+b2X2hX.+b6A6

where the coefficients are given in table I for several values of the dirrensionless outer diameter b. The values or

s are assumed, while the values of an and b are computed

TAJ3LE I, coefficients of K'(X)

and.

by collocation. The relative deviation of K*(A) from K(7t)

is

less than L% for the whole real axis. By determining the roots of the polynomials in (3.5.1) we can easily obtain the factorization (2.3.8) of K*(7). Using these functions in the

integral (5.2.19) we find the values of table II for the approximate shrirJ-f it stresses.

TABLE II, p(x)/5.

From (3.2.19) we

obtain

by (3.2.2) and (3.2.22)

p(x),

p*(x)

= -

2(i-)

(b2-1) - x b2(1+v) +

(i-(3.3.1)

+const.asx-O, (3.3.2) b s 2 a a6 b0 b2 b 66 3 1.25 1.00 -0.725 0.290 0.00553 1.00 -0.756 0.3314 0.00553 2 5.00 5.11 2.32 0.332 0 1.00 0.3142 0.332 0 1.5 6.00 9.12 0.2144 0.0265 0 1.00 0.0178 0.0285 0 1.2 10.0 30.9 0.0963 0.00303 0 1.00 0.00685 0.00303 0 b X0.03125 0.0625 0.125 0.1675 0.25 0.375 0.5 1 3 2.83 2.19 1.75 1.1i2 1.30 .25 2 2.140 1.80 1.39 1.01 0.921 0.9814 1.5 1.77 1.18 0.7143 0.538 0.619 0.658 1.2 1.1i1 0.716 0.2142 0.133 0.1149 0.253 0.308 0.3214 0.3214

22.

,->i 2 , 2

p(x), p*(x)

1V i

b -lj as x - co

+ (ivfl

Formula (3.3.3) agrees with the result which we obtain from the elementary theory when an infinite tube is shrunk on to an infinite shaft of uniform diameter and contact occurs for their whole length.

.L. THE LIMITING CASE OF IINITELY THICK TUBE

We now consider the limit as b oo. We shall again have

to determine Green's function u0(x) of the inner wall. In

this case we shall have to take the unknown functions B (X) and B9(X) in (3..i5) equal to zero in order to avoid

singuíarity at infinity. This problem was discussed in ref. 13 so that we may write down at once the integral equation with the dimensionless variables (3.2.1)

00 I

k(x-)

_{p() d,}

(3.14.1) where +0G -jAx 2(f7) d?.. e

k(x) i

jt

X2K(IX) +[X2+2(1-v)

Ic(IAI)

c0 K(X)

KA)

-A2K( ¡Xl) + [x2+2(1-v)}K(IAi) (3.L.2)

Here a complication arises. The kernel k(x), for finite b, possesses a strip of regularity around the real axis in the X plane. This is no longer true in the limit as b -. c The function

(3.14.3)

is not analytic.

Proni the point of view of mechanics, however, it is not necessary to consider these difficulties in detail. For it is evident that the shrink fit stresses of an infinitely

thick tube are the limit of the stresses produced by tubes

of increasing thickness. Then, from continuity consideratis we can calculate the approximate stresses of the infinitely

thick tube by approximating the function KÇX) in (3.14.3).

Also the asymptotic relations (3.3.2) and (3.3.3) remain

valid.

Now the course of the calculation is the same as before; the coefficients of (3.3.1) are

s a0 a2 a a6 b0 b2 b b5

the error in this case being less than 1 .5%. We find for the stresses

x = 0.0625 0.125 0.25 0.5 1

p*(x)/

_{= 3.13}

2.L4

1.97

1.73

_1.58 1.50.

3.5

DISCUSSION OF THE RESULTS

The contact pressures are plotted in fig. 3.5.1, which

shows p*(x)/ for various thickness ratios b/a. The ratios

chosen are b/a = 1

.2;

1 .5;

2; 3 and.

. The stresses tend to iníinity as x + 0, the order of the singularity being

independent of the thickness of the tube, as is clear from

(3.3.2). The interesting region for the variable x, inwhich

the stresses change rapidly, is approximately O x 1.

Here we see from fig.

_3.5.1

that for b/a =

2, 3

and c the

shrink-fit pressure is a monotonically decreasing function

for increasing values of x ( 1. However, for b/a = 1

.2

and

1

.5

there exists a minimum which becomes negative for

suf-ficiently thin tubes. Since negative shrink-fit stresses cannot exist, this means that our theory ceases to be

applicable. This behaviour is to be expected from the thecxy of beams on an elastic foundation.

16

.2

08

0,4

Flg.3.5.1. The contact pressux'e p(x)/8 for various thickness ratios b/a.

23.

2L1.

Finally we want to make a remark on the accuracy of the

numerical results given here, On the basis of observations made in ref. L the relative deviation of the approximate values p(x) fran the exact values p(x) ought to be

approxi-mately one half of the corresponding relative deviation of

K*(X) from K(A). For this reason the calculated shrink - fit

Chapter L

THE HOMOGENEOUS FIRST ORDER

INTEGRO-DIFFERENT IAL EQUATION

OF THE WIENER-HOPP TYPE.

We shall discuss an integro-differential equation with first order derivatives of the unknown function, which occur

both under and outside the integral sign. Equations of this

kind sometimes arise in physics (ref. 5). In this chapter the method of solution is exposed for this simple form,it is an extension of methods described in chapter 2. A solution can be obtained even when the kernel k(x) decreases

algebraicly for lxi -+co while at the same time the solution

does not tend to zero for x - + oo.

The following cases are treated separately:

Case 1. The kernel k(x) decreases exponentially ik(x)l<e with a > O, for lxi - co and the solution has the asymptotic1

behaviour lf(x)i =Û(eC),

with

y < a for x -. + co

Case 2. The kernel decreases algebraicly k(x) =ixl n > i for lxi - co.

Whenever we need explicite information about the

singularity of the kernel

at

x O we shall assume that k(x)

is an even function with respect to x (para. L.2.).

L.i

THE GEIthiRAL EQUATION, CASE i

Our integro-differential equation has the following form co

a0 f(x)a1 fT(x)

=01

f()+b1 f'()

k(x-)de. (L.1.i)

When we have found a function f(x) for x > O which satisfies this equation we have to solve a simple first order differen-tial equation in order to find f(x) for x < O. From the assumption of the exponential behaviour of k(x) at infinity

we find

la0 f(x)-fa1 f'(x)i < C x - - co. (LL.i .2)

Hence if we consider the left hand side of

(1.í .1) as

_one

function h(x) we can apply a Fourier-transformation to this

26.

A remains

on a line L with Im A = , within the atri;

_ß,

y

< Im X < a.

Applying a Fourier-transformation to (L..1) we obtain

H_(A)+F(X)a0-ia1A - 'T K(A)(b0-ib1A)}= 4a1-b1 K(Afl

(L.i

.3)

where

H_(A) =

f

f(x)+aA f?(x)}eAX (L4..1 .L)

00

First we consider the related homogeneous equation

y(X)+Y(X)

a-ja1X - K(X)(b

-ib1

X)} = O.

_(L,.1.5)

This means that we seek solutions of (L..i.i) with

f(0) = O.

We assume further that L has been chosen in such a way that

there lies no zero of the factor of

Y+(A)

on L.

Analogous to the

procedure in para 2.2,

we consider

the

expression

(X-a)

in G(X)=ln {.-4a0-ia1A-

K(A)(b_ib1X)

(L1..i .6)

where a is in and b in S. The integer n can be

determined in sich a way that the principal value of in G(X) for ReX-p on L tends to zero.

We then

find,

for the solution of

(L.i

_.5)

Y₊(A) =

L (A-b)'

(A) P(A),

(L.i

.7)

a1 +

Y_(A)

= (A-a)

A) P(A), (L.i.8)

where

-i

rlflG)

exp

_-

dï7, A in S

and P(X) is

an

arbitrary

polynomial. Again we consider the zero's, A ₌ A (m=1

....

q) of

- 'T

K(A)(b0-ib1A)

trip

ß and above the line L. Then also the

Y(A)

y(A)

q - (.1.11)(1.i.12) JI (A_Am) q Ji _{(A_Am)} m=1 mxl within the s

functions

(A) -(LL.l .9) (L..'!.i o)

are solutions of (L..i.5), but now on a line L

with

Im X <

Im A

_{< ß.}

In the case (n+q) I we find a soiBtion of q

(Lj.î .5) which can be interpreted as a Fourier-transform,while

an arbitrary polynomial

P(A) of degree (n+q-l) can be taken

as a factor.

Now we return to the irthomogeneous equation (Lp.i .3) and

try to obtain a solution. The general solution can then be

found by adding solutions of the homogeneous equation. We

choose some solution Y(X) and

Y_(X),

assuming P(X) -i in

(L.i

.7)

and (L4..i .8) in order to avoid zero's of Y(A) in

S(A), from the set provided by (L.1.l1) and (L..1:12) and

wfite instead of (L-.i.3)

H_(X) (x)

f(o) a1-b.1 2n K(Afl

X - -

e (x)-.e

(A).(L..I.I3)

Y_( )

Y(A)

Y_(A) +

Here we have introduced the functions e (A) and

e_(x)

which

are discussed in para. 1 .2 and which reult from splitting

the right hand side of (L-.1.13) into two parts regular in 5 and S respectively. They are determined within an additive

term consisting of an arbitrary polynomial

and

can easily

be

calculated without recourse to the general integral

representation (1.2.8),

in terms of

Y(X) and

Y_CA). By definition we have

Y(A)+Y(X) k0-ia1 A - i.f2 K(A)(b -ib1 A ) = O (Li..i .1L4.)

hence

K(A) (a-ia1 A)

Y(A)

=

(b0-ib1A)Y(A)

+ Y_(A)

}

(.i.i5)

Combining (L4..i.i3) and

ib

_o

in b, we find

I

(L..i

.1 5) and assuming

for instance

since Y(X) have no zero's in S

b1

y(-i

b0

(.i.i6)

e (A)-

i(o)

+

(b0-ib1A)

Y()

+

Y(-i

) +

Q(A) ,

(.1.17)

ib

where Q(A) is an arbitrary polynomial. The case

_-

_b

°

in S_

can be treated entirely analogous. The solution of (L.i.13)

and

hence of

(L.i .3)

becomes

r(o)

(a1b0-a0b1)

28. f(o) _a1b0-a0b1- b1 Q (A)Y_ (A) *f(b0-ib1X) (4.-i .1 8) b Y (X)

F(X)= -(A)y(A)=-

f + - Q(A)y(A). [r(b -ibi) o 1 + b1 (L.i.19)

In this result we may take Q(X) O because this term only

furnishes a solution of the homogeneous equation. Aparent1 when Y_(X) is bounded at infinity we may interprete (1.i .1 8 and (L-.1.19) as Fourier-transforms.

L.2 CASE I WITH a0 # O, a1 = O ASD b1 * O

Equation (L..1.3) now reads

H(X)+F(X)1a0-T

K(A)(b0-ib fLO) b1 K(X) (L.2.1) and the homogeneous eauation

Y(A)+Y(A).a0-f

K(A)(b0-ib1A)= Y_(A)+Y(A)T(X)=O,(L.2.2)

where we have introduced the abbreviation T(A). There is a

difference between the equations and (Li.2.2). In

(L..i .5) the factor of F (A) is of the order A when

Re A -' ± while in (14..2) the asymptotic behaviour of this function, T(A), depends on the behaviour of x(X) for

Re A -+ ± . We now remind the assumption that in such cases

we shall consider only kernels k(x) which are even functions of x, hence also K(A) is even in A.

The asymptotic behaviour of K(X) for Re X -. ± depends

on the behaviour of k(x) in the neighbourhood of X = O. We

assume k(x) to be continuously differentiable for O < e1 < x for each e1. We consider several possibilities for the

behaviour of k(x) in the neighbourhood of X = O.

Let k(x) be continuously differentiable for O x <

e1 < _then

um

K(A)

=IAim

, i < m. (L1..2.3)

Re A-±oo

Let k(x) be the sum of a logarithm rd a continuously

um

AIAL1

Re

A-+±

e) Let k(x) be the sum of x

(o <

6' < i) and a continuously

differentiable function for O X < e2 then

um

IÇ(X) BIAI_1 (L4..2.)

Re À-*±

The formulae (L4.2.3) and (L.2.5) follow directly from ref.

1L. while (Li.2.L-) can be deduced by subtracting from k(x) the function C K (x) where K (x) is a modified Bessel-function and C some agitable consant, then we arrive again at case a.

The difficulties which can occur in the cases a, b and e are concentrated in finding appropriate functions aniogous to (Li..i.6). We shall now state these functions.

a) In this case the factor T(?) of YÇA) in (Li..2.2) tendB

to a0 for Re X -. ± , hence instead of (L.i .6) we can use

in G(X) = in T(X) a0

(X-b'

(i-6')

T(A)

(A-a)

2 in G-(A)=lnE (i-*6')

ib1B

_(A)r+

₂ (L.2 6)

T(X) tends to a0 ± i A b1 for Re

A ,

± co we take

a

j*[

Ab n- i in G(A)=1n T(X) X-a

2i

a-i2'Ab

(-)

,(L.2.7) a0+i Ab1

T(A) tends to ± i b. BlAt6', O < 6' < i for Re

A-±co

we take

(L.2.8)

The values of the multivalued functions which are the factors of T(A) are to be fixed in such a way that the principal

value of in G(X) for Re A - ± co on L tends to zero.

The essence of the fornilae above is that we have multi-piled T(X) by functions which can be factorised by inspection

into functions reguiar and without zero's in S

and S.

We

find the following soiutions for the homogeneos equation

( L1.. 2.2)

a) Y (A) =

_(L.2.9)

+ (X b) n

(x)

P(A)

30.

b)

(a0+i Ab1)

_(L.2.11)

i

b

2ai

in

(ai

Ab1)

PCA) a

_L.2.i2)

y_(A) = e) = n

(i±6')

(A b 2 ¶(A) P(A), y_(A) a'

where ¶(X) is defined in each case by the integral (Li..i

.9)

with the corresponding G(A).

Following the treatment of the preceeding paragraph below

form (14.1

.9)

we arrive again at formulae analogous to (L.i.i8)

and

(14.1.19).

The case a =0, b =0, a0 # 0, b * O is the homogeneous

Wiener-Hopf euatioi with exponenially decreasing kernel and

is discussed already in chapter

2.

14.3

THE E1AI EQUATION CASE 2

The treatment given in para. 14.1 is also largely applicable

to the case of equations with kernels which are only

trans-formable on a line L. However, there are differences, for in

the latter case we cannot translate L and hence (14.1.11) and

(14.i.12)

cannot be used. Further if there is a zero of (14.i.10)

on L this cannot be avoided by translating L slightly.

We now discuss a method to obtain solutions of (14.1 .3)when

(14.i.10)

has zero's on L. To demonstrate the procedure we shall treat the case of one zero A = p of the first order. An extension to several zero's and zero's of higher order does not offer principal difficulties.

From

_(14.i.11)

and (14.i.12) we see that a zero X of (14.1.10)

iXmX

m

introduces a term of the form c, e In the solution of the integro-differential ecuation. In our case with a zero y on the line of transformation it seems reasonable that the

solution of the integral behaves as c9e for x -+ + cc. Hence

we try to find a constant A in order that the function -ipx

v(x) = f(x) - A e -. 0, x - + co

o

(L1..3.i)

The integro-differential equation for v(x) can be derived easily from (14.i.i), 'e find

av(x)+av'(x)=

_{Îbov()+biv'(}

k(x-)d

-O cc

(14.3.2)

-

A(a0-ia1v)e1 +A(b0-ib1v)

f

e' k(x-)d

, X > 0. (L..2.i

3)

(L1.2.1L)

are

where

W_(X) a1 (Ä-v)V(X)

-n

Y(X)

(X b\

_a'

Under the assumption that V is a zero from (L..1.1O) we see

that the last two terms of the right hand side of (L..3.2)

cancel each other for x - + Applying a

Fourier-transfor-mation to (L1.3.2) we obtain

(A)-i-V(A) a0-ia1X - K(X)(b0-ib1 X)} =

v(O ) Al

= + ía -b. '1 K(A) ía -ja v- K(X)(b -ib vfl,

i

*[(Xv)

O I O I

where

_v(O)

is the value of v(x) when x tends to zero throi

positive values and

w(X)

f

(3.)

The solutions y(X) of the homogeneous equation

a0_ia-if K(X) (b0-ib1 X)

Y_(A)+Y(A)

(A-v)

-

°

(L.3 3)

(L.3.6)

a0-ia117.-*Tr K(Îi)(b0-ib1ri)

(l?_a)n

1

in ii

i

(n-v)

(n-b)

4.3.7)

Herewith (L.3.3) can be written in the form

[y(0)

_{ía -b}

K(X)}+

Ai

ía

Y(A)

i

Y(A)

-

YA)

I

r(Xv)

-ia1v - K()(b0-ib1 y )}] =

0(X) -

(L.3.8)

where we have introduced G(A) and

O(A)

which result from

the splitting of the right hand side. Under the assumption

that

p = i b ¡b ,

for instance, in S

we find analogous to

the procedure ii para. Li..1

ív(o)ò

(A-v)--Ai(ò

-ib1v)}E1-Y(X)/Y(p)

i. P(X)y(X)

1 +

.T

(À-v)(b -ib X)

a1 (A-v)

32.

(a0b-a1b0) Y_(X)

w (A) _{(A_v(ofl(b_ibX)}

(b0-1b1X)Y(p)

-.v(0)ib1 a1(A-v)-A a1(b0-ib1vfl+P(X) Y_(X). (L..3.iO)

When n > O we can interprete (.3.9) and (L.3.1O) as

Fourier-transforms while we can admit an arbitrary polynomial of the

degree (n-1). For the ease n = O we can consider these

functions as Fourier transforms when y(0) a

(L.3.11)

'

yC)

From (Li.3.9) we observe that we have to choose A in such a

way that the singularity X = V disappears, this yields

-'T p(v) y (y) y (p)

A _-

_(Y(p)Y(t'))

(.3.1 2)

Hence we have determined a solution of (L1.3.3) and by the inverse transformation and (L.3.1 ) also a solution of (L.i .1) with in general r(o) # O. An analogous treatment can be given

for p = -i b,,/b in S.

If we want t solve an equation with several and higher

order zero's of (L4..î.iO) the analysis becomes much more

com-plicated. Assuming zero's X= y (r=î ... N) of the order p

we shall have to consider the ntegro-differential ecivatio

f or the function

N

r

-ivx

v(x) = f(x) - E ( E _Arm xm)e r (L.3.13)

r=i m=1

By an analogous reasoning as before we can obtain relations for the coefficients A

The discussion of pa. L.2 can also be extended in the

way of para. L..3 to kernels which decrease algebraicly at

infinity, this we shall not do here. However, to show the

applicability of the ideas of this paragraph we shall discuss an example.

EXAMPLE

We consider the ordinary Wiener-Hopf integral equation

which we have discussedto some extent in para. 2.L1.

=

r

_r(E)

f(x) =

m)

i(x-f)2

'

riowever, the range of the parameter m is different we take here m > 1/it. Formal transformation of (Li..L.i) yields

F(X) + F(A)

i-

meX

= O.

The function Eirnze possesses two zero's on the real

axis which we call A = ±v The asmptotical behaviour of the

I solution is then assumed to be

where

f(x) Ae-ivx + BeÌVX, 11m x

Hence we consider instead of (Li.L4..i) the equation for the unknown function i -ivx ivx v(x) = f(x) - Ae + Be ), which reads -ive iv

v(x)_mL

v(E)dE

_(Ae+

iVX (e +Be )

-

1+(x)2

- Be )+mj

1(x)2

o

Transformation of this equation yields

W(A)V(X)E1-nirre7'=

E Ai Bi

_Eimire

.fj(A-v) '.[t(X+v) o 1vX iVX)

W (A)=

f

tv(x)Ae +Be dx.

We now write (L.6) in the form

w_(X)

v(A)

(A2-v2)= _-

Ai

+

Bi

Y(A) - Y(X)

(A-v) r(A+v)

(L1..L.8)

where Y(A) are the solutions of

IAl

(x)+

1-mae

= o,

- (A2-v2)

Y(A)

=

(A+i) w(A),

Y_(A)=(A-i1

w_(x) (..1o)(L.L..ii)

and

(L..i4.3)

34.

± exp

-f

+

ln{(1-e

(2)

From (L.L.8) we obtain with the use of (L..L.9)

w_(A)

v(A)

(A2v2) ± A(A+v) +

B(A-v). yiX)13)

Y_(A) -

Y(A)

He nc e

v(A)

f(A2-v2)

ABP()

y(Afl,

(L.t.iL)

w_(A) =

p(A)

y(A).

(4.L.15)

In connection with

(L.)4.i1.)

we see that we have to take

P(A) C. Further we have to compensate the singularities

A = ± y in

(!.L.iLi.).

This yields

A

= -

-

y(v) ,

(L.L.i6)

B =

Y(-v).

(L.L.17)

Hence the solution of

(Lj.Lj.i)

is determined.

We have given two different discussions of eQuation (L.L.i), depending on the values of ni. For rn < 1/t the equation was discussed in para. 2.L,

m > The case m = i/z cannot be solved directly, however

it is possible to give an asjmptotic expansion of A

(L.L.i6

and B (L..L.17) for small values of g-'.

d?7, Ain

2)

Chap ter

5

AWIENER-HOPF TYPE

INTEGRO-DIFFERENTIAL EQUATION WITH

FOURTH ORDER DERIVATIVES

We shall discuss an integro-differential equation, which

is an extension of an equation discussed by Reuter and

Sonciheirir (ref.

6)

which arises in considerations about the anomalous skixua effect (ref.

_5).

Although the derivatives

occurring in this equation are of the second and fourth oida' we can follow closely the line of thought of para.Li-.1 and para 11.2. It was asked to expand the quotient of the

solution and its first derivative for the value zero of the

independent v4riable in terms of a small parameter /3. The

case that this parameter is zero is discussed by Reuter and Sondheimer. It turns out that this quotient does not depend analytically on jS, there arises a term of the form /3 in /3.

5.1 !OURIER TRANSFORMATION OF THE EQUATION

The integral equation has the form

(2)

(Li.)

f (x)=ia

j

f(t)-iß f (tflk(x-t)dt (5.1.1)

o (2) (Li.)

where a and _/3 are positive real numbers. By f (x) and f (x

are denoted the second and fourth order derivatives of f(x The kernel k(x) reads

k(x)

=

f (

- -)

exp (-clxls)ds,

I

where o is in general a cplex number with a positive real part. We shall determine the physically important quantity

f(0)/f(0) ,

and obtain its asymptotical expansion for _{/3 -} O.

(2)

(L1.)

It will be assumed that f(x), f (x) and f (x)are bounded

(2)

for x -

+ .

From

_(5.1

_{.1) we deduce that f (x) is of the}

(5.1 .2)

36.

order e when X -* Then it is allowed to apply a

Fourier-transformation to (5.1 .1) when A remains on a line L above

and

sufficiently close to the real

axis

in the complex A plane.We

find

H_(A)-F(A)tA2+ia'T K(A)(i-iß A =

(i)

(3)

(2)

=

Ef (o)-f(ofl+

K(A)-f (o)+f

(o)+A2f)_iA3f(o)

(5.1.L)

whee

_°

(2)

H

(A)=

.._ f

f

(x) e1

dx,

-

_O0 and

--f

k(x)e" dx= {--- i(

+

£)

in(_fl,

c

_{(5.1 .6)}

the logarithm is defined by

in (_j) - -ai,

Re A-+ co A on L. (5.1.7)

First we consider the homogeneous part of (5.1 .)

Y (A)-Y (A)A2+ia*T

K(A)(1-ißA)

=Y_(A).-Y(X) T(A)=0,

-+

(5.1.8)

where, as in para. L.2, we have introduced the abbreviation

T(A). In the following we shall use the notations of that

paragraph. The asymptotic behaviour of K(A) is

L.

a A - +

-

-,Re A-+ 00,A on L,

n=i n A A3 3X

hence

T(A)czßxA2IAI, ReA-*±c»onL.

The function in O(A) can be taken in the form

A2+ia K(A)(i-ij3 A) ln O(A) = ln

a132r(A2+p2) (2±2)

(5.1 .9)

(5.i.10)

(s.i.ii)

where p is an arbitrary real positive number. We find for

where

in G(ri)

= exp - _Çi7-Xj d77.

+ _{Y_(-j)ÇA+.t)}

Y_(-i)(A+1)

(A' -R (- -i i (-i u) + R - (ii')

+

-

S

+

t Y_(A+

_{Y (-i)(X+i) y)2(A)}

i R(i)

+ 2

h

( (A) y (A)-. + , Y_(A)=W_(A)(X-ip)1i', (5.1.12) + c3i(X+ip)*IX+ip (5.1 .13) We now write (5.1 .L-)

H_(A)

F(A)

(i)

(3)

Y(A)

Y(A) =

y(7)[

f (0)-iXf(0) +aß K(A)-f (0) +

(2)

2

_{f) - j}

f(0)fl (A)- e(A).(5.1.1)

+ iX ±

(o) +

A

The functions

e

(A) and G (A) can be calculated exactly in

the same way as1n para.

I.i

by splitting K(A)/Y_(A) with

the aid of (5.1.8). We find.

(3)

(2) (i)

-x2j.r

(0)+ix3ß

r (o)+ ir

(o)+A

r(ofl +

A -ii)

Y(A)

-R1 (-u) +1 R1 (-ip) + R2(1)

i R(ij4

+ y(i)2(A-i)

(s.i .1

5)

1 (3) (2) 2

-r

(0)+i

(o)+A f)-iA r(o)}

+

(5.1 .16)

where

()

(2) (i)

38.

(3)

(2) ₂

R2(X)= - f

(o)iAf (o)

A f (0)-i x

r(o)

(5.I.18)

iit

and j = e is a zero of the function (1-ißA). The

solution, (5.1.15) and (5.1.16) are essentially the same as (4.i.18)

and

(4.1.19). Because we

wa\to

obtain a function

f(x) whose fourth order derivative f") possesses a one

sided Fourier-transform for O x

()

(2)

i iAx

c= -f

(o)+iA

f (o)+A2

f)-1Af(0)A)

-If

(x)e

o

(5.1.I) it follows that the second

(5.1.16) must tend to zero Re A - ± co In te general

to zero as IAI /2 Hence

following conditions

-R1 (-12)

i

R1

(-1/2)

_R2(/2)

i R(ií1)

Y(12)

Y(-i)

+ 2 + Y(j/2)122

-R1 ()

R1

(-1/2)

_R2(bL)

R2(i/1)

- 0.

Y_(-ii) +

Y(/2)ìi2

Y(1/2)/22

When (5.1.20) and (5.1.21) are satisfied, the second term of

the right hand side of (5.1.16) tends to zero asRr /2 for

ReX-i ± . ¶ÇJs, however, generates a singularity of the form

x for f'x) as x _-+ 0+, which cannot be tolerated for

physical reasons, hence we have to go one step further and

demand

-R1(-/2) i R1 (-1/2) R2(/2) i R(iji)

=

0.

Y_(-i/2)

_Y(/2)/22

_Y(i/2)/22

These three linear equations, in the four unknowns f(0)

(i) (2)

(3)

f (o), f (0) and f (o), determine the ratio

f(0) 122 Y(i12)-i y(/2fl +122Y_(-i12)i y_(-12)

12

( )

-. (5-.1-.23)

f

(o) 12

(j/2) 'Ç(12fl

12

term of the right haTld side of

more quickly than A4 for

form in which it stands it

tends we shall have to satisfy the

00

where we take instead of L the real axis in the A plane

as

line of integration.

5.2

THE ASYMPTOTICAL EXPÂiSION OF E(A)

Instead of (5.1 .2L1.) we can write

i 172+ia

K(ii)(1-iTh)

l._I_n

a ß Zi-)

¿

(rj2-A2)

39.

From (5.1.12), (5.1.13) and (5.1.23) we see that the

asymp-totical expansion for ¡3 -+ 0 of r(0)/f (o) can be calculated

directly from the asymptotical expansion of

+00 *[

K(ii)(1-iß

E(A)=f

aJ3Jr(772+iD2) (i2+p2)

w,

(5.1.2Li.)

J

(ii-A)

(5.2.1)

where we have taken for p the value zero. This limit process

must also be made in (5.1.12). We now consider

(

ue

i7\_

1-e

i

u1

7{-ln(a7zli)-ln

₂

77+ln1+ja

₂

_2i

x(ii)?72(1iuri4fl1

77 U e

(5.2.2)

where u

¡34

and

assumes the values

, ,

or

and

it is asked to determine E(u e1) for large values of u.

We shall treat the three terms of the integrand of(5.2.2)

separately. First we consider

-2 e1u

ln(ai)f

₂

_2i

=

ilnanu, uein S,

17 -U e

_(5.2.3)

next

_{00 +00 0} d77

in

d77--euI

ln ii

-2 euf

_{ii2_u2e2}

-2-u2e21

di7

f

2u2e2

O 00

The path of integration in the first integrai on the

right-hand side of (5.2.Li.) is the real axis with a small semicircle

above the point 17 = O. We find

-2

euf

₂ =

(±- )in in u,

ue

in S.

(5.2.5)

-Lo.

The last term in the integrand of (5.2.2) causes more troitie. We divide the interval of integration as follows:

JùlH

+ir

K(r,)772(1 -iu%

77

=

f

...

d27 +

f

...

dJ7 +

f

...

d37 = I + II + III,

(5.2.6)

where s = + ¿ and r = 3 + 6', . and 6' being arbitrary but

su'ficient1y small real quantities. We shall treat the integrals I, II and III separately

us - r

-772_U2e2h -us =

_u_2e_2J [1n1+iafK(77)il2

+ o 2 2

a1uK(r)

(IÌ_ e_2)n

d77.(5.2.7)

+ ln 1±

nO u2

The expansion of the denominator is possible on account of

the special range of integration. The first logarithm of

the integrand in (5.2.7) multiplied by the general term of

the expansion gives

e -2(n+1)ìp U e 2(n+1)

f

772n ln E1+iaJK(77)77 2d77. (5.2.8) o

For a further reduction of this form we need the expansion of the logarithm for 77 - +

a;

using (5.1 .9) we obtain

in 1+ia'[K(77)7721

bi7', t

real (5.2.9)

with b = jan ; b = iaL.c b5 = lanc2; Hence we

can wrte instead of (5.2.8S

_e2+1

f

772n Ein(i+jx()2)

b21

u

ni)

m=3

(i+i)

us _2n +

f

2n bm77_m + 772n(771)

Li.

Go 2n

b21

f

2n in(1+ia'[ K(17)172)- Z bml7_m

-

17 m=3 172n(171) 5.2.10) us

where we have supposed n 2. For n = O or n = I we have to

change (5.2.10) slightly, because then we need fewer conver-ence-producing ternis or none at all, see (5.2.13) and

(5.2.1L1.). The first integral is independent of u. After inte-gration we find for the second integral terms of the form

s s

i

_I,

i i 2n - 2 and ln(77-fl) (5.2.11)

or

u

and

in (u5+1 ) = s in u - (-i )m (5.2.12)

m=1

Because the final result

_must

be independent of s, we can

omit these terms from the beginning. If now we expand the

logarithm in the third integral of (5.2.10) and integrate,we

see that only terms of the type of (5.2.12) arise, hence we

can also neglect this integral. It willturn out that for our

purpose t is sufficient to consider quantities up to the

order u; then we obtain from (5.2.10) for the values n O

I and 2 Go A,

n=0,

-e2u2

f

in 1+ia K(17)

2dî7 =-e2u2

(5.2.13) o 00

-eu

f

ln1 1

2ivK(77)2-

=-e u B, n=1 (5.2.1L1)

-e6u6 J [in

1

+iaK(17) 2ia77+ic

--6ip -6

=-e

u C,n=2.

2 1cx7Vc d -'7+1 (5.2.15)

The second logarithm in (5.2.7) gives, with the generai term of the exransion of the denominator

_-2(ni)i

f

172n in

H+

auK('7)'72

d17. _(5.2.16)

u21+1)

_i