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IMM-NYU 307

JANUARY 1963

NEW YORK UNIVERSITY

COURANT INSTITUTE OF

MATHEMATICAL SCIENCES

The Solution of Some Non-Linear Integral

Equations with Cauchy Kernels

A. S. PETERS

NEW YORK UNIVERSITY

COURANT INSTITUTE - LIBRARY

4 Washington Place, New York 3, N. Y.

PREPARED UNDER

CONTRACT NO. NONR-285(06)

WITH THE

OFFICE OF NAVAL RESEARCH

IMM-NYU 307

January I963

New York University

Courant Institute of Mathematical Sciences

THE SOLUTION OF SOME NON-LINEAR INTEGRAL EQUATIONS

WITH CAUCHY KERNELS

A. S. Peters

This report represents results obtained at the

Courant Institute of Mathematical Sciences,

New York University, with the Office of Naval

Research, Contract No. Nonr-2o5{06) .

Reproduction in xvhole or in part is permitted

for any purpose of the United States Government,

1. Introduction

In the theory of radiative transfer there are several problems

which can be solved by finding the solution H(u) of the non-linear

integral equation

1

n 1 \ 1 _ T ^ r g(u)H(u)du

which is called Chandrasekhar ' s equation. The books by

Chandrasekhar [1] and Kourganoff [2] contain discussions of this

important equation; and these books also present the contributions

of various mathematicians who have shown that (1.1) can be solved

explicitly by using function theory techniques based on analytic

continuation. Recently, (1961 ) C. Fox [5] has shown that (1.1) can

be converted into the linear equation

1

(1.2) H(x)G(x) = 1 + x/ ^i^

)du

X

where G(x) is known and g(x) is prescribed. The equation (1.2) is a

singular Integral equation with a Cauchy kernel and it can be

solved for H(u) by using an extension of Carleman's method as shown

for example in Muskhelishvili ' s book [4].

Chandrasekhar ' s equation can be linearized by first writing

it in the form

do)

r k (u)du

xg(x) = AT(x)dx

-j -hi J -irâ€” d-

and after multiplying (lA) by (i)-j^(l) and using (1.5) we find

(1.5)

>;L(e)G^(e) = '^ +f

X- ii

This is essentially the procedure that was used by Pox to pass from

(1.1) to (1.2). It suggests the possibility of solving

(1.6)

n (|), (u)dU

A^^(x)+^l(x) j -inrr^^ h

(x)

which is both singular and non-linear. Equation (1.6), in turn^

suggests an investigation of the more general equation

A^(c)+Mo/i^^= f(a

where C is an interior point of the simple smooth arc L which

connects the points t and t, in the complex r-plane.

One of the purposes of this paper is to show in Section 2

that the equation I can be solved explicitly by using elementary

function theory techniques. It turns out that the solution of I

is in some ways simpler than the solution of (1.3). We will also

be concerned with the solution of some other non-linear equations.

In Section J> we show how to solve

II

7r2i2(a

r c|)(T)dT

J T-C

â– - L

f(a

and Section ^ is devoted to the solution of

2

III

TT^^^iO +

/^

)d'

f(a .

In Section 5 we show that there is a connection between equations

I, II, III and certain problems in potential theory.

The equations I, II, III may be of interest for at least two

reasons. In the first place, they are of interest in themselves as

Cauchy singular, non-linear integral equations which can be solved

explicitly. In the second place, they present a formulation of

certain non-linear boundary value problems. Equation III, for

example, is intimately related to a problem in two-dimensional

potential theory which has a number of physical applications. This

Is the problem of finding a potential function in a domain D when

its normal derivative is prescribed on one part of the boundary C;

and the magnitude of its gradient is given on the remaining part

of C. In Section 5 we show how an explicit formula for the solution

of this problem can be found.

The final Section 6 is concerned with a brief discussion of

the non-linear system

L

and some other systems which can be linearized by the method

developed in Section 2.

We state here the main conditions and assumptions upon which

our analysis is based. If t = T(t) is the equations of the simple

smooth arc L directed from t to t, ^ t let L[t ,t, ] denote the

set of points T=T(t), t+ (t +t, ) we have

r Ux)p ^ r

Ut^,x^] L[-1,1]

(l>Q(v)dv

which shows that the transformation does not change the form of

equations I, II, III. Thus there is no loss of generality if we

assume, as we will hereafter, that L in I, II, III is L[-l,l].

2. Equation I

We proceed to show how

(2.1) 4(o + oo^l^^^ " const. ,

^^ ^^^^ G^(z)

Z â€” = .

z-!^oo I ^

z + LJ 1-z

The general solution of (4.9) is

G(z) = G^(z) + /l-z^ p(z)

where p(z) must satisfy

}-^

21

(4.11) P'^IO-P'IC) = .

The function p(z) must be taken so that the properties of

e^^^V 2 +ijl-z^ match those of

This function is analytic for z not on L and it vanishes like c /z

as z â€” > 00, provided c = / (j)(T)dT ^ 0. Furthermore, in accordance

L

with the assumptions about h^Kx) admitted in the introduction, the

behavior of F(z) in the neighborhood of an endpoint a of L is such

that Â£ _^^(a-z)P(z) = 0; and the limit values P"''(C), F"(0 must

satisfy a uniform Holder condition. These properties and the

condition (4.11) imply that p(z) must be analytic everywhere and it

must vanish at infinity, i.e., p(z) = 0.

We have now found that

,..is, . exp {lii! / Â±y'^-

L Wi-t'^J(t-z)

This gives

(4.15) F^(c) = Â± (^-iiT^)yf(r7 .exp|%f. \ .^iilzMi_|

L yi-T^(T-a J

(4.14) F-(0 = Â± k .i/IIFj/fTTT . exp \- ^ f AlLlMl.)

^ L ji-T2(T-aj

The solution of (4.1) is obtained by subtracting (4.14) from (4.13).

It is

,x^Â« .=

:. jtv=,i::*;:nj oo .

It will be noticed that in the above analysis we have assumed

If

this is not the case, that is, if for example

(4.16)

while

(4.17)

r (l)(T)dT =

L

:^=-/xi(

T)dT =

then F(z) vanishes like c^/z , c^ ^ 0, as z â€” > oo and the solution

of (4.6) as we have given it has to be adjusted. However, it is

clear from the above analysis that if (4.l6) and (4.17) prevail then

the adjustment is easily made by taking F(z) = e V(z+i^l-z )

instead of (4.8), which leads to

4.18) F(z) = Â± [z-i/ll?]^ expj l^Ff- /

â– ^-^ f(T)dT

(i^^y

(T-Z)

23

5. Some Non-linear Problems in Potential Theory

The equations vie have analyzed can be identified v.'ith certain

problems in potential theory. For example, consider the problem of

finding ^(x,y) such that

^xx^^'^^ ^^yy^^'^^ = , y < ,

^y(x,0) = , |x| > 1

and subject to an additional condition on y = 0, [x] < 1 which is

specified below. Let us suppose that ij/ {x,0) , |x| < 1, satisfies

the conditions imposed on (j)(t) in Section 1, and that each of

f (x,y) and ^ (x,y) vanishes as z = x + iy â€” > oo . The harmonic

function ^ (x,y) can then be written

^ -'_^ (t-x) +y

1

= -h^ /^a(t-x)2+y2]^^(t,o)dt

-1

1

^' (x,y) = - J- f

from which

â€¢*â– (t-x)^ (t,0)dt

â€¢^ {t-x)2 + y2

and by integration

1

r

_^ (t-x)'' + y'

, r (t-x)^ (t,0)dt

^^(x,y) = i / ^ ^â€”

r^ ^ (t,0)dt

^x(->Â°) = I j \-X

-1

Now if we impose the additional condition

2k

Ayx,0)+^^(x,0)^y(x,0) =Ii2L)

x| < 1

then (l)(t) = ^ [t,0), \t\ < 1, must satisfy

7.cl)(x)+(l)(x) f MtJ|t = f(x)

-1

If instead of I we impose

|x| < 1

II

^l{x,o)-r^u,o) = lifl

w\ < 1

then f(t) = ^ (t,0), must satisfy

r 1

7r^c|)2(x)

r (|)(t)dt

J t- X

f(x)

1x1 < 1 .

Finally, if the additional condition is

III

^J(x,0) + ^^(x,0) =

1x1 < 1

then (|)(t) = ^ (t,0), must satisfy

Tr^cp^{x) +

1 12

r (|)(t)dt

J t -X

-1

f(x)

1x1 < 1

We proceed to show that problem III can be solved for domains

more general than the half plane. For the half plane problem we

can write

^(x,y) = (I Jiz)

and then

25

9\z) = (Â£9'(2) +1 ciJ^^'(z) = ^^(x,y) -i^y(x,y)

1 /o r (!)(t)dt . Â± 'i r (i)(t)dt

which is the same as

(5.1) .T/,.,=i/i(SJ|i .

-1

The integral on the right hand side of (5.1) has been determined in

connection with the problem of solving the integral equation III,

in Section h, and we have

(5.2) ?^'(z) = Â± (^-^fl^)^ exp[4# / -^^(t)at 1

^^ -1 (il-t^J'^(t-2) -^

where y is 1, or 2. The function f{:i,y) can be obtained from (5.2)

by an integration.

Let z = m( C ) = ni( s +iTi) be a function which maps the domain

D, with boundary C in the C-plane, conformally into the lower half

of the z-plane. Let the image of C, a part of C, be the segment

y = 0, |x| < 1; and let the image of Cg, the remaining part of C,

be y = 0, |x| =* 1. Under this mapping, M,(x,y) is transformed into

^(x,y) = tr(e,ri) = ^2 > [m(a] ,

a function harmonic in D.

Let s be the arc length measured along C from say the initial

point of C, . If r= 4(s) +iri(s) is a point on C, then the normal

derivative of ^(s,!!) at the boundary is

26

(5.3) -^3^râ€” = lT^^^t-(^)5

dm(.T) dx

nen -

Hence for cT on C

If cr is on Cg, then ^^j^ = if ^^ ^Â®^^ ^"^ "J^'Lmlcr)] is real.

2

111 (4,Ti) =

n

The tangential derivative of ^{^,r\) at the boundary is

From (5.5) and (5.4) we see that for (Ton C,

â€” n ~"s I as I I 1

=

dm{o')

ds

^[i'il-T[m(

[â– r

=

dm{cr)

ds

â€¢

Therefore

if

we

take

h(

s) = â–

am(

JANUARY 1963

NEW YORK UNIVERSITY

COURANT INSTITUTE OF

MATHEMATICAL SCIENCES

The Solution of Some Non-Linear Integral

Equations with Cauchy Kernels

A. S. PETERS

NEW YORK UNIVERSITY

COURANT INSTITUTE - LIBRARY

4 Washington Place, New York 3, N. Y.

PREPARED UNDER

CONTRACT NO. NONR-285(06)

WITH THE

OFFICE OF NAVAL RESEARCH

IMM-NYU 307

January I963

New York University

Courant Institute of Mathematical Sciences

THE SOLUTION OF SOME NON-LINEAR INTEGRAL EQUATIONS

WITH CAUCHY KERNELS

A. S. Peters

This report represents results obtained at the

Courant Institute of Mathematical Sciences,

New York University, with the Office of Naval

Research, Contract No. Nonr-2o5{06) .

Reproduction in xvhole or in part is permitted

for any purpose of the United States Government,

1. Introduction

In the theory of radiative transfer there are several problems

which can be solved by finding the solution H(u) of the non-linear

integral equation

1

n 1 \ 1 _ T ^ r g(u)H(u)du

which is called Chandrasekhar ' s equation. The books by

Chandrasekhar [1] and Kourganoff [2] contain discussions of this

important equation; and these books also present the contributions

of various mathematicians who have shown that (1.1) can be solved

explicitly by using function theory techniques based on analytic

continuation. Recently, (1961 ) C. Fox [5] has shown that (1.1) can

be converted into the linear equation

1

(1.2) H(x)G(x) = 1 + x/ ^i^

)du

X

where G(x) is known and g(x) is prescribed. The equation (1.2) is a

singular Integral equation with a Cauchy kernel and it can be

solved for H(u) by using an extension of Carleman's method as shown

for example in Muskhelishvili ' s book [4].

Chandrasekhar ' s equation can be linearized by first writing

it in the form

do)

r k (u)du

xg(x) = AT(x)dx

-j -hi J -irâ€” d-

and after multiplying (lA) by (i)-j^(l) and using (1.5) we find

(1.5)

>;L(e)G^(e) = '^ +f

X- ii

This is essentially the procedure that was used by Pox to pass from

(1.1) to (1.2). It suggests the possibility of solving

(1.6)

n (|), (u)dU

A^^(x)+^l(x) j -inrr^^ h

(x)

which is both singular and non-linear. Equation (1.6), in turn^

suggests an investigation of the more general equation

A^(c)+Mo/i^^= f(a

where C is an interior point of the simple smooth arc L which

connects the points t and t, in the complex r-plane.

One of the purposes of this paper is to show in Section 2

that the equation I can be solved explicitly by using elementary

function theory techniques. It turns out that the solution of I

is in some ways simpler than the solution of (1.3). We will also

be concerned with the solution of some other non-linear equations.

In Section J> we show how to solve

II

7r2i2(a

r c|)(T)dT

J T-C

â– - L

f(a

and Section ^ is devoted to the solution of

2

III

TT^^^iO +

/^

)d'

f(a .

In Section 5 we show that there is a connection between equations

I, II, III and certain problems in potential theory.

The equations I, II, III may be of interest for at least two

reasons. In the first place, they are of interest in themselves as

Cauchy singular, non-linear integral equations which can be solved

explicitly. In the second place, they present a formulation of

certain non-linear boundary value problems. Equation III, for

example, is intimately related to a problem in two-dimensional

potential theory which has a number of physical applications. This

Is the problem of finding a potential function in a domain D when

its normal derivative is prescribed on one part of the boundary C;

and the magnitude of its gradient is given on the remaining part

of C. In Section 5 we show how an explicit formula for the solution

of this problem can be found.

The final Section 6 is concerned with a brief discussion of

the non-linear system

L

and some other systems which can be linearized by the method

developed in Section 2.

We state here the main conditions and assumptions upon which

our analysis is based. If t = T(t) is the equations of the simple

smooth arc L directed from t to t, ^ t let L[t ,t, ] denote the

set of points T=T(t), t+ (t +t, ) we have

r Ux)p ^ r

Ut^,x^] L[-1,1]

(l>Q(v)dv

which shows that the transformation does not change the form of

equations I, II, III. Thus there is no loss of generality if we

assume, as we will hereafter, that L in I, II, III is L[-l,l].

2. Equation I

We proceed to show how

(2.1) 4(o + oo^l^^^ " const. ,

^^ ^^^^ G^(z)

Z â€” = .

z-!^oo I ^

z + LJ 1-z

The general solution of (4.9) is

G(z) = G^(z) + /l-z^ p(z)

where p(z) must satisfy

}-^

21

(4.11) P'^IO-P'IC) = .

The function p(z) must be taken so that the properties of

e^^^V 2 +ijl-z^ match those of

This function is analytic for z not on L and it vanishes like c /z

as z â€” > 00, provided c = / (j)(T)dT ^ 0. Furthermore, in accordance

L

with the assumptions about h^Kx) admitted in the introduction, the

behavior of F(z) in the neighborhood of an endpoint a of L is such

that Â£ _^^(a-z)P(z) = 0; and the limit values P"''(C), F"(0 must

satisfy a uniform Holder condition. These properties and the

condition (4.11) imply that p(z) must be analytic everywhere and it

must vanish at infinity, i.e., p(z) = 0.

We have now found that

,..is, . exp {lii! / Â±y'^-

L Wi-t'^J(t-z)

This gives

(4.15) F^(c) = Â± (^-iiT^)yf(r7 .exp|%f. \ .^iilzMi_|

L yi-T^(T-a J

(4.14) F-(0 = Â± k .i/IIFj/fTTT . exp \- ^ f AlLlMl.)

^ L ji-T2(T-aj

The solution of (4.1) is obtained by subtracting (4.14) from (4.13).

It is

,x^Â« .=

:. jtv=,i::*;:nj oo .

It will be noticed that in the above analysis we have assumed

If

this is not the case, that is, if for example

(4.16)

while

(4.17)

r (l)(T)dT =

L

:^=-/xi(

T)dT =

then F(z) vanishes like c^/z , c^ ^ 0, as z â€” > oo and the solution

of (4.6) as we have given it has to be adjusted. However, it is

clear from the above analysis that if (4.l6) and (4.17) prevail then

the adjustment is easily made by taking F(z) = e V(z+i^l-z )

instead of (4.8), which leads to

4.18) F(z) = Â± [z-i/ll?]^ expj l^Ff- /

â– ^-^ f(T)dT

(i^^y

(T-Z)

23

5. Some Non-linear Problems in Potential Theory

The equations vie have analyzed can be identified v.'ith certain

problems in potential theory. For example, consider the problem of

finding ^(x,y) such that

^xx^^'^^ ^^yy^^'^^ = , y < ,

^y(x,0) = , |x| > 1

and subject to an additional condition on y = 0, [x] < 1 which is

specified below. Let us suppose that ij/ {x,0) , |x| < 1, satisfies

the conditions imposed on (j)(t) in Section 1, and that each of

f (x,y) and ^ (x,y) vanishes as z = x + iy â€” > oo . The harmonic

function ^ (x,y) can then be written

^ -'_^ (t-x) +y

1

= -h^ /^a(t-x)2+y2]^^(t,o)dt

-1

1

^' (x,y) = - J- f

from which

â€¢*â– (t-x)^ (t,0)dt

â€¢^ {t-x)2 + y2

and by integration

1

r

_^ (t-x)'' + y'

, r (t-x)^ (t,0)dt

^^(x,y) = i / ^ ^â€”

r^ ^ (t,0)dt

^x(->Â°) = I j \-X

-1

Now if we impose the additional condition

2k

Ayx,0)+^^(x,0)^y(x,0) =Ii2L)

x| < 1

then (l)(t) = ^ [t,0), \t\ < 1, must satisfy

7.cl)(x)+(l)(x) f MtJ|t = f(x)

-1

If instead of I we impose

|x| < 1

II

^l{x,o)-r^u,o) = lifl

w\ < 1

then f(t) = ^ (t,0), must satisfy

r 1

7r^c|)2(x)

r (|)(t)dt

J t- X

f(x)

1x1 < 1 .

Finally, if the additional condition is

III

^J(x,0) + ^^(x,0) =

1x1 < 1

then (|)(t) = ^ (t,0), must satisfy

Tr^cp^{x) +

1 12

r (|)(t)dt

J t -X

-1

f(x)

1x1 < 1

We proceed to show that problem III can be solved for domains

more general than the half plane. For the half plane problem we

can write

^(x,y) = (I Jiz)

and then

25

9\z) = (Â£9'(2) +1 ciJ^^'(z) = ^^(x,y) -i^y(x,y)

1 /o r (!)(t)dt . Â± 'i r (i)(t)dt

which is the same as

(5.1) .T/,.,=i/i(SJ|i .

-1

The integral on the right hand side of (5.1) has been determined in

connection with the problem of solving the integral equation III,

in Section h, and we have

(5.2) ?^'(z) = Â± (^-^fl^)^ exp[4# / -^^(t)at 1

^^ -1 (il-t^J'^(t-2) -^

where y is 1, or 2. The function f{:i,y) can be obtained from (5.2)

by an integration.

Let z = m( C ) = ni( s +iTi) be a function which maps the domain

D, with boundary C in the C-plane, conformally into the lower half

of the z-plane. Let the image of C, a part of C, be the segment

y = 0, |x| < 1; and let the image of Cg, the remaining part of C,

be y = 0, |x| =* 1. Under this mapping, M,(x,y) is transformed into

^(x,y) = tr(e,ri) = ^2 > [m(a] ,

a function harmonic in D.

Let s be the arc length measured along C from say the initial

point of C, . If r= 4(s) +iri(s) is a point on C, then the normal

derivative of ^(s,!!) at the boundary is

26

(5.3) -^3^râ€” = lT^^^t-(^)5

dm(.T) dx

nen -

Hence for cT on C

If cr is on Cg, then ^^j^ = if ^^ ^Â®^^ ^"^ "J^'Lmlcr)] is real.

2

111 (4,Ti) =

n

The tangential derivative of ^{^,r\) at the boundary is

From (5.5) and (5.4) we see that for (Ton C,

â€” n ~"s I as I I 1

=

dm{o')

ds

^[i'il-T[m(

[â– r

=

dm{cr)

ds

â€¢

Therefore

if

we

take

h(

s) = â–

am(

1 2

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