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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 -ir— 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

■- 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 — = .

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 — > 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 £ _^^(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! / ±y'^-

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



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

L yi-T^(T-a J

(4.14) F-(0 = ± 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^« .=



:. 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 — > 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) = ± [z-i/ll?]^ expj l^Ff- /



■^-^ 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 — > 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



•*■ (t-x)^ (t,0)dt
•^ {t-x)2 + y2



and by integration



1

r

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



, r (t-x)^ (t,0)dt
^^(x,y) = i / ^ ^—



r^ ^ (t,0)dt

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



1 /o r (!)(t)dt . ± '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) = ± (^-^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^r— = lT^^^t-(^)5

dm(.T) dx
nen -

Hence for cT on C



If cr is on Cg, then ^^j^ = if ^^ ^®^^ ^"^ "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,
— n ~"s I as I I 1











=


dm{o')
ds


^[i'il-T[m(


[■r










=


dm{cr)
ds







Therefore


if


we


take
















h(


s) = ■


am(


1

Online LibraryArthur S PetersThe solution of some non-linear integral equations with Cauchy kernels → online text (page 1 of 2)