The uniqueness of the solution of the inverse problem in the asymptotic theory of radiation transfer

M. V.

MASLENNIKOV (r~oscow)

(Received

1.

19 l/arch

Statement

of

tile

f?62)

proltlem

In transfer theory 3 larcd place is occllpied by tke problem of tile space-nr+gilrrr distrilluticn of monocl~~roc~atic flow of rn,liat.ion at a c!eptli of a tliin plane layer of matter. In this Frobler! the elencntary event of the iliteraction of rndiation with tlie medium is copFletel,v ?cscri:)eci hy the tliffllsion indicnt.or ,;(v). Expresser: in t!;e lang;;:e of neutror! physics, B,;(p) ilcl is the Drol;ahil ity th:it as a result of one collision 3 neutron is r’.eflecteil from its initial clirection of motion !~y an :ingle i; such tllat p < cos ;< < p t , !p. The <;llantity 1

s

1--2n g(P)+ -1

is the probability

that the neut,ron is il)sorl)eL’ ir, one collision.

Ry ivw of example, let us consider the classical flilne problem of tlie diffusion of radiation across 3 semi-infinite lwer. Suppose the l~onogeneous and isotropic tliffusin& metliw fills tbc !ialf-space -r > 0. Id p be the cosine of the angle between t.1.c props&,ion of rn;liation nnr! id c interior normal to the !)ound:rry of tllis half-space. Then the liensit?- of radiation ~Y(T, p) obeys the quation 03 I) (z, p) ==

1

‘i* c-F’Y (t

-

pp) (I? \ g (t-1,p’) 4 (t - pp, p’) t/p ’ -1

0

W:ere y(x)

*

” ‘I

= r)

,‘“C’!L.

for

mat.

x<

2,

0,

No.

Y(x)

6.

x

1

1044-lc153.

for

x > 0. Tlie kernt?l .,(v,

1968.

ti’)

is

1248

M. V.

las

lenn i kov

connected with the indicator L(P) by a simple relation (see below). It can be shown that for any indicator the principal term of the solution of this equation has the form Jrns(t, p) = eat cp(p), for large ? where the parameter h > 0 and the function q(p) satisfy the “characteristic integral equation”

(1.1) The non-negative

solution

cp(p) of

this

equation

is unique

(see

cl]).

Thus, by setting the indicator r;(u) we uniquely determine the asymptote of the radiation distribution. This article is devoted to an examination of the inverse problem: given the asymptotic characteristics of density of radiation (i.e. the function q(u). satisfying (1.1)) it is required to find the indicator d(u). The existence of a solution to this problem follows from the fact that it is being set. The proof of the uniqueness of this solution is non-trivial. It is not difficult to show that for the solution of the inverse poblem to be unique there must be absorption of radiation by the medinm, the q(u) is always a constant). however small (if there is no absorption, Therefore we exclude the case of pure dispersion from the subsequent examination. This means that we restrict ourselves beforehand to those indicators g(u) for which the quantity

is positive. I,et us consider the class of indicators r to which belong those nnd only those, each of which Possesses the following property:* c:(p)

is defined

and non-negative

on L-1,

11; g E f<,(-1,

g(p)

1) with

2T((W 1) = do. n < (i.< 1.

l

From the physical

obliged

to

instead

standpoint,

more natural to make the The corresponding theory V. X. Vladimirov

(tZl,

that ;: E I.:, greatly narrow the class of

simplifies physically

condition 2) superfluous.

article

in the

of

requirement that can be developed. [3!,

putting g is For

p. 151).

g E la2 it

would

be

summable (g E L,). this comment we are However

the

assumption

all the constructions and does not real indicators, Se note further that have to refer, is /4! , to which we shall

Asymptotic

theory

Here we denote the scalar symbol (f,, f,), as we shall

of

rndiat

1249

transfer

ion

product of the functions do below.

jl

2 E I,, by the

Let us fix an arbitrary function ,; E r. It can be expaniled in a series of Legendre polynomials which is convergent in the mean:

(1.2) zS are uniquely

The coefficients ing properties:

The kernel relation

g(cl,

II’) of equation

defined

(1.1)

by L’(D) and nossess

is connected

the follow-

with S(I.I) by the

which gives

i!Y(P! P’)=C

* 2k+l 2

gkPk

@)

pk

b’)*

(1.6)

k=o

The series on the right converges in the mean with respect variable P * for any fixed value of ~1 E r-1, 11. *To the function g(p) let us relate kernel’ ;(P, P’) acting in L,(-1, 1):

the integral

operator

to the

2 with

Let A E (-1, 1). The operator (1 + Acl)-‘z possesses in L, exactly one non-negative eigenfunction cph(~), (qq, I) = 2. This function corresponds to the simple positive eigenvalue of /‘g(h) and is continuously dependent on P E c-1, 13. P (A) and 1~ are regular functions of h in some neighbourhood of the in . f erval (-1, 1) (where q,, is considered as an abstract function of the parameter h with values in iA,(-l, 1). The function kfg(h) is even, and for h > 0 -$ MJk)>

0, Ms((l) = go <

1 and IirnM, (h) = + o0. ;.=I-0

3. V. ?tr.s lrnn ibov

12X

The eigenfunction

(p,+ I)ossesses the properties:

T-a(P)= ‘pak-c1);%A4 = 1 ([4ll ill). Thlls, tf,e clenent

i of

the clnss

r is related

with the function of two variables Q(V)

in a one-to-one manner

defined

for A E

(-1, 1) and

iI EE I-1, 11. Let Q be t!le set of all functions Q(U) each of which carreslX.UId:; in the sense indicated to some g E r. The rule sccordirrg to WfiiCh to ear% G E

Ry cO0st~~ction ISet 5 E

r we associate

a Q(V)

G is a single-valued

I- and u)h = ii& There

S!lCh tf:at !',(A,) = 1. Put

= R(P).

is denoted by G:G_

happing of r on R.

exists exactly one h, = h,(g) E

$01 = q~,(~). Civen d, the function

(%I)

#al

is

uniquely de P* med. T;ie pair

is tf;e uriirllIe solution 9)) With A > 0, fp >c, (ho(g), cpfgf) satisfies

(h,(g), q@l)

iI? tfle Cl3S.Y Of !XIirS (h, illdi c&or ,; E r the pair

of the direct (9,

1)

=

2:

equation

prn!de~

giver!

the

(1.1).

It turns out that tf,e converse is :tlso true: if the pair (A, ‘rp) of t!ie given form is a solution of (1.1) for SOPX? ir,dicatOr g E r, LLen t9e indicator (, is uniquely defined by t3is con&tion.

!.:oreover,as

sliows, tfii:; iti,ticator and the pnrmeter TLeOres: 1, which follows, uniquely tlefinei? I>:: Mxin;: OnI:* One function ‘i’* I& let

1’,, ck3ot.e

9, iknote

tf,e

r. ny

011 Q,, :.LIIilQ, is

vurinblc

CI E

c$:“f

(&,,,q

functions

constniction,

-

rpfgf)

qJPf,

I-I.

is

9,

il

class

of

?VftiCfi

arise

solutions

r. normrrlisetl1,~ t!!e condition

R,

introfi!uced

Go is a single-valued

tfle class of all non-negative fulictions

drove,

:md

iw,v:\er, ;: runs mamin2

of r

‘!J Of euuatiotls

(q), 1) L: 2. 'Ye r10te that,

wf,icf, dcrmkl 011 Ol?lV one

11.

‘ll;iwr~m 1. iSo ~3;s

r i;!to rZ, in ;t one-to-otrc

T:!e prOof of TheOrem classes 9 md

g-9

tile clx4r, of all

tf,rou& tf,e set

X1.1) for ;> E ir, c.mt~trns%t.0

msppini;

jt, are

I is obtained

mnner.

!p tke partial description Of the

Q. to lv!iic?l we come at Once.

r,r!tus fix t?ic: nrbitr.%ry flr!tCti.Ori ~ph(~) f?Z O. Tl~er.there exists e.5 I- SlICi,tll;Lt(13\= ;s,,.This naeans tf;nt for a11 p 15% (-I, 1) and ;; 65: (-I, 1)

The kernel I;(~, II’) is connected with the indicator I; by tbr? reldion (1.5). Ming a continuous function of p, ~(11) can he c?x~7sntleclinto a series of Iqcndre polynomials :d.lich conver;;es, at, lmst in t1.r: neon, for each fixed value of A r-1 (-1, 1): ‘i’). (I”) --: 2 t(J’T)(3”) I’,; fp). I /, Since Q tFfe Fourier

is an abstract coeffi ciertts

analytic

1p (h)

function

21: -i 1 7..

---y--

(X.2)

of the parameter h ,7 (-I, lj

(‘I,),, I’,;)

tfcnend snalyticslly on A on the same interval (-1, 1). Poreover, on the basis of t11c resol ts ohtainetl in L41, Section 5, we c;tl; conclude that tile functions up(‘) (A) can ?e represented in the form

where the IV&(A) nre even,

It will

I)e of

especial y:-r1

btting

connecting

(2.2)

importance I]:,; (0) --- (.-

in (2.1)

the ids

I[_,

re::ulsr

we arrive

on f-1,

later

I),

iy,(h)

-

1 and

that +J~(O) f 0 and

,1)‘i,

Ii--

0,1,2,..,

at 3n infinite

system of equdions

:

(1.) 5 (!,,

I;.

(I,!,:!

,...)

i.c,(-

1,l).

(2.4)

C’e can examine system (2.4) hy the metliod of cotitinued fractions. This method !~its heen use:! more than once to find ttle solution of (2.4) ([Sl , 153, 171 and oti~ers), Fe are interested not in tile form of the soliltion of this system, I,ut in infornotioL abut tb,e nature of tk,e deIWncJence of lyl, on A, w:,ich is contained in it,. I et,

2. V.

1351

/l,((h)” Then we can write $k

h2’qLZ&, I

-

(2.4)

that,

uniformly

lim bk (A) = -

Jet us compare the continued =k+l bk+l(h)

+

$+2

ak+l

way: k>,O.

(a)$k+2 (a),

with respect

2;

k==cu

(1) = bk(a) +

k>l.

in the following

(a) = bk(a)‘b-b (a) +

It is clear

‘!k

las lennikov

to all

lim ek (h) = -

(-1,

1)

he.

k=m

fractions

A E

(2.5)

P-6)

(k > 1)

ih)

&)/bk_t2(A)+ . . .

S

bk’ (h) +

i $$ k’=k+l k’ (2.7)

with system (2.5). The quantity convergents:


is the limit

(if

it

exists)

of the sequence

of

(2.8) The numerator and denominator of a convergent to the we11-known recurrence relations

are calculated

according

and for v ,I- 2

The convergence of a sequence of convergent3 ties of the roots p, 2 (A) of tl,e cJ.aracteristic cocfficicnts of whicL are the limits (2.6) :

tleJ)entls on the pronerquadratic equation, tl,e

Asymptotic

PI(h)=

theory

of

-(1+1/1-4),

Let ,qo be a circle A=oandradiusuE(O,

radiation

1253

transfer

pz(h)=-(I-pT=3?).

in the complex A-plane 1). ThenforAES

with centre

at the point

a

a) l
It will fractions*

values

be sufficient. using certain theorems in the theory of continued to study the convergence of the fractions (2.7) for large of k.

7heorem 2. For each u E (0, 1) there possessing the following properties: 1) if k > h’(u) then we can find v >V”, h E *cc; 2) for k>.\‘(a) respett to all 3)

tfie

functions

4) we t;sve,

the limiting h EVE,C,; ik(h),

uniformly

k >

exists

v0 = v,(k,

relation

Bb” (1) #0

is satisfied

uniformly

(3.X)

to all

number N(a)

0) such that

P/(o) are analytic

with respect

a natural

in the circle

for

with

.qo;

h CZ .q,

lim ik (A) = p1 (A). k-03 There exists a simple connection between the coefficients Q(A) and the fractions ck(A). It .is obvious in the case A = 0. F’or ak_+.r(0) = (1,
Relation (2.11) remains true for h f 0 3150, at least for sufficiently large k. In order to show this, it is necessary to study the behaviour of yb(A) as k - CG.Let h E (-1, l), h f 0. Ye liave

This gives

l

See,

for

example.

181, Theorems

40.

41,

42.

hf. Y. %QSlenn i kov

1254

At the same time,

for A E

(-1,

I),

h + o

ThUS

We use one more fact:

Theoren

3. Let o E

(0,

1).

mr

k > .“(N

and h e

(--Cr: o)

$k (a) = tk (a) $ki-1 (a). This theorem is a consequence of conditions (a), (h), (c), (3) snd Theorem 2. The proof is not cocplicated, and we sk,all not give it. WE note only that it is essentiall:~ the same as the proof of Theorem 4c; in the book t81. corollary. For, $a,+

For k

if

7

$k, (&) = 0,

(h,) = 0. Eut it including

k
b(u)

Now we shall IAem+nrl. Let $k(.h)#O

and A EZ f-o,

y”(h,)

for

O
0.

61, then,

from (2.5)

that

from Theorem 3,

~k(&)=o

for

all

= 0, which is not true.

show that no coefficient a~(O,l),

yk(h) *

hr E (-o,

kr > N (o),

then follows

o)

k,>O,

t~k(h) is zero. for

&,(h)#O

A=(--6,

a).

Then

and h~(-&a).

The proof is obtained hy induction on k,. For k, = 0 and k, = I the statement is obviously true, since yQ(h) 5 1. Suppose the lemma is true for k, 1 md arbitrary (I & (0, 1). Ye shall prove it for =n+1. k* Be have

$,+t (A) +O

for A 62 t-u.

a).

Let us suppose that y,(h)

fias

zeros in the interval (-CT, a). Due to the fact that y,,(h) is even and y,(O) i( 0 the set N of zeros of l?,,(A) be!onginl: to (0, a) is not empty. $n (A) #0 for Let or = inf N. Then *)n (or) = 0, q E (ci, a), A =

(--or,

0,).

%.I

(A) #:O

for

On the other

By the induction

Ilypothesis,

it

h E (-- ai, e1). hand, turning

to (2.4)

we find

follows

from this

that

1255

$n+l (sl) # 0, -

and tt,erefore

sgn %+1 W.

But

[O,G,] c

51oreover,

$n_1 (5]) + 0.. (-

G, cj,

so

that

sgn I&+_~(~5~)=

I#,,,, (h)

floes not become

zero

on [(I, G1]. Ry the SNOWtolte1: sgn $n+l (al) = sgn IJ,,__~ (0) - sgn $._, (0)

(cf.

(2.3)).

is zero

Thns

sgn -&_1 (5,) = -

at some point

sgn $+~ (0) and, ttyerefore,

of t!,e interval

(0,

$n-1 (h)

aI).

This contradiction means t!lnt ,II,,(A) !-GE no zeros in tk,e interval from this that l?,(h) KY the in:Juction hypnt,llesis, it foI!ow (-a, 0). also is not zero ir. (47, u), 0 \(L
4. The coefficients

Tuwrem

of m(p) (-1, 1):

tl,e lemma.

E

!! in L.ecendre polynomials

$k (A)+o; For any u E

gh. (h) = ‘k$

(0,

sgn gk (A) = (1) we have,

(cp,., Pk)

of tile expansion

are r.ot zero in tile interval

l)“,

uniformly

AE(-l,l),

k=O,l

with respect

to h E

3 .... ,-, (4,

a)

(2.12) Prooj. Let u E (0, 1). From the corollary of ‘Pleorem 3, we know that for k>I~f(O + 1 yk(h) is not zero on (-u, a). Fixin:, k, = P.‘(u) + 1 using the lemma we find that yap f 0 on (-a, a) for 0
qk th) #O, Thus, sgn $k (h) = sgn ‘$k (0) =

(-I)*.

k=O, 1,2, ..,.

me 1 imiting

relation

follows

from

Theorems 3 and 2. The results we have obtained the elements of the class R,.

enable

us to find

certain

properties

of

‘Tlleorem 5. Let 9 E R,. Then 9 is a positive continuous function of Let qk be the coefficients of the expansion PEI(--,ll, (cpt 1)=2. of 9 in a series of Legendre polynomials;

7. V.

1256

so

that

and the series

of

Alas lenhikov

is converge&-+

the mean at least.

The coefficients ‘pk form an alternating which are different from zero: (Pk #

There exists

0,

sgn

(Pk =

sequence,

k=0,1,2

(-ljkj

all

,...;

the elements

‘~,,=i.

the limit lim

'k-1 qk

-=--_

k=cu where a > 1 and

%---l _ _(Pk

k=

+&)+~k,

1, 2, 3, .

(2.13)

. t

with 2

IiS%<

(2.14)

CC.

k --1

= p

,%te.

It is obvious

?roo/.

There exists

L. l3y the definition

Put Q(P)

Then AO(g) = 2u(a*

= I;g.

that Theorem 1 follows a function

2 E

of the mapping

&,

from this

r such that

foil ows that all

theorem.

&g :-= CF. Ilet us fix

C+(p) I= c+-,,,:~,(I_I), where

9, (p) -= kg, and h= h,,(g) is defined by the condition 0 c ho(g) < 1. Since m(1.1) E Q ail the previous results to it. From the relation

ant! from Theorem 4 it

+ 1)-l,

Fg(h) = 1, can be extended

the qk are different

from zero,

1257

wit].

Moreover,

from (2.12)

where

I/

1 -i-

-__ k;1,(a)

1-

%=I

X,, (2)

Solving

this

for ho(;)

we fincl h,, (8) --=;;-i

Ei~llalities

(2.15)

1.

->

now folloiu

.

immediately

from (2.4)

with A = hO((,).

Thus h,(d) and ul. are uniquely rlefined, giver, the function This conpietes the ;&-oof of T!,eorem 1. It remains to estal,lis!l relations (2.13), (?. 14). Put

Then

lirn 6,, r= 0 and for all

Ii--CO

It is not difficult, using tion..(2.15) to the form

~2 E ?“. t!-,e ”

k >l

identical

transformations,

li >

to bring

1

rela-

(E.ICi)

where h

k-

_a2+1

uk = (2kIt is clear

that

(2.17)

,-gk+

for

I>-‘; k > 1. 1 + uk 7 0, 1 + uk 3 0 anti

.‘, t’.

1258

For

every

:lar frrnntkot

n > 1

+L-_.. p-t‘2

n 1 -+++j rI-i=l

1 i- v k-f-n+1

I+

1 -+-Uk

uk+j

This equality is easily proved by induction take tlte limit ir: (2.18) as ti - m we find

*k+n+l*

on n, usin:,: (2.16).

(2.18) If ‘Nre

From (2.17) IilL: <

fj

00

k=l

But the latter

condi%ion

is satisfier1

(see

(I.?)).

00 al,d iirn& k=w

Rince u > 1, there for k >kO

exist

66%

Thus =O.

10, 1) rend a natural

J1 + t&l-'< 2 and jcx-2 pJ
This gives

for k >ko

on the bi3sis of

(2.19)

dye find

number k, such that

Asymptotic

This completes

the proof

theory

of

radiation

1259

transfer

of Theorem 5.

Theorem 5 does not give a complete description of the class R,. However, suppose the function 9 possesses the properties listed in Theorem and the Fourier coefficients 5: ‘p is continuous on L-1, 11 and positive,

are different from zero and form an alternating sequence. Suppose relation (2.13) is satisfied with a > 1 and the numbers Sk possess the property (2.14). Put A = 2a/(a2 + l), and define the numbers gk. k = 0, according to (2.15). Then 1, 2, ..-,

and relations (2.16), (2.17) are satisfied. It is clear vanish as k - a), and therefore from condition (2.14) it

5 khZk<00,

so that,

also

enables

&
k=o

k=l

This last fact the relation

$

that Uk and uk follows that

us to define

g @L) =

the

w 2k+l

2

7

function

L(M) E

L,(-1,

1) by

gkPk@L)*

k=O

This function g(D) is not, in general, contained in the class r but if we define the kernel g(~, P’) according to formula (1.5). the characteristic equation of transfer theory

will

still

be satisfied.

In conclusion we note that we can apparently introduce in the class R, a topology with respect to which GO will be a homeomorphism of r on %. I acknowledge my gratitude to AN. Tikhonov, direct cause for carrying out this study.

whose question

Translated

by R.

was the

Feinstein

M. Y. Has lcnnikov

1260

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2.

Vladimirov,

V.S.,

Ref.

3.

Vladimirov,

V-S.,

Tr.

4.

lifaslennikov, 255-266.

M. V.,

Dokl.

1.

M.V.,

Zh.,

Inst.

vychisl.

No. 2, Ref.

Akad.

matent.

NO.&

118,

Mateaatika,

Matem.

Zh.

Nauk SSR,

Akad.

895-898,

28322,

1962.

Nauk SSSR, No. 61, i

maten.

fiz.,

1958.

1961.

1, NO. 2,

1961.

5.

Wailer, 1948.

I.,

6.

HoLte,

G.,

7.

Wick,

G.C.,

8.

Perron,

0..

Ark.

Ark.

Phys.

Aitr.

Hat.

Mat.

Astr.

Rev.,

Die Lehre

75,

Fyr.,

Fyz.,

34A, Hiifte

35A.

736-737,

Hiifte

1, Nos.

4,

No.

3.4‘5,

36,

1948.

1949.

von den Kettenbriicken.

Leipzig-Berlin,

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